🧠:[<think>Okay, let's tackle this problem step by step. It's a triangle problem with some given conditions, and we need to find cos C and the area when a=5. Let me start by writing down what's given.First, we have triangle ABC with angles A, B, C opposite sides a, b, c respectively. We know that angle B is π/3, which is 60 degrees. Then there's this equation: (a - b + c)(a + b - c) = (3/7)bc. We need to find cos C in part I and the area when a=5 in part II.Let me start with part I. The given equation looks a bit complicated, but maybe we can simplify it. Let's expand the left-hand side:(a - b + c)(a + b - c) = [a + (-b + c)][a + (b - c)]. Hmm, this resembles the difference of squares formula. Wait, (x + y)(x - y) = x² - y². Let's see if we can apply that here.Let me set x = a and y = (b - c). Then, the product becomes (a + ( -b + c ))(a + (b - c)) = (a - (b - c))(a + (b - c)) = a² - (b - c)². Wait, let's check:(a - b + c)(a + b - c) = [a + (c - b)][a + (b - c)] = a² - (b - c)². Yes, that works. So expanding that:Left-hand side (LHS) = a² - (b - c)² = a² - (b² - 2bc + c²) = a² - b² + 2bc - c².So the equation becomes:a² - b² + 2bc - c² = (3/7)bc.Hmm. Let me rearrange terms:a² - b² - c² + 2bc - (3/7)bc = 0Combine the bc terms: 2bc - (3/7)bc = (14/7 - 3/7)bc = (11/7)bcSo:a² - b² - c² + (11/7)bc = 0Wait, but in the standard cosine law, we have a² = b² + c² - 2bc cos A. But angle B is given as π/3, so maybe we can use the Law of Cosines for angle B.Law of Cosines for angle B: b² = a² + c² - 2ac cos B.Since B = π/3, cos B = 0.5, so:b² = a² + c² - 2ac*(1/2) = a² + c² - ac.Therefore, b² = a² + c² - ac. Let's keep that in mind.Going back to our earlier equation:a² - b² - c² + (11/7)bc = 0But from the Law of Cosines, we can substitute b². Let's substitute b² = a² + c² - ac into the equation.So replacing b² with a² + c² - ac:a² - (a² + c² - ac) - c² + (11/7)bc = 0Simplify term by term:a² - a² - c² + ac - c² + (11/7)bc = 0Combine like terms:(-c² - c²) + ac + (11/7)bc = 0Which is:-2c² + ac + (11/7)bc = 0Factor out c:c*(-2c + a + (11/7)b) = 0Since c is a side length, it can't be zero. So:-2c + a + (11/7)b = 0Let's solve for a:a = 2c - (11/7)bHmm. So a = (14c - 11b)/7. Not sure if that helps directly. Let's see. Maybe we can use the Law of Sines? Since we have angles and sides related.Law of Sines says that a/sin A = b/sin B = c/sin C = 2R, where R is the circumradius.Given that angle B is π/3, so sin B = sin(π/3) = √3/2. Therefore, b = 2R*(√3/2) = R√3. So R = b / √3.Similarly, a = 2R sin A = (2b / √3) sin A, and c = 2R sin C = (2b / √3) sin C.Alternatively, ratios: a / sin A = b / sin B = c / sin C.So, a = (b / sin B) * sin A = (b / (√3/2)) sin A = (2b / √3) sin ASimilarly, c = (2b / √3) sin CMaybe expressing a and c in terms of b and angles A and C could help. Let's see.We have angle B = π/3, so angles A + C = π - π/3 = 2π/3. Therefore, A = 2π/3 - C.So, sin A = sin(2π/3 - C) = sin(2π/3)cos C - cos(2π/3)sin C = (√3/2)cos C - (-1/2)sin C = (√3/2)cos C + (1/2)sin C.Therefore, a = (2b / √3)[ (√3/2)cos C + (1/2)sin C ] = (2b / √3)( (√3 cos C + sin C)/2 ) = (2b / √3)(√3 cos C + sin C)/2 = (b / √3)(√3 cos C + sin C) = b cos C + (b / √3) sin C.Similarly, c = (2b / √3) sin C.So, let's write a in terms of b and angle C:a = b cos C + (b / √3) sin C.And c = (2b / √3) sin C.Earlier, we had from the equation: a = 2c - (11/7) b.So let's substitute a and c expressions into this equation.Left side: a = b cos C + (b / √3) sin C.Right side: 2c - (11/7) b = 2*(2b / √3 sin C) - (11/7) b = (4b / √3) sin C - (11/7) b.Set them equal:b cos C + (b / √3) sin C = (4b / √3) sin C - (11/7) b.Divide both sides by b (since b ≠ 0):cos C + (1/√3) sin C = (4/√3) sin C - 11/7.Bring all terms to the left side:cos C + (1/√3) sin C - (4/√3) sin C + 11/7 = 0Combine the sin C terms:cos C + [ (1/√3 - 4/√3) sin C ] + 11/7 = 0Which simplifies to:cos C - (3/√3) sin C + 11/7 = 0Simplify 3/√3 = √3:cos C - √3 sin C + 11/7 = 0Hmm. So we have an equation involving cos C and sin C. Let's write it as:cos C - √3 sin C = -11/7This looks like a linear combination of cos and sin. Maybe we can express the left side as a single trigonometric function using the formula:A cos C + B sin C = R cos(C - φ), where R = √(A² + B²) and tan φ = B/A.In our case, A = 1, B = -√3.So, R = √(1 + 3) = 2, and tan φ = -√3 / 1 = -√3. Therefore, φ = -π/3 (since tan(-π/3) = -√3).Therefore, cos C - √3 sin C = 2 cos(C - (-π/3)) = 2 cos(C + π/3)So our equation becomes:2 cos(C + π/3) = -11/7Therefore, cos(C + π/3) = -11/(14)So, C + π/3 = arccos(-11/14). Then C = arccos(-11/14) - π/3.But we need to find cos C. Let's use the cosine addition formula.Let me set θ = C + π/3. Then θ = arccos(-11/14). Then cos θ = -11/14, so θ is in the second quadrant.We need to find cos C = cos(θ - π/3) = cos θ cos π/3 + sin θ sin π/3.We know cos θ = -11/14, and cos π/3 = 1/2, sin π/3 = √3/2.We need sin θ. Since θ is in the second quadrant, sin θ = √(1 - cos² θ) = √(1 - (121/196)) = √(75/196) = (5√3)/14.Therefore, cos C = (-11/14)(1/2) + (5√3/14)(√3/2) = (-11/28) + (15/28) = (4/28) = 1/7.So, cos C = 1/7. That's part I done. Let me verify that.Wait, let's check the calculations again. Let's go step by step.Starting from cos C - √3 sin C = -11/7.Express as R cos(C + φ) = -11/7.Wait, actually, in the standard formula:A cos C + B sin C = R cos(C - φ), where R = √(A² + B²), and φ = arctan(B/A).But in our case, it's A = 1, B = -√3. So, R = 2, and φ = arctan(-√3/1) = -π/3. Therefore,cos C - √3 sin C = 2 cos(C + π/3). So 2 cos(C + π/3) = -11/7 ⇒ cos(C + π/3) = -11/14.Then, to find cos C, we use the angle addition formula:cos C = cos[(C + π/3) - π/3] = cos θ cos π/3 + sin θ sin π/3, where θ = C + π/3.Given that cos θ = -11/14 and θ is in the second quadrant (since cos θ is negative and arccos(-11/14) is between π/2 and π), so sin θ = √(1 - (121/196)) = √(75/196) = 5√3/14.Therefore:cos C = (-11/14)(1/2) + (5√3/14)(√3/2) = (-11/28) + (15/28) = (4/28) = 1/7.Yes, that checks out. So cos C = 1/7. So part I's answer is 1/7.Now, part II: If a = 5, find the area of triangle ABC.We need to find the area. The area of a triangle can be given by (1/2)ab sin C or other similar formulas. Let's recall that we know angle B is π/3, and we just found cos C = 1/7. Let's see how we can relate the sides.First, let's recall that from the Law of Cosines, we had:b² = a² + c² - ac. Wait, no. Wait, angle B is π/3, so Law of Cosines gives:b² = a² + c² - 2ac cos B = a² + c² - 2ac*(1/2) = a² + c² - ac. So that's correct.We also found expressions for a and c in terms of b and angle C earlier. But maybe now that we know cos C, we can find sin C, and then use the Law of Sines to relate sides.Given cos C = 1/7, so sin C = √(1 - (1/7)²) = √(48/49) = 4√3/7.So sin C = 4√3/7.From the Law of Sines, we have a/sin A = b/sin B = c/sin C.But angle A is π - B - C = π - π/3 - C = 2π/3 - C.So sin A = sin(2π/3 - C) = sin(2π/3)cos C - cos(2π/3)sin C = (√3/2)(1/7) - (-1/2)(4√3/7) = (√3/14) + (4√3/14) = 5√3/14.Therefore, sin A = 5√3/14.From Law of Sines, a/sin A = b/sin B = c/sin C.Given a = 5, sin A = 5√3/14, sin B = √3/2, sin C = 4√3/7.Compute the ratios:a/sin A = 5 / (5√3/14) = 5 * (14)/(5√3) ) = 14 / √3.Similarly, b/sin B = b / (√3/2) = 2b / √3.And c/sin C = c / (4√3/7) = 7c / (4√3).Since all these ratios are equal to 14/√3, so:2b / √3 = 14 / √3 ⇒ 2b = 14 ⇒ b = 7.Similarly, 7c / (4√3) = 14 / √3 ⇒ 7c = 14 * 4 ⇒ 7c = 56 ⇒ c = 8.So sides are a =5, b=7, c=8.Now, with sides known, we can compute the area. Alternatively, since we have two sides and the included angle, we can use (1/2)ab sin C, but let's check.Wait, angle B is π/3, sides a, b, c. Wait, if we have sides a, b, c, and know all three sides, we can use Heron's formula. But maybe using the formula (1/2)ab sin C is easier.Wait, area = (1/2)ab sin C. Let's check: sides a and b are adjacent to angle C? Wait, no. In triangle ABC, side a is opposite angle A, side b opposite angle B, and side c opposite angle C.So if we want to use (1/2)ab sin C, actually, that would be sides a and b with angle between them... Wait, the angle between sides a and b is angle C, because side a is opposite angle A, side b opposite angle B, so the angle between sides a and b is angle C. Wait, is that correct? Let me visualize the triangle.Imagine triangle ABC, with side a opposite angle A, side b opposite angle B, and side c opposite angle C. So vertices are labeled A, B, C with opposite sides a, b, c.Therefore, the angle between sides a and b is angle C. Similarly, angle between sides a and c is angle B, and angle between sides b and c is angle A.Therefore, yes, area can be expressed as (1/2)ab sin C.Alternatively, (1/2)bc sin A, or (1/2)ac sin B.But since we know sides a=5, b=7, c=8, maybe Heron's formula is straightforward.Compute the semi-perimeter: s = (a + b + c)/2 = (5 +7 +8)/2 = 20/2 =10.Area = √[s(s - a)(s - b)(s - c)] = √[10(10 -5)(10 -7)(10 -8)] = √[10*5*3*2] = √(300) = 10√3.Alternatively, check with (1/2)ab sin C:(1/2)*5*7*sin C. Since sin C = 4√3/7.So (1/2)*5*7*(4√3/7) = (1/2)*5*4√3 = (1/2)*20√3 =10√3. Same result. So area is 10√3.Alternatively, using angle B: (1/2)*a*c sin B.Given a=5, c=8, sin B=√3/2. So (1/2)*5*8*(√3/2)= (1/2)*40*(√3/2)=10√3. Same answer.Therefore, the area is 10√3 when a=5.Wait, but let me confirm if the sides a=5, b=7, c=8 are consistent with the given condition (a - b + c)(a + b - c) = 3/7 bc.Let's compute (5 -7 +8)(5 +7 -8) = (6)(4) =24.Now compute (3/7)*b*c = (3/7)*7*8=24. Yes, 24=24. So that checks out. Good.Therefore, part II's answer is 10√3.Final AnswerⅠ. boxed{dfrac{1}{7}} Ⅱ. boxed{10sqrt{3}}</think>Given a triangle ( triangle ABC ) with sides ( a ), ( b ), and ( c ) opposite to angles ( A ), ( B ), and ( C ) respectively, where ( B = frac{pi}{3} ) and the equation ( (a - b + c)(a + b - c) = frac{3}{7}bc ) holds, we need to find ( cos C ) and the area of the triangle when ( a = 5 ). Part I: Finding ( cos C )1. Expand the given equation: [ (a - b + c)(a + b - c) = a^2 - (b - c)^2 ]2. Simplify using the given condition: [ a^2 - (b^2 - 2bc + c^2) = frac{3}{7}bc ] [ a^2 - b^2 - c^2 + 2bc = frac{3}{7}bc ]3. Rearrange terms: [ a^2 - b^2 - c^2 + frac{11}{7}bc = 0 ]4. Use the Law of Cosines for ( B = frac{pi}{3} ): [ b^2 = a^2 + c^2 - ac ]5. Substitute ( b^2 ) into the equation: [ a^2 - (a^2 + c^2 - ac) - c^2 + frac{11}{7}bc = 0 ] [ -2c^2 + ac + frac{11}{7}bc = 0 ]6. Express ( a ) in terms of ( b ) and ( c ): [ a = 2c - frac{11}{7}b ]7. Use the Law of Sines and angle relationships to express ( a ) and ( c ) in terms of ( b ) and angles: [ a = b cos C + frac{b}{sqrt{3}} sin C ] [ c = frac{2b}{sqrt{3}} sin C ]8. Substitute these into the equation and solve for ( cos C ): [ cos C - sqrt{3} sin C = -frac{11}{7} ] Using trigonometric identities and solving, we find: [ cos C = frac{1}{7} ] Part II: Finding the Area when ( a = 5 )1. Using the Law of Sines and given ( a = 5 ): [ frac{a}{sin A} = frac{b}{sin B} = frac{c}{sin C} ]2. Calculate sides ( b ) and ( c ): [ b = 7, quad c = 8 ]3. Use Heron's formula or the formula for area with sides and included angle: [ text{Area} = frac{1}{2}ab sin C = frac{1}{2} times 5 times 7 times frac{4sqrt{3}}{7} = 10sqrt{3} ] Final AnswersⅠ. (boxed{dfrac{1}{7}}) Ⅱ. (boxed{10sqrt{3}}),<think>Okay, so I have this triangle ABC, and angle B is π/3, which is 60 degrees. The problem gives me this equation: (a - b + c)(a + b - c) = (3/7)bc. I need to find cos C first, and then if a is 5, find the area.Alright, let's start with part (Ⅰ). Let me recall some triangle laws. The Law of Cosines might be helpful here. Since angle B is 60 degrees, maybe I can relate sides a, b, c using that. The Law of Cosines states that for any triangle, c² = a² + b² - 2ab cos C. Similarly, for angle B, which is 60 degrees, we have b² = a² + c² - 2ac cos B. Since B is 60 degrees, cos B is 0.5, so that simplifies to b² = a² + c² - ac.Now, the given equation is (a - b + c)(a + b - c) = (3/7)bc. Let me expand the left side. Let's see:(a - b + c)(a + b - c) = [a + (c - b)][a - (c - b)] = a² - (c - b)² = a² - (c² - 2bc + b²) = a² - c² + 2bc - b².So that simplifies to a² - b² - c² + 2bc. But from the Law of Cosines for angle B, we know that b² = a² + c² - ac. Let me substitute that into the equation.So substituting b² = a² + c² - ac into a² - b² - c² + 2bc, we get:a² - (a² + c² - ac) - c² + 2bc = a² - a² - c² + ac - c² + 2bc = -2c² + ac + 2bc.So the left-hand side simplifies to -2c² + ac + 2bc. The equation is then:-2c² + ac + 2bc = (3/7)bc.Let me write that down:-2c² + a c + 2b c = (3/7)b c.Hmm, let's subtract (3/7)bc from both sides to bring all terms to one side:-2c² + a c + 2b c - (3/7)b c = 0Simplify the terms with bc:2b c - (3/7)b c = (14/7 - 3/7)bc = (11/7) bcSo now, the equation is:-2c² + a c + (11/7) b c = 0Hmm, this seems a bit messy. Maybe I can factor out a c?So, c(-2c + a + (11/7) b) = 0But c can't be zero since it's a side length. So:-2c + a + (11/7) b = 0Which gives:-2c + a + (11/7) b = 0Let me rearrange:a + (11/7) b = 2cMultiply both sides by 7 to eliminate denominators:7a + 11b = 14cSo 14c = 7a + 11b => c = (7a + 11b)/14Hmm. So c is expressed in terms of a and b. Now, maybe we can relate this to the Law of Cosines equation we had earlier: b² = a² + c² - a c.Since we have c in terms of a and b, let's substitute c into this equation.So, substituting c = (7a + 11b)/14 into b² = a² + c² - a c.First, compute c²:c² = [(7a + 11b)/14]^2 = (49a² + 154ab + 121b²)/196Then compute a c:a c = a*(7a + 11b)/14 = (7a² + 11ab)/14Now plug into the equation:b² = a² + (49a² + 154ab + 121b²)/196 - (7a² + 11ab)/14Multiply all terms by 196 to eliminate denominators:196b² = 196a² + 49a² + 154ab + 121b² - (7a² + 11ab)*14Compute each term:Left side: 196b²Right side:196a² + 49a² = 245a²154ab remains.121b² remains.Now subtract (7a² + 11ab)*14:14*(7a²) = 98a²14*(11ab) = 154abSo subtract 98a² + 154abTherefore:Right side = 245a² + 154ab + 121b² - 98a² - 154abSimplify:245a² - 98a² = 147a²154ab - 154ab = 0121b² remains.So total right side: 147a² + 121b²Therefore, the equation is:196b² = 147a² + 121b²Subtract 121b² from both sides:75b² = 147a²Simplify:Divide both sides by 3:25b² = 49a² => (b/a)² = 49/25 => b/a = 7/5 => b = (7/5)aSo the ratio of b to a is 7:5. Therefore, if we let a = 5k, then b = 7k for some k. Wait, but in part (Ⅱ), a is given as 5, so maybe k is 1 there. But let's keep going.So we have b = (7/5)a.Now, recall earlier we had c = (7a + 11b)/14. Let's substitute b = (7/5)a into this.c = (7a + 11*(7/5)a)/14 = (7a + (77/5)a)/14 = ( (35/5 + 77/5)a )/14 = (112/5 a)/14 = (112a)/(5*14) = (8a)/5.So c = (8/5)a.So now we have expressions for b and c in terms of a: b = (7/5)a and c = (8/5)a.Now, the problem asks for cos C. Let's use the Law of Cosines for angle C.Law of Cosines: c² = a² + b² - 2ab cos CWe can solve for cos C:cos C = (a² + b² - c²)/(2ab)We have b and c in terms of a. Let's substitute those:b = (7/5)a, c = (8/5)aSo:a² + b² - c² = a² + (49/25)a² - (64/25)a² = a² + (49/25 - 64/25)a² = a² - (15/25)a² = a² - (3/5)a² = (2/5)a²Then, 2ab = 2a*(7/5)a = (14/5)a²Therefore, cos C = (2/5 a²)/(14/5 a²) = (2/5)/(14/5) = 2/14 = 1/7.So cos C = 1/7. That seems to be the answer for part (Ⅰ).Let me check if this makes sense. If cos C is 1/7, then angle C is acute, which is plausible. Let me verify with the values.If a = 5, then b = 7/5 *5 =7, c =8/5 *5=8. Then check the original equation: (a - b + c)(a + b - c) = (5 -7 +8)(5 +7 -8)= (6)(4)=24. On the right side, (3/7)bc=(3/7)*7*8=24. So 24=24. That works. Also, check Law of Cosines for angle B: b²=49, a² + c² -ac=25 +64 -5*8=89 -40=49. Correct. So everything checks out. So cos C is indeed 1/7.Now part (Ⅱ): If a =5, find the area of triangle ABC.We have a =5, so from previous results, b =7 and c=8.We can compute the area using various methods. Since we know all three sides, Heron's formula is an option. Alternatively, since we know two sides and the included angle, we can use the formula (1/2)ab sin C. Wait, but we know angle B is 60 degrees, so maybe use that?Wait, angle B is 60 degrees, and sides a=5, c=8, with b=7. So sides a, b, c are 5,7,8. Alternatively, we can use the formula (1/2)*a*c*sin B, since angle B is between sides a and c. Wait, actually, angle B is between sides a and c? Wait, in triangle ABC, angle B is between sides a and c. Because in standard notation, side a is opposite angle A, side b opposite angle B, side c opposite angle C. So angle B is between sides a and c. So yes, sides a and c are adjacent to angle B. Therefore, the area can be calculated as (1/2)*a*c*sin B.Since angle B is 60 degrees, sin 60 is √3/2. Therefore, area is (1/2)*5*8*(√3/2)= (1/2)*40*(√3/2)= 10*(√3/2)=5√3. Wait, but let me check with Heron's formula to be sure.Heron's formula: area = √[s(s-a)(s-b)(s-c)], where s=(a+b+c)/2.s = (5 +7 +8)/2=20/2=10.Area = √[10*(10-5)(10-7)(10-8)] = √[10*5*3*2] = √(300)=10√3. Wait, that's different. Wait, Heron's formula gives 10√3, but the other method gave 5√3. Which is correct?Wait, no, the other method: (1/2)*a*c*sin B is (1/2)*5*8*(√3/2)= (1/2)*40*(√3/2)= 20*(√3/2)=10√3. Wait, that's the same as Heron's. Wait, I must have miscalculated earlier. Let me check again:(1/2)*a*c*sin B: a=5, c=8, angle B=60°. So (1/2)*5*8*sin(60)= (20)* (√3/2)=10√3. Yes, correct. So the area is 10√3.Alternatively, since we have sides 5,7,8, and angle B=60, which is between sides a and c. So that formula is valid. Heron's formula also confirms 10√3.Alternatively, using another formula: area = (1/2)*ab sin C, but we need angle C. Since we found cos C=1/7, then sin C=√(1 - (1/7)^2)=√(48/49)=4√3/7. Then area=(1/2)*a*b*sin C=(1/2)*5*7*(4√3/7)= (1/2)*5*4√3=10√3. Same result.So the area is 10√3.Therefore, answers are (Ⅰ) 1/7 and (Ⅱ) 10√3.Final AnswerⅠ. boxed{dfrac{1}{7}} Ⅱ. boxed{10sqrt{3}}</think>Given a triangle ( triangle ABC ) with angles ( A, B, C ) opposite sides ( a, b, c ) respectively, where ( B = frac{pi}{3} ) and the equation ( (a - b + c)(a + b - c) = frac{3}{7}bc ): Part (Ⅰ): Finding ( cos C )1. Law of Cosines for Angle ( B ): [ b^2 = a^2 + c^2 - 2ac cos B ] Given ( B = frac{pi}{3} ), ( cos B = frac{1}{2} ), so: [ b^2 = a^2 + c^2 - ac ]2. Expanding the Given Equation: [ (a - b + c)(a + b - c) = frac{3}{7}bc ] Expanding the left-hand side: [ a^2 - (c - b)^2 = a^2 - c^2 + 2bc - b^2 ] Substituting ( b^2 = a^2 + c^2 - ac ): [ a^2 - (a^2 + c^2 - ac) - c^2 + 2bc = -2c^2 + ac + 2bc ] Equating to the right-hand side: [ -2c^2 + ac + 2bc = frac{3}{7}bc ] Simplifying and solving for ( c ): [ -2c + a + frac{11}{7}b = 0 implies 7a + 11b = 14c implies c = frac{7a + 11b}{14} ]3. Substituting ( b = frac{7}{5}a ) and ( c = frac{8}{5}a ): Using the Law of Cosines for angle ( C ): [ cos C = frac{a^2 + b^2 - c^2}{2ab} ] Substituting ( b = frac{7}{5}a ) and ( c = frac{8}{5}a ): [ cos C = frac{a^2 + left(frac{49}{25}a^2right) - left(frac{64}{25}a^2right)}{2 cdot a cdot frac{7}{5}a} = frac{frac{2}{5}a^2}{frac{14}{5}a^2} = frac{1}{7} ] Part (Ⅱ): Finding the Area when ( a = 5 )1. Sides Calculation: Given ( a = 5 ), we find ( b = 7 ) and ( c = 8 ).2. Area Calculation: Using the formula for the area with sides ( a = 5 ), ( c = 8 ), and angle ( B = frac{pi}{3} ): [ text{Area} = frac{1}{2} cdot a cdot c cdot sin B = frac{1}{2} cdot 5 cdot 8 cdot sin frac{pi}{3} = frac{1}{2} cdot 5 cdot 8 cdot frac{sqrt{3}}{2} = 10sqrt{3} ] Final AnswersⅠ. boxed{dfrac{1}{7}} Ⅱ. boxed{10sqrt{3}}]

🔑:Solution: (Ⅰ) From (a-b+c)(a+b-c)= frac {3}{7}bc, we get: a^{2}-(b-c)^{2}=a^{2}-b^{2}-c^{2}+2bc= frac {3}{7}bc, therefore a^{2}=b^{2}+c^{2}- frac {11}{7}bc, therefore cos A= frac {b^{2}+c^{2}-a^{2}}{2bc}= frac {11}{14}, therefore sin A= sqrt {1-cos ^{2}A}= frac {5 sqrt {3}}{14}, Thus, cos C=-cos (A+B)=-cos Acos B+sin Asin B=- frac {11}{14}× frac {1}{2}+ frac {5 sqrt {3}}{14}× frac { sqrt {3}}{2}= frac {1}{7}; (Ⅱ) From (Ⅰ), we get sin C= sqrt {1-cos ^{2}C}= frac {4 sqrt {3}}{7}, In triangle ABC, by the Law of Sines frac {a}{sin A}= frac {b}{sin B}= frac {c}{sin C}, we get: c= frac {asin C}{sin A}= frac {5× frac {4 sqrt {3}}{7}}{ frac {5 sqrt {3}}{14}}=8, Then, S= frac {1}{2}acsin B= frac {1}{2}×5×8× frac { sqrt {3}}{2}=10 sqrt {3}.Thus, the answers are boxed{frac {1}{7}} for (Ⅰ) and boxed{10 sqrt {3}} for (Ⅱ).

🧠:[<think>Okay, let's try to tackle this problem. So, we have three positive real numbers x, y, z such that their sum is 1. The equation given is a/(xyz) = 1/x + 1/y + 1/z - 2. We need to find the range of the real number a. Hmm, interesting. Let me think step by step.First, let's rewrite the given equation to express a in terms of x, y, z. If we multiply both sides by xyz, we get:a = (1/x + 1/y + 1/z - 2) * xyzSimplifying each term:1/x * xyz = yzSimilarly, 1/y * xyz = xzAnd 1/z * xyz = xySo, substituting back:a = (yz + xz + xy - 2xyz)Wait, so a is equal to yz + xz + xy - 2xyz. Okay, that simplifies things a bit. So, the problem reduces to finding the range of a = xy + yz + zx - 2xyz, given that x, y, z are positive real numbers with x + y + z = 1.Alright, so we need to find the possible values of a. That is, the minimum and maximum values of a given the constraint x + y + z = 1 and x, y, z > 0.Let me recall that when dealing with symmetric functions of variables under a sum constraint, methods like Lagrange multipliers or using inequalities (AM-GM, Cauchy-Schwarz) can be helpful. Since this is an Olympiad problem, maybe inequalities would be the way to go.Alternatively, since the variables are symmetric, perhaps we can assume two variables are equal to simplify the problem. For example, set y = z, then express x in terms of y (since x + 2y = 1) and then express a as a function of y. Then find its extrema. Let me try both approaches.First, let's try using Lagrange multipliers. Let me set up the function a = xy + yz + zx - 2xyz with the constraint x + y + z = 1.Define the Lagrangian:L = xy + yz + zx - 2xyz - λ(x + y + z - 1)Take partial derivatives with respect to x, y, z, and set them to zero.Partial derivative with respect to x:∂L/∂x = y + z - 2yz - λ = 0Similarly, ∂L/∂y = x + z - 2xz - λ = 0∂L/∂z = x + y - 2xy - λ = 0And the constraint x + y + z = 1.So, we have three equations:1. y + z - 2yz = λ2. x + z - 2xz = λ3. x + y - 2xy = λSo, all three expressions equal to λ. Therefore, we can set them equal to each other.From equations 1 and 2:y + z - 2yz = x + z - 2xzSubtract z from both sides:y - 2yz = x - 2xzFactor:y(1 - 2z) = x(1 - 2z)If 1 - 2z ≠ 0, then y = x.If 1 - 2z = 0, then z = 1/2. Let's check both possibilities.Case 1: 1 - 2z ≠ 0, so y = x.Similarly, from equations 1 and 3:y + z - 2yz = x + y - 2xySubtract y from both sides:z - 2yz = x - 2xyBut since we already have y = x (from the first case), substitute x for y:z - 2xz = x - 2x^2Bring all terms to one side:z - 2xz - x + 2x^2 = 0Factor:z(1 - 2x) - x(1 - 2x) = 0So, (z - x)(1 - 2x) = 0So either z = x or 1 - 2x = 0 => x = 1/2.Subcase 1a: z = x. Then since y = x, all variables are equal: x = y = z. Then x + y + z = 1 => 3x = 1 => x = 1/3. So, x = y = z = 1/3. Then compute a:a = xy + yz + zx - 2xyz = 3*(1/3 * 1/3) - 2*(1/3)^3 = 3*(1/9) - 2*(1/27) = 1/3 - 2/27 = 9/27 - 2/27 = 7/27 ≈ 0.259.Subcase 1b: 1 - 2x = 0 => x = 1/2. Then since y = x, y = 1/2. But x + y + z = 1, so z = 1 - 1/2 -1/2 = 0. But z must be positive. So this is invalid. Therefore, Subcase 1b is not possible.Case 2: 1 - 2z = 0 => z = 1/2. Then from the constraint x + y + z = 1, x + y = 1 - 1/2 = 1/2. Let me check equations 1 and 2 in this case.From equation 1: y + z - 2yz = λ => y + 1/2 - 2y*(1/2) = y + 1/2 - y = 1/2 = λFrom equation 2: x + z - 2xz = λ => x + 1/2 - 2x*(1/2) = x + 1/2 - x = 1/2 = λFrom equation 3: x + y - 2xy = λ. Since x + y = 1/2, then 1/2 - 2xy = λ. But λ is 1/2, so 1/2 - 2xy = 1/2 => -2xy = 0 => xy = 0. But x and y are positive, so this is impossible. Hence, Case 2 leads to a contradiction. Therefore, the only critical point is when x = y = z = 1/3, which gives a = 7/27.But we need to check the boundaries as well. Since x, y, z are positive, approaching the boundaries would mean one of them approaches 0. Let's see what happens when one variable approaches 0.Suppose z approaches 0. Then x + y approaches 1. Let's set z = 0 (even though it must be positive, but approaching 0). Then a = xy + 0 + 0 - 0 = xy. Since x + y = 1, the maximum of xy is 1/4 (when x = y = 1/2), and the minimum approaches 0 (when one variable approaches 1 and the other approaches 0). Therefore, as z approaches 0, a approaches the interval (0, 1/4]. But wait, we need to check if a can actually reach up to 1/4.Wait, if z approaches 0, then a approaches xy, and since x + y =1, the maximum of xy is indeed 1/4. So, in this case, a can approach up to 1/4. However, at z = 0, it's not allowed, but as z approaches 0, a approaches 1/4.Similarly, if two variables approach 0, say y and z approach 0, then x approaches 1. Then a = xy + yz + zx - 2xyz approaches 0. So, a can approach 0.Wait, but if two variables approach 0, then a = xy + yz + zx - 2xyz. Let's say y and z approach 0, then all terms xy, yz, zx, and xyz approach 0. So, a approaches 0. So, the minimal value of a is 0 (but not achieved since variables must be positive). However, when we set variables to be equal, we get a =7/27 ≈0.259, which is higher than 0. So, does that mean that the minimum is 0 and the maximum is 1/4? But wait, when z approaches 0, a approaches 1/4, which is about 0.25, which is actually less than 7/27≈0.259. Wait, this is conflicting. Wait, 1/4 is 0.25, which is less than 7/27≈0.259. So, actually, the maximum would be 7/27, and the minimum approaches 0? Hmm, but this contradicts the previous thought.Wait, let's compute 7/27. 7 divided by 27 is approximately 0.259. 1/4 is 0.25, so 7/27 is larger. Therefore, when variables are equal, a is 7/27, which is higher than the 1/4 when one variable approaches 0. Therefore, that suggests that the maximum is 7/27, and the minimum approaches 0. However, we need to verify this.Wait, but in the case when two variables approach 0, a approaches 0, but when one variable approaches 0, a approaches up to 1/4. Wait, but 1/4 is less than 7/27. So, then 7/27 is actually higher. Therefore, perhaps the maximum value is 7/27, and the minimum is approaching 0.But let's check with another case. Suppose two variables are equal, and the third is different. Let me set y = z, so x + 2y =1 => x =1 - 2y. Then, express a in terms of y.Compute a = xy + yz + zx - 2xyz. Since y = z and x =1 - 2y:a = (1 - 2y)y + y^2 + y(1 - 2y) - 2(1 - 2y)y^2Simplify:First term: (1 - 2y)y = y - 2y^2Second term: y^2Third term: y(1 - 2y) = y - 2y^2Fourth term: -2(1 - 2y)y^2 = -2y^2 + 4y^3So, adding up the first three terms:(y - 2y^2) + y^2 + (y - 2y^2) = y - 2y^2 + y^2 + y - 2y^2 = 2y - 3y^2Then subtract the fourth term:2y - 3y^2 - (-2y^2 + 4y^3) = 2y - 3y^2 + 2y^2 - 4y^3 = 2y - y^2 - 4y^3Thus, a = 2y - y^2 - 4y^3, where y must satisfy x =1 - 2y >0 => 1 - 2y >0 => y <1/2. Also, y >0. So, y ∈ (0, 1/2).Now, let's find the maximum of a(y) = 2y - y^2 -4y^3 in the interval (0, 1/2). To find critical points, take derivative:a’(y) = 2 - 2y -12y^2Set equal to zero:2 - 2y -12y^2 =0Multiply both sides by -1:-2 + 2y +12y^2 =0 => 12y^2 +2y -2 =0Divide by 2:6y^2 + y -1 =0Solve quadratic equation:y = [-1 ± sqrt(1 +24)]/(12) = [-1 ±5]/12Thus, y = (4)/12 =1/3 or y = (-6)/12 = -1/2. Disregard negative solution. So y=1/3.Check if y=1/3 is within (0, 1/2). 1/3 ≈0.333 <1/2=0.5. Yes.So, critical point at y=1/3. Compute a(1/3):a(1/3) =2*(1/3) - (1/3)^2 -4*(1/3)^3 =2/3 -1/9 -4/27= (18/27 -3/27 -4/27)=11/27≈0.407.Wait, that's different. Wait, hold on. Wait, earlier when we considered x=y=z=1/3, a was 7/27≈0.259. But here, with y=1/3, x=1 -2*(1/3)=1 -2/3=1/3. So, x=1/3, y=z=1/3. Wait, but then a=7/27≈0.259. But according to this calculation, a=11/27≈0.407. That can't be. There must be a mistake in my substitution.Wait, let me check again. When x=1 -2y, and y=1/3, then x=1 -2/3=1/3. So, x=y=z=1/3. Therefore, a should be 7/27. But according to the expression a=2y - y^2 -4y^3, substituting y=1/3:2*(1/3)=2/3, (1/3)^2=1/9, 4*(1/3)^3=4/27.So, a=2/3 -1/9 -4/27 = (18/27 -3/27 -4/27)=11/27. Wait, this contradicts the previous value. So, where is the mistake?Wait, no, when we set y = z and x =1 -2y, then a = xy + yz + zx -2xyz. Let's compute that directly when x=y=z=1/3:xy + yz + zx =3*(1/3*1/3)=3*(1/9)=1/3. Then 2xyz=2*(1/3)^3=2/27. Therefore, a=1/3 -2/27=7/27≈0.259. But according to the substitution when x=1 -2y and y=1/3, we get a=11/27. That suggests a mistake in the substitution steps.Wait, let's re-derive a(y). Let me do that again. If x=1 -2y, and y=z, then:a = xy + yz + zx -2xyzCompute each term:xy = (1 -2y)yyz = y^2zx = (1 -2y)ySo, xy + yz + zx = (1 -2y)y + y^2 + (1 -2y)y = 2*(1 -2y)y + y^2 = 2y -4y^2 + y^2 =2y -3y^2Then, subtract 2xyz:2xyz=2*(1 -2y)*y*y=2*(1 -2y)y^2=2y^2 -4y^3Therefore, a= (2y -3y^2) - (2y^2 -4y^3)=2y -3y^2 -2y^2 +4y^3=2y -5y^2 +4y^3Ah! I see, earlier I must have made a mistake in expanding the terms. The correct expression is a=2y -5y^2 +4y^3, not 2y -y^2 -4y^3. That was the error. So, let me redo the derivative.So, a(y)=4y^3 -5y^2 +2yCompute derivative:a’(y)=12y^2 -10y +2Set to zero:12y^2 -10y +2=0Solve quadratic equation:y=(10±sqrt(100 -96))/24=(10±sqrt(4))/24=(10±2)/24Thus, y=(10+2)/24=12/24=1/2 or y=(10-2)/24=8/24=1/3.So critical points at y=1/2 and y=1/3. But y must be in (0,1/2). So y=1/3 is in (0,1/2), y=1/2 is the endpoint.Compute a(1/3):a=4*(1/3)^3 -5*(1/3)^2 +2*(1/3)=4/27 -5/9 +2/3= (4/27 -15/27 +18/27)=7/27≈0.259.Compute a(1/2):But y=1/2 would make x=1 -2*(1/2)=0, which is invalid since x must be positive. Therefore, the maximum occurs at y=1/3, giving a=7/27. Then, as y approaches 0, what happens to a(y)?a(y)=4y^3 -5y^2 +2y. As y approaches 0, the dominant term is 2y, so a approaches 0.As y approaches 1/2, x approaches 0. So, we can analyze the behavior as y approaches 1/2. Let me take the limit as y approaches 1/2 from below.Compute a(1/2):But x=1 -2y, so as y→1/2, x→0.Then, in the original expression, a=xy + yz + zx -2xyz. If x approaches 0 and y=z approaches 1/2, then:xy ≈0*(1/2)=0yz≈(1/2)*(1/2)=1/4zx≈0*(1/2)=0xyz≈0*(1/2)^2=0Therefore, a approaches 0 +1/4 +0 -0=1/4. So, as y approaches 1/2, a approaches1/4≈0.25.But 7/27≈0.259>0.25. So, in this case, when y approaches1/2, a approaches1/4, which is less than the critical point value at y=1/3. Therefore, the maximum value of a in this case is 7/27, and the minimum approaches0. But wait, but when two variables approach0, a approaches0 as well. So, in this setup, the maximum is7/27, and the minimum is approaching0.But earlier, when we considered the case of two variables approaching0, a approaches0, and when one variable approaches0, a approaches up to1/4. But 1/4 is still less than7/27. Therefore, the maximum overall is7/27, achieved when x=y=z=1/3, and the minimum approaches0. But wait, but when we have one variable approaching0, a approaches1/4, but when two variables approach0, a approaches0. So, perhaps the minimal value is0, and the maximum is7/27. But then, in between, there are values between0 and7/27. Wait, but when one variable approaches0, a approaches1/4≈0.25, which is between0 and7/27≈0.259. Hmm, so there is a higher value at7/27.Wait, but let me confirm. Suppose we take another case where variables are not equal. For example, set x=0.5, y=0.5, z=0. But z must be positive. So, let me take x=0.5, y=0.5 -ε, z=ε, where ε approaches0. Then compute a:a=xy + yz + zx -2xyz=0.5*(0.5 -ε)+(0.5 -ε)*ε +0.5*ε -2*0.5*(0.5 -ε)*εSimplify:=0.25 -0.5ε +0.5ε -ε^2 +0.5ε - [ (0.5 -ε)*ε ]=0.25 -ε^2 +0.5ε -0.5ε +ε^2=0.25So, as ε approaches0, a approaches0.25. Therefore, in this case, a approaches1/4 from below.But if we take x approaching1, and y,z approaching0, then a approaches0.Therefore, the range of a is from0 up to7/27. But we need to confirm whether there are any higher values than7/27. Since the critical point at x=y=z=1/3 gives a=7/27, and when variables are unequal, the value is lower, then7/27 is indeed the maximum. The minimum is0, but since variables must be positive, a can get arbitrarily close to0 but cannot be0.Wait, but when two variables approach0, a approaches0. When one variable approaches0, a approaches1/4. Therefore, the possible values of a are from0 up to7/27. But when you set variables unequal, can a be greater than7/27?Wait, let's check with x=1/2, y=1/4, z=1/4. Then compute a=xy + yz +zx -2xyz= (1/2)(1/4) + (1/4)(1/4) + (1/2)(1/4) -2*(1/2)(1/4)(1/4)=1/8 +1/16 +1/8 -2*(1/32)= (2/16 +1/16 +2/16) - (2/32)=5/16 -1/16=4/16=1/4=0.25.Which is less than7/27≈0.259.Alternatively, take x=0.4, y=0.3, z=0.3. Then compute a=0.4*0.3 +0.3*0.3 +0.4*0.3 -2*0.4*0.3*0.3=0.12 +0.09 +0.12 -2*0.036=0.33 -0.072=0.258≈7/27≈0.259. Close, but still less. So, seems like7/27 is indeed the maximum.Therefore, the range of a is (0,7/27]. But wait, but when variables are equal, a=7/27, so that's achievable. And a can approach0 but not reach it. So, the range is (0,7/27]. But wait, when two variables approach0, a approaches0, but can a actually reach0? No, because variables must be positive. So, the infimum is0, but not achieved. Similarly, the supremum is7/27, which is achieved. So, the range is0 <a ≤7/27.But let's verify with another example. Take x approaching1, y and z approaching0. Let x=1 -2ε, y=ε, z=ε, where ε approaches0. Then compute a=xy + yz + zx -2xyz= (1 -2ε)ε +ε^2 + (1 -2ε)ε -2*(1 -2ε)*ε^2=2ε -4ε^2 +ε^2 +2ε -4ε^2 -2ε^2 +4ε^3=4ε -7ε^2 +4ε^3. As ε approaches0, this approaches4ε, which approaches0. Therefore, a approaches0.Thus, the minimum is0 (exclusive), and the maximum is7/27 (inclusive). Therefore, the range of a is (0,7/27].But the problem states "the range of the real number a". Since a can approach0 but not reach it, and the maximum is7/27. So, the answer should be that a is in (0,7/27].But let me check once more if there's a case where a can be higher than7/27. Suppose we take x=0.25, y=0.25, z=0.5. Then a=0.25*0.25 +0.25*0.5 +0.25*0.5 -2*0.25*0.25*0.5=0.0625 +0.125 +0.125 -2*0.03125=0.3125 -0.0625=0.25. Still 0.25.If I take x=0.3, y=0.3, z=0.4. Then a=0.3*0.3 +0.3*0.4 +0.3*0.4 -2*0.3*0.3*0.4=0.09 +0.12 +0.12 -0.072=0.33 -0.072=0.258≈7/27≈0.259.So, close to7/27 but not higher. So, it seems the maximum is indeed7/27.Therefore, the range of a is all real numbers from0 (exclusive) up to7/27 (inclusive).But let me check if there's any other critical points when variables are not equal. Wait, in the Lagrange multiplier method, we found only one critical point at x=y=z=1/3. So, since the function is symmetric and smooth, and the only critical point inside the domain gives the maximum, and the boundaries give lower values approaching0, then the maximum is7/27 and the infimum is0.Therefore, the range of a is (0,7/27].But the problem says "the range of the real number a". Since a is determined by x, y, z positive with x + y + z=1, and given that a can approach0 but not reach it, and can reach7/27, the answer should be that the range is (0,7/27].But I need to confirm if there are other possible critical points when variables are unequal. Wait, using Lagrange multipliers, we saw that the only critical point is when x=y=z=1/3. So, that is the only critical point. Therefore, the maximum is at that point, and minima at the boundaries.Hence, the range of a is (0,7/27].But wait, the problem statement says "the range of the real number a". However, in Olympiad problems, sometimes intervals are given with fractions. Let me compute7/27. 7 divided by27 is irreducible. So, as a fraction, it's7/27. But let me check if the answer expects it in boxed form.Wait, the problem says "the range of the real number a is ___". So, they expect an interval. Since a can be as close to0 as desired (but not0) and up to7/27. So, the range is0 <a ≤7/27. In mathematical notation, this is (0, 7/27].But in some contexts, especially in Olympiads, they might write the interval as 0 <a ≤7/27, or using parentheses and brackets. But since the problem mentions "the range of the real number a", and the answer expects it in boxed form, I need to write it properly.But in China, for Olympiad problems, sometimes they write intervals using parentheses for open ends and brackets for closed ends. So, the answer should be (0, 7/27].Alternatively, if they require using fractions, 7/27 is already in simplest form.Alternatively, let me check if there's any mistake in my reasoning.Wait, suppose we have three variables x, y, z approaching each other, then a approaches7/27. But is this indeed the maximum? For example, take x=0.4, y=0.3, z=0.3. Then a=0.4*0.3 +0.3*0.3 +0.4*0.3 -2*0.4*0.3*0.3=0.12 +0.09 +0.12 -0.072=0.33 -0.072=0.258≈0.259≈7/27≈0.25926. So, very close. Therefore, seems like7/27 is the upper bound.Alternatively, if I take x=0.333, y=0.333, z=0.334. Then a≈0.333*0.333 +0.333*0.334 +0.333*0.334 -2*0.333*0.333*0.334≈0.1109 +0.1112 +0.1112 -2*0.037≈0.3333 -0.074≈0.2593, which is≈7/27≈0.25926. So, confirming that when variables are close to1/3, a is close to7/27.Therefore, based on this, the maximum value is7/27, and the minimum is0 (exclusive). So, the range is0 <a ≤7/27.Thus, the answer is boxed{(0, dfrac{7}{27}]}Final AnswerThe range of the real number ( a ) is boxed{left(0, dfrac{7}{27}right]}.</think>Given that ( x, y, z ) are positive real numbers such that ( x + y + z = 1 ), we need to find the range of the real number ( a ) given by the equation:[frac{a}{xyz} = frac{1}{x} + frac{1}{y} + frac{1}{z} - 2]First, we rewrite the given equation to express ( a ) in terms of ( x, y, z ):[a = (xy + yz + zx - 2xyz)]We need to find the range of ( a ) under the constraint ( x + y + z = 1 ). Using the method of Lagrange multipliers and analyzing critical points, we find that the maximum value occurs when ( x = y = z = frac{1}{3} ). Substituting these values into the expression for ( a ):[a = 3 left( frac{1}{3} cdot frac{1}{3} right) - 2 left( frac{1}{3} cdot frac{1}{3} cdot frac{1}{3} right) = frac{1}{3} - frac{2}{27} = frac{7}{27}]Next, we consider the boundary cases where one or more variables approach 0. If one variable approaches 0, the expression ( a ) approaches ( frac{1}{4} ). However, this value is less than ( frac{7}{27} ). If two variables approach 0, the expression ( a ) approaches 0. Therefore, the minimum value of ( a ) is 0 (exclusive), and the maximum value is ( frac{7}{27} ) (inclusive).Thus, the range of the real number ( a ) is:[boxed{left(0, dfrac{7}{27}right]}],<think>Okay, so I have this problem here where I need to find the range of the real number ( a ) given that ( x, y, z ) are positive real numbers satisfying ( x + y + z = 1 ), and the equation ( frac{a}{xyz} = frac{1}{x} + frac{1}{y} + frac{1}{z} - 2 ). Hmm, let's try to parse this step by step.First, let's restate the problem in my own words. We have three variables ( x, y, z ) that add up to 1. They are all positive, so none of them can be zero. The equation given relates ( a ) to the reciprocals of ( x, y, z ) and their product. The question is asking for the possible values of ( a ), given these constraints.So, to find the range of ( a ), I probably need to express ( a ) in terms of ( x, y, z ) and then find its maximum and minimum values under the constraint ( x + y + z = 1 ). Let me see.Starting with the given equation:[frac{a}{xyz} = frac{1}{x} + frac{1}{y} + frac{1}{z} - 2]If I multiply both sides by ( xyz ), I can solve for ( a ):[a = xyz left( frac{1}{x} + frac{1}{y} + frac{1}{z} - 2 right)]Simplify the right-hand side. Let's distribute ( xyz ):[a = xyz cdot frac{1}{x} + xyz cdot frac{1}{y} + xyz cdot frac{1}{z} - 2xyz]Simplifying each term:- ( xyz cdot frac{1}{x} = yz )- ( xyz cdot frac{1}{y} = xz )- ( xyz cdot frac{1}{z} = xy )So substituting back:[a = yz + xz + xy - 2xyz]Therefore, ( a = xy + yz + zx - 2xyz ). Now, we need to find the range of ( a ) given that ( x, y, z ) are positive real numbers summing to 1.So, the problem reduces to finding the maximum and minimum possible values of ( a = xy + yz + zx - 2xyz ) with ( x + y + z = 1 ) and ( x, y, z > 0 ).Okay, so now the task is to optimize ( a = xy + yz + zx - 2xyz ) under the constraints ( x + y + z = 1 ) and ( x, y, z > 0 ). This seems like a problem that can be approached using Lagrange multipliers, but maybe there's a smarter way given that it's symmetric in variables. Also, since it's an Olympiad problem, perhaps there's an inequality-based approach.First, let's consider the symmetry. The expression for ( a ) is symmetric in ( x, y, z ), so maybe the extremum occurs when two variables are equal, or all three are equal. Let's test some cases.Case 1: All variables equal. If ( x = y = z ), then each is ( frac{1}{3} ). Let's compute ( a ):[a = 3 left( frac{1}{3} cdot frac{1}{3} right) - 2 left( frac{1}{3} right)^3 = 3 cdot frac{1}{9} - 2 cdot frac{1}{27} = frac{1}{3} - frac{2}{27} = frac{9}{27} - frac{2}{27} = frac{7}{27} approx 0.259]So when all variables are equal, ( a = frac{7}{27} ).Case 2: Two variables equal, third different. Let's suppose ( x = y ), then ( z = 1 - 2x ). Since ( z > 0 ), ( 1 - 2x > 0 implies x < frac{1}{2} ), and ( x > 0 ).Express ( a ) in terms of ( x ):First, compute ( xy + yz + zx - 2xyz ).Since ( x = y ), ( z = 1 - 2x ):[xy = x^2][yz = x(1 - 2x)][zx = x(1 - 2x)]So,[xy + yz + zx = x^2 + 2x(1 - 2x) = x^2 + 2x - 4x^2 = -3x^2 + 2x]Then,[xyz = x^2(1 - 2x)]So,[a = (-3x^2 + 2x) - 2x^2(1 - 2x) = -3x^2 + 2x - 2x^2 + 4x^3 = 4x^3 -5x^2 + 2x]So now ( a ) is a function of ( x ): ( a(x) = 4x^3 -5x^2 + 2x ), where ( 0 < x < frac{1}{2} ).To find the extrema, take derivative with respect to ( x ):[a'(x) = 12x^2 -10x + 2]Set derivative equal to zero:[12x^2 -10x + 2 = 0]Quadratic equation: ( 12x^2 -10x + 2 = 0 )Discriminant: ( D = 100 - 96 = 4 )Solutions:[x = frac{10 pm 2}{24} = frac{12}{24} = frac{1}{2} quad text{or} quad x = frac{8}{24} = frac{1}{3}]But ( x ) must be less than ( frac{1}{2} ), so critical points at ( x = frac{1}{3} ) and ( x = frac{1}{2} ). However, ( x = frac{1}{2} ) would make ( z = 1 - 2x = 0 ), which is not allowed. So the critical point is at ( x = frac{1}{3} ).Compute ( a ) at ( x = frac{1}{3} ):First, ( z = 1 - 2 times frac{1}{3} = frac{1}{3} ). So actually all variables are equal here. Which is the case we already considered. So ( a = frac{7}{27} ).Wait, but if we take ( x = frac{1}{3} ), then all variables are equal. So in this case, the critical point in the two-variable case coincides with the all-equal case. Hmm. So perhaps we need to check endpoints?But since ( x ) approaches 0, let's see what happens when ( x ) approaches 0. Let me compute the limit as ( x to 0^+ ):In the case where ( x = y to 0 ), then ( z to 1 ). Then ( xy = x^2 to 0 ), ( yz = x z to 0 ), ( zx = x z to 0 ). So ( xy + yz + zx to 0 ). Also, ( xyz = x^2 z to 0 ). Therefore, ( a to 0 - 0 = 0 ). So as ( x to 0 ), ( a to 0 ).But since ( x ) must be positive, ( a ) can get arbitrarily close to 0 but not equal to 0. So maybe 0 is a lower bound?But wait, maybe there's another case where two variables approach 0, but then the third approaches 1. Wait, but in that case, same thing. Let me check another case where two variables are not equal. Maybe set one variable approaching 1 and the other two approaching 0. Let me see.Suppose ( x to 1 ), ( y, z to 0 ). Then ( xy + yz + zx approx 0 ), ( xyz approx 0 ), so ( a approx 0 ). So again, approaching 0.Alternatively, if two variables are approaching some value, but maybe not equal. Let's think. But in the two-variable case, when we set two variables equal, we found that the critical point is at all variables equal. So maybe the maximum is at ( frac{7}{27} ), and the minimum approaches 0. But wait, the problem says "positive real numbers", so maybe 0 is not included. But the question is about the range of ( a ), so it might be ( (0, frac{7}{27}] ). Wait, but maybe there's a higher value?Wait, but let's test another case. Let's suppose two variables are equal, but not necessarily in the case where the third is 1 - 2x. Wait, no, if we fix two variables equal, then the third is determined. But maybe if we set variables not equal. Let's try an example.Take ( x = frac{1}{2} ), ( y = frac{1}{3} ), ( z = frac{1}{6} ). Then compute ( a ):Compute ( xy + yz + zx ):( frac{1}{2} times frac{1}{3} = frac{1}{6} )( frac{1}{3} times frac{1}{6} = frac{1}{18} )( frac{1}{6} times frac{1}{2} = frac{1}{12} )Sum: ( frac{1}{6} + frac{1}{18} + frac{1}{12} = frac{6}{36} + frac{2}{36} + frac{3}{36} = frac{11}{36} approx 0.305 )Compute ( 2xyz = 2 times frac{1}{2} times frac{1}{3} times frac{1}{6} = 2 times frac{1}{36} = frac{1}{18} approx 0.055 )So ( a = frac{11}{36} - frac{1}{18} = frac{11}{36} - frac{2}{36} = frac{9}{36} = frac{1}{4} = 0.25 ). So this is lower than ( frac{7}{27} approx 0.259 ). So even in this case, the value of ( a ) is lower.Wait, so maybe the maximum is indeed ( frac{7}{27} ), achieved when all variables are equal, and the minimum approaches 0 when one variable approaches 1 and the others approach 0.But the problem states that ( x, y, z ) are positive real numbers, so they can't actually be 0. Therefore, ( a ) can't be 0, but can get as close to 0 as desired. So the range would be ( (0, frac{7}{27}] ).But wait, before concluding, let's check another case where one variable is different. Let's take ( x = frac{1}{2} ), ( y = frac{1}{2} ), but then ( z = 0 ), which is invalid. So instead, take ( x = y = frac{1}{2} - epsilon ), ( z = 1 - 2(frac{1}{2} - epsilon) = 2epsilon ). Then as ( epsilon to 0^+ ), ( z to 0 ).Compute ( a = xy + yz + zx - 2xyz ).( xy = (frac{1}{2} - epsilon)^2 )( yz = (frac{1}{2} - epsilon)(2epsilon) = epsilon - 2epsilon^2 )Similarly, ( zx = epsilon - 2epsilon^2 )So total ( xy + yz + zx = (frac{1}{4} - epsilon + epsilon^2) + 2(epsilon - 2epsilon^2) = frac{1}{4} - epsilon + epsilon^2 + 2epsilon - 4epsilon^2 = frac{1}{4} + epsilon - 3epsilon^2 )Then ( xyz = (frac{1}{2} - epsilon)^2 times 2epsilon approx frac{1}{4} times 2epsilon = frac{epsilon}{2} ) as ( epsilon to 0 )So ( a approx frac{1}{4} + epsilon - 3epsilon^2 - 2 times frac{epsilon}{2} = frac{1}{4} + epsilon - 3epsilon^2 - epsilon = frac{1}{4} - 3epsilon^2 )Therefore, as ( epsilon to 0 ), ( a to frac{1}{4} ). Wait, so here ( a ) approaches ( frac{1}{4} ), which is 0.25. But earlier, when variables are equal, ( a approx 0.259 ). So even higher. So in this case, as ( epsilon to 0 ), ( a to frac{1}{4} ), which is less than ( frac{7}{27} approx 0.259 ). Therefore, this suggests that the maximum is indeed at the symmetric point where all variables are equal.But wait, when variables are equal, we get ( a = frac{7}{27} approx 0.259 ), which is higher than 0.25. So maybe this is indeed the maximum. Let's check another configuration. Let's suppose ( x = 0.4 ), ( y = 0.4 ), ( z = 0.2 ). Then compute ( a ):Compute ( xy = 0.16 ), ( yz = 0.08 ), ( zx = 0.08 ). Sum: ( 0.16 + 0.08 + 0.08 = 0.32 ). Then ( xyz = 0.4 times 0.4 times 0.2 = 0.032 ). Therefore, ( a = 0.32 - 2 times 0.032 = 0.32 - 0.064 = 0.256 ). Which is approximately 0.256, which is still less than ( frac{7}{27} approx 0.259 ).Another example: ( x = 0.35 ), ( y = 0.35 ), ( z = 0.3 ). Then ( xy = 0.1225 ), ( yz = 0.105 ), ( zx = 0.105 ). Sum: ( 0.1225 + 0.105 + 0.105 = 0.3325 ). ( xyz = 0.35 times 0.35 times 0.3 = 0.03675 ). Then ( a = 0.3325 - 2 times 0.03675 = 0.3325 - 0.0735 = 0.259 ). That's exactly ( frac{7}{27} approx 0.259 ). Wait, but here ( x, y, z ) are 0.35, 0.35, 0.3, which are not equal. But the calculation gives ( a approx 0.259 ), same as when variables are equal. Wait, that can't be. Wait, let me recalculate:Wait, ( 0.35 + 0.35 + 0.3 = 1 ). So:( xy = 0.35 times 0.35 = 0.1225 )( yz = 0.35 times 0.3 = 0.105 )( zx = 0.3 times 0.35 = 0.105 )Sum: ( 0.1225 + 0.105 + 0.105 = 0.3325 )( xyz = 0.35 times 0.35 times 0.3 = 0.35 times 0.105 = 0.03675 )Thus, ( a = 0.3325 - 2 times 0.03675 = 0.3325 - 0.0735 = 0.259 ). And ( frac{7}{27} approx 0.259259... ), so 0.259 is approximately that. So perhaps in this case, even with unequal variables, we get the same value? That seems odd. Wait, maybe there's a miscalculation here.Wait, 0.35, 0.35, 0.3. Let's compute ( a = xy + yz + zx - 2xyz ).So, xy = 0.1225, yz = 0.105, zx = 0.105, so sum = 0.3325. Then, xyz = 0.35 * 0.35 * 0.3. Let's compute that:0.35 * 0.35 = 0.1225, then 0.1225 * 0.3 = 0.03675. So 2xyz = 0.0735. Then, 0.3325 - 0.0735 = 0.259, which is exactly 259/1000. And 7/27 is approximately 0.259259..., so it's slightly higher. Therefore, this is a close value but not exactly 7/27. Hmm, so maybe there's a different configuration where a is higher?Wait, perhaps the maximum occurs when variables are equal, so at 7/27 ≈ 0.259, but in the example above with variables 0.35, 0.35, 0.3, we get approximately 0.259, which is very close. But maybe that's due to rounding. Let's compute exactly.Let me compute 7/27:7 divided by 27: 27*0.25=6.75, 7-6.75=0.25, 0.25/27≈0.009259, so total is 0.25 + 0.009259≈0.259259...So 0.259259... is 7/27. The value 0.259 is just an approximate decimal. So in reality, when variables are equal, we get exactly 7/27, which is approximately 0.259259, which is a bit higher than 0.259. So in the previous example with 0.35, 0.35, 0.3, we got 0.259, which is slightly less than 7/27. Therefore, the maximum seems indeed to be at the symmetric point.Alternatively, maybe there's a maximum when variables are equal. Let's try to prove that.Let me consider the function ( a = xy + yz + zx - 2xyz ), under the constraint ( x + y + z = 1 ).We can use the method of Lagrange multipliers. Let's set up the Lagrangian:[mathcal{L} = xy + yz + zx - 2xyz - lambda (x + y + z - 1)]Take partial derivatives with respect to x, y, z, and set them equal to zero.Partial derivative with respect to x:[frac{partial mathcal{L}}{partial x} = y + z - 2yz - lambda = 0]Similarly for y:[frac{partial mathcal{L}}{partial y} = x + z - 2xz - lambda = 0]And for z:[frac{partial mathcal{L}}{partial z} = x + y - 2xy - lambda = 0]And the constraint:[x + y + z = 1]So, from the partial derivatives, we have:1. ( y + z - 2yz = lambda )2. ( x + z - 2xz = lambda )3. ( x + y - 2xy = lambda )Therefore, equations 1, 2, and 3 must all equal the same ( lambda ). So set equations 1 and 2 equal:( y + z - 2yz = x + z - 2xz )Simplify:( y - 2yz = x - 2xz )Factor:( y(1 - 2z) = x(1 - 2z) )If ( 1 - 2z neq 0 ), then we can divide both sides by ( (1 - 2z) ), getting ( y = x ).If ( 1 - 2z = 0 ), then ( z = frac{1}{2} ). Let's check that case later.Similarly, setting equations 1 and 3 equal:( y + z - 2yz = x + y - 2xy )Simplify:( z - 2yz = x - 2xy )Factor:( z(1 - 2y) = x(1 - 2y) )Similarly, if ( 1 - 2y neq 0 ), then ( z = x ). If ( 1 - 2y = 0 ), then ( y = frac{1}{2} ).So, assuming ( 1 - 2z neq 0 ) and ( 1 - 2y neq 0 ), we get ( y = x ) and ( z = x ), so all variables equal. Then ( x = y = z = frac{1}{3} ), which gives the critical point we found earlier.Now, check the case where ( 1 - 2z = 0 implies z = frac{1}{2} ). Then, from the constraint ( x + y + z = 1 ), we have ( x + y = frac{1}{2} ). Let's substitute ( z = frac{1}{2} ) into the expression for ( a ):[a = xy + y cdot frac{1}{2} + x cdot frac{1}{2} - 2xy cdot frac{1}{2}]Simplify:[a = xy + frac{1}{2}y + frac{1}{2}x - xy = frac{1}{2}(x + y)]Since ( x + y = frac{1}{2} ), then:[a = frac{1}{2} times frac{1}{2} = frac{1}{4}]So in this case, ( a = frac{1}{4} ), which is less than ( frac{7}{27} approx 0.259 ). Therefore, this is not the maximum. Similarly, if ( y = frac{1}{2} ), then ( x + z = frac{1}{2} ), and similar calculations would lead to ( a = frac{1}{4} ). Therefore, the maximum occurs at the symmetric point where all variables are equal.Hence, the maximum value of ( a ) is ( frac{7}{27} ).Now, for the minimum. We saw earlier that as variables approach 0, ( a ) approaches 0. But we need to verify if ( a ) can get arbitrarily close to 0. Suppose we fix one variable approaching 1, and the other two approaching 0. Let's take ( x = 1 - epsilon - delta ), ( y = epsilon ), ( z = delta ), where ( epsilon, delta ) approach 0.Compute ( a = xy + yz + zx - 2xyz ).Compute each term:- ( xy = (1 - epsilon - delta)epsilon approx epsilon )- ( yz = epsilon delta approx 0 )- ( zx = (1 - epsilon - delta)delta approx delta )- ( xyz = (1 - epsilon - delta)epsilon delta approx 0 )Therefore, ( a approx epsilon + delta - 0 = epsilon + delta ). If we let ( epsilon ) and ( delta ) approach 0, such that ( epsilon + delta = t ), where ( t ) approaches 0, then ( a approx t ), which can be made as small as desired. Therefore, the infimum of ( a ) is 0. However, since ( x, y, z ) are positive, ( a ) cannot be 0. Thus, the range of ( a ) is ( (0, frac{7}{27}] ).But let's confirm there isn't a lower bound higher than 0. Suppose we take ( x ) approaching 1, and ( y = z ). Let ( x = 1 - 2epsilon ), ( y = z = epsilon ), with ( epsilon to 0^+ ). Then compute ( a ):[xy + yz + zx = (1 - 2epsilon)epsilon + epsilon^2 + (1 - 2epsilon)epsilon = 2(1 - 2epsilon)epsilon + epsilon^2 = 2epsilon - 4epsilon^2 + epsilon^2 = 2epsilon - 3epsilon^2][xyz = (1 - 2epsilon)epsilon^2 approx epsilon^2]Therefore,[a = (2epsilon - 3epsilon^2) - 2(1 - 2epsilon)epsilon^2 approx 2epsilon - 3epsilon^2 - 2epsilon^2 = 2epsilon - 5epsilon^2]As ( epsilon to 0 ), ( a approx 2epsilon ), which approaches 0. So even in this case, ( a ) can be made arbitrarily small. Hence, the infimum is 0, but not attainable.Therefore, combining the maximum and the infimum, the range of ( a ) is ( (0, frac{7}{27}] ).However, the problem statement says "the range of the real number ( a ) is _." In Olympiad problems, sometimes intervals are expressed with their bounds, even if they are not attained. But in this case, since ( x, y, z ) are positive, ( a ) can't be 0, so the range is ( 0 < a leq frac{7}{27} ).But let's check once more for any possible mistakes. For instance, could there be a case where ( a ) is negative?Suppose ( x, y, z ) are such that ( xy + yz + zx < 2xyz ). Then ( a ) would be negative. Let's see if this is possible.Take ( x = y = z = frac{1}{3} ). Then ( xy + yz + zx = 3 times frac{1}{9} = frac{1}{3} ), ( 2xyz = 2 times frac{1}{27} = frac{2}{27} approx 0.074 ). So ( frac{1}{3} > frac{2}{27} ), so ( a = frac{1}{3} - frac{2}{27} = frac{7}{27} ), positive.If we take ( x ) very large, but wait, ( x + y + z = 1 ), so variables can't be large. All variables are less than 1. Let's try ( x = 0.5 ), ( y = 0.5 ), ( z = 0 ). But z must be positive. Let’s take z = 0.0001, so x = y = 0.49995.Compute ( xy + yz + zx approx 0.49995^2 + 2 times 0.49995 times 0.0001 approx 0.24995 + 0.00009999 approx 0.25005 )Compute ( 2xyz approx 2 times 0.49995 times 0.49995 times 0.0001 approx 2 times 0.24995 times 0.0001 approx 0.00004999 )Thus, ( a approx 0.25005 - 0.00004999 approx 0.25 ), which is positive.Alternatively, take x=0.9, y=0.05, z=0.05. Compute ( xy + yz + zx = 0.9*0.05 + 0.05*0.05 + 0.9*0.05 = 0.045 + 0.0025 + 0.045 = 0.0925 ). ( 2xyz = 2*0.9*0.05*0.05 = 2*0.00225 = 0.0045 ). Thus, ( a = 0.0925 - 0.0045 = 0.088 ), still positive.Wait, what if two variables are very small? Let's take x=0.5, y approaching 0.5, z approaching 0. Let x=0.5, y=0.4999, z=0.0001. Then:xy = 0.5*0.4999 ≈ 0.24995yz ≈ 0.4999*0.0001 ≈ 0.00004999zx ≈ 0.5*0.0001 ≈ 0.00005Sum ≈ 0.24995 + 0.00004999 + 0.00005 ≈ 0.25005xyz ≈ 0.5*0.4999*0.0001 ≈ 0.000024995Thus, 2xyz ≈ 0.00004999So, a ≈ 0.25005 - 0.00004999 ≈ 0.25Still positive. Alternatively, take x=0.6, y=0.3, z=0.1:xy=0.18, yz=0.03, zx=0.06. Sum=0.27. xyz=0.6*0.3*0.1=0.018. 2xyz=0.036. a=0.27 - 0.036=0.234. Still positive.It seems that ( a ) is always positive. Let me check if there's a case where ( a ) could be negative.Suppose x, y, z are such that ( xy + yz + zx < 2xyz ). Let's see if this is possible.Let’s rearrange: ( xy + yz + zx < 2xyz )Divide both sides by xyz (since x, y, z >0):( frac{1}{z} + frac{1}{x} + frac{1}{y} < 2 )But given that x + y + z =1, and x, y, z >0, can the sum of reciprocals be less than 2?Wait, let's try to see. Take x = y = z = 1/3. Then sum of reciprocals is 3/(1/3) =9, which is way larger than 2. If x approaches 1, then reciprocals approach infinity. If one variable is very small, say z approaches 0, then 1/z approaches infinity. So in all cases, sum of reciprocals is at least... Wait, how?Actually, since x, y, z are positive and sum to 1, each of them is at most 1, so their reciprocals are at least 1. Hence, sum of reciprocals is at least 3. So the equation ( frac{1}{x} + frac{1}{y} + frac{1}{z} - 2 = frac{a}{xyz} ) implies that ( frac{a}{xyz} ) is at least 3 -2 =1. Therefore, ( a geq xyz times 1 ), but since xyz can be very small, a can be very small. However, from the earlier analysis, ( a = xy + yz + zx - 2xyz ), which is always positive. Wait, so maybe my previous thought was wrong. Let's check.Wait, in the problem statement, it's given that ( frac{a}{xyz} = frac{1}{x} + frac{1}{y} + frac{1}{z} - 2 ). If ( frac{1}{x} + frac{1}{y} + frac{1}{z} - 2 ) must equal ( frac{a}{xyz} ). Since ( x, y, z ) are positive, ( frac{a}{xyz} ) must be a real number, so the right-hand side must be positive. But earlier analysis shows that even when variables are approaching extremes, ( a ) remains positive. But let's check if ( frac{1}{x} + frac{1}{y} + frac{1}{z} - 2 ) is positive.Indeed, since ( frac{1}{x} + frac{1}{y} + frac{1}{z} geq frac{9}{x + y + z} = 9 ) by Cauchy-Schwarz inequality (AM-HM inequality), so ( frac{1}{x} + frac{1}{y} + frac{1}{z} geq 9 ), so ( frac{1}{x} + frac{1}{y} + frac{1}{z} -2 geq 7 ). Wait, this contradicts previous analysis where ( a ) approaches 0. Wait, clearly something is wrong here.Wait, hold on. The Cauchy-Schwarz inequality states that ( frac{1}{x} + frac{1}{y} + frac{1}{z} geq frac{9}{x + y + z} ). Since ( x + y + z =1 ), then ( frac{1}{x} + frac{1}{y} + frac{1}{z} geq 9 ). Therefore, ( frac{a}{xyz} = frac{1}{x} + frac{1}{y} + frac{1}{z} -2 geq 9 - 2 =7 ). Therefore, ( frac{a}{xyz} geq 7 implies a geq7xyz ).But earlier, we saw that as variables approach 0, ( a ) approaches 0. However, according to this inequality, ( a geq7xyz ). Since ( xyz ) can approach 0, then ( a ) can approach 0 as well, but with ( a geq7xyz ). So if ( xyz ) approaches 0, then ( a ) approaches 0 as well, but bounded below by 7xyz. However, this does not prevent ( a ) from approaching 0. For example, if ( xyz = epsilon ), then ( a geq7epsilon ), but if ( a ) is exactly (7epsilon ), then ( epsilon ) can be made as small as desired, hence ( a ) can approach 0.But wait, but according to the problem, ( a ) is given by ( a = xy + yz + zx - 2xyz ). So perhaps the inequality ( a geq7xyz ) is true, but since ( xyz ) can be very small, ( a ) can still be very small. Hence, the infimum is 0.But there's a contradiction with the Cauchy-Schwarz result. Wait, because if ( frac{1}{x} + frac{1}{y} + frac{1}{z} geq9 ), then ( frac{a}{xyz} geq7 ), so ( a geq7xyz ). However, even if ( xyz to 0 ), ( a ) can be made as small as desired, but with ( a geq7xyz ). So, for example, if ( xyz = epsilon ), then ( a geq7epsilon ). But if ( a =7epsilon ), then ( epsilon = a/7 ). But in reality, ( a = xy + yz + zx - 2xyz ). How does this relate?Wait, perhaps there's a misunderstanding here. Let me re-examine.We have ( frac{a}{xyz} = frac{1}{x} + frac{1}{y} + frac{1}{z} -2 ). From Cauchy-Schwarz, ( frac{1}{x} + frac{1}{y} + frac{1}{z} geq9 ), so ( frac{a}{xyz} geq9 -2 =7 implies a geq7xyz ). But ( xyz leq left( frac{x + y + z}{3} right)^3 = left( frac{1}{3} right)^3 = frac{1}{27} ), so ( a geq7 times frac{1}{27} = frac{7}{27} ). Wait, this contradicts previous analysis where ( a ) can be as low as approaching 0. But this must be wrong.Wait, actually, the maximum of ( xyz ) occurs when ( x = y = z =1/3 ), so ( xyz leq1/27 ). Therefore, ( a geq7 times xyz leq7 times1/27=7/27 ). Wait, no. If ( a geq7xyz ), and ( xyz leq1/27 ), then ( a geq7 times0 ), which is just non-negative. Wait, no. This inequality says that ( a ) is at least (7xyz ), but since ( xyz ) can be as small as approaching 0, then ( a ) can also approach 0, but bounded below by 0. Hence, the earlier analysis holds. There must be a mistake in the reasoning here.Wait, let's step back. The Cauchy-Schwarz gives ( frac{1}{x} + frac{1}{y} + frac{1}{z} geq9 ), so ( frac{a}{xyz} geq7 implies a geq7xyz ). This is a valid inequality. So, ( a ) is at least seven times the product ( xyz ). But since ( xyz ) can be made arbitrarily small (by making one variable approach 1 and others approach 0), the lower bound for ( a ) is 0. Because even though ( a geq7xyz ), as ( xyz to0 ), ( a to0 ). Hence, there's no contradiction.Therefore, ( a ) can be made arbitrarily close to 0 but remains positive, and the maximum value is ( frac{7}{27} ).Thus, the range of ( a ) is ( (0, frac{7}{27}] ).Final AnswerThe range of the real number ( a ) is boxed{left(0, dfrac{7}{27}right]}.</think>Given that ( x, y, z ) are positive real numbers such that ( x + y + z = 1 ), we need to find the range of the real number ( a ) given by the equation:[frac{a}{xyz} = frac{1}{x} + frac{1}{y} + frac{1}{z} - 2]First, we express ( a ) in terms of ( x, y, z ):[a = xyz left( frac{1}{x} + frac{1}{y} + frac{1}{z} - 2 right)]Simplifying the right-hand side:[a = yz + xz + xy - 2xyz]We need to find the range of ( a ) under the constraint ( x + y + z = 1 ). Key Steps:1. Symmetric Case Analysis: - When ( x = y = z = frac{1}{3} ): [ a = 3 left( frac{1}{3} cdot frac{1}{3} right) - 2 left( frac{1}{3} right)^3 = frac{7}{27} ] This gives the maximum value of ( a ).2. Boundary Case Analysis: - When one variable approaches 1 and the other two approach 0, the value of ( a ) approaches 0. For example, if ( x to 1 ) and ( y, z to 0 ): [ a approx 0 ] This shows that ( a ) can get arbitrarily close to 0 but remains positive.3. Inequality Verification: - Using the Cauchy-Schwarz inequality, we confirm that the sum of reciprocals ( frac{1}{x} + frac{1}{y} + frac{1}{z} geq 9 ), leading to ( frac{a}{xyz} geq 7 ). However, since ( xyz ) can be made arbitrarily small, ( a ) can approach 0 but remains positive. Conclusion:The range of ( a ) is from values just above 0 up to ( frac{7}{27} ). Therefore, the range of the real number ( a ) is:[boxed{left(0, dfrac{7}{27}right]}]]

🔑:5. Fill left(0, frac{7}{27}right]. Reason: It is easy to know that x, y, z in (0,1) and 1-x, 1-y, 1-z in (0,1). On one hand, it is easy to get a = xy + yz + zx - 2xyz > xy > 0. When x, y rightarrow 0, z rightarrow 1, a rightarrow 0. On the other hand, it is easy to get a = (1-x)x + (1-2x)yz.When 0 < x < 1, (1-2x)yz > 0, so, a < (1-x)x leqslant frac{1}{4}.In summary, we have a in left(0, frac{7}{27}right].

🧠:[<think>Okay, so I need to find the volume of the common part of two spheres. Their radii are 5 and 3 units, and the distance between their centers is 4 units. Hmm, this sounds like a problem involving the intersection of two spheres. I remember that when two spheres intersect, their common part is called a lens-shaped region, and its volume can be calculated using the formula for the volume of a spherical cap, multiplied by two because there are two caps involved. Wait, but I need to recall exactly how that works.First, let me visualize the problem. There's a larger sphere with radius 5 and a smaller sphere with radius 3. The distance between their centers is 4 units. So the spheres definitely intersect because the sum of the radii is 8, and the distance between centers is 4, which is less than 8. Also, since the distance is greater than the difference of the radii (5 - 3 = 2), the smaller sphere isn't entirely inside the larger one. So they intersect each other, forming a lens shape.To find the volume of the intersection, I need to compute the combined volume of the two spherical caps from each sphere that lie within the other sphere. The formula for the volume of a spherical cap is ( V = frac{pi h^2}{3}(3r - h) ), where ( h ) is the height of the cap, and ( r ) is the radius of the original sphere. So, I need to find the heights of the caps for both spheres.Let me denote the two spheres as Sphere A (radius 5) and Sphere B (radius 3). The distance between their centers is 4 units. Let me set up a coordinate system where the center of Sphere A is at the origin, and the center of Sphere B is along the positive x-axis at (4, 0, 0). The intersection of the two spheres will form a lens, and the line connecting their centers will intersect the lens at two points. The distance from each center to the plane where the spheres intersect is needed to compute the height of each cap.Let’s call the distance from the center of Sphere A to the intersection plane as ( d_1 ), and from Sphere B as ( d_2 ). Since the centers are 4 units apart, ( d_1 + d_2 = 4 ). Also, using the Pythagorean theorem for each sphere's radius, the radius of the intersection circle (which is the same for both spheres) can be found. For Sphere A: ( r_1 = sqrt{5^2 - d_1^2} ). For Sphere B: ( r_2 = sqrt{3^2 - d_2^2} ). Since the intersection circle's radius must be the same for both spheres, ( r_1 = r_2 ), so:[sqrt{25 - d_1^2} = sqrt{9 - d_2^2}]Squaring both sides:[25 - d_1^2 = 9 - d_2^2]Simplify:[25 - 9 = d_1^2 - d_2^2][16 = d_1^2 - d_2^2]But we also know that ( d_1 + d_2 = 4 ). Let me denote ( d_1 = x ), so ( d_2 = 4 - x ). Substitute into the equation:[x^2 - (4 - x)^2 = 16]Expand ( (4 - x)^2 ):[x^2 - (16 - 8x + x^2) = 16]Simplify:[x^2 -16 +8x -x^2 =16][8x -16 =16][8x =32][x=4]Wait, that can't be right. If ( d_1 =4 ), then ( d_2 =0 ). But if ( d_1 =4 ), then the radius of the intersection circle would be ( sqrt{25 -16}=3 ), and for Sphere B, ( sqrt{9 -0}=3 ). So that does check out. Wait, but if ( d_1 =4 ), that means the plane of intersection is 4 units away from Sphere A's center, which is exactly at the center of Sphere B. That seems possible. Let me confirm.If the two spheres intersect such that the plane of intersection is at Sphere B's center, then the radius of the intersection circle on Sphere A would be ( sqrt{5^2 -4^2} =3 ), and on Sphere B, the radius would be ( sqrt{3^2 -0^2}=3 ). So yes, that works. But wait, Sphere B's center is 4 units away from Sphere A's center, so if the intersection plane is at Sphere B's center, then Sphere B's cap height would be from its center to the edge of the intersection. Wait, but Sphere B's radius is 3, and the distance from Sphere B's center to the intersection plane is 0, so the height of the cap for Sphere B would be 3 units? Wait, maybe I need to think again.Wait, the height h of the spherical cap is the distance from the intersection plane to the "top" of the sphere. For Sphere A, the distance from its center to the intersection plane is 4, so the height of the cap (the part of Sphere A that is inside Sphere B) would be ( h_A = 5 - d_1 =5 -4=1 ). Similarly, for Sphere B, the distance from its center to the intersection plane is 0, so the height of its cap (the part inside Sphere A) would be ( h_B =3 - d_2 =3 -0=3 ). But that seems like Sphere B's entire volume would be inside Sphere A, but Sphere B is only 3 units in radius, and the center is 4 units away from Sphere A. Wait, that can't be. If the center of Sphere B is 4 units away from Sphere A, and Sphere B has radius 3, then the farthest point of Sphere B from Sphere A's center is 4 +3 =7, which is within Sphere A's radius of 5? Wait no, 4 +3 =7, which is larger than 5, so Sphere B is partially outside Sphere A. Therefore, the intersection is a lens shape where part of Sphere B is inside Sphere A and vice versa.But according to the calculation above, the intersection plane is at Sphere B's center. Wait, that seems strange. Let me double-check the equations.We had:( sqrt{25 - d_1^2} = sqrt{9 - d_2^2} )and( d_1 + d_2 =4 )Solving these, we found ( d_1 =4 ), ( d_2=0 )But let's verify if this is correct.If ( d_1=4 ), then the radius of the intersection circle is ( sqrt{25 -16}=3 )For Sphere B, ( d_2=0 ), so the radius of the intersection circle is ( sqrt{9 -0}=3 ). So yes, they agree.But this seems counterintuitive because the intersection plane is at the center of Sphere B, which is 4 units from Sphere A. So the intersection circle on Sphere A is 3 units, which is the same as Sphere B's radius. But Sphere B's radius is 3, so Sphere B is exactly reaching up to that intersection plane. Wait, but the intersection plane is at Sphere B's center. Hmm.Wait, maybe another way to think about it: if the two spheres intersect, the set of points common to both is all points that satisfy both sphere equations.Sphere A: ( x^2 + y^2 + z^2 =25 )Sphere B: ( (x -4)^2 + y^2 + z^2 =9 )Subtracting the two equations:( x^2 + y^2 + z^2 - [(x -4)^2 + y^2 + z^2] =25 -9 )Simplify:( x^2 - (x^2 -8x +16) =16 )( x^2 -x^2 +8x -16=16 )( 8x -16=16 )( 8x=32 )( x=4 )So, the intersection of the two spheres is the plane x=4. That is, all points where x=4, and the distance from Sphere A's center (origin) is 4 units along the x-axis, and from Sphere B's center (4,0,0) is 0 units. So, the intersection is a circle lying on the plane x=4, with radius 3 (as calculated earlier). So, indeed, the intersection is a circle of radius 3 at x=4. Therefore, the spherical caps are the portions of each sphere beyond this plane.For Sphere A, the cap is the region where x >=4 (since Sphere A is centered at the origin). But the distance from the center of Sphere A to the plane is 4, which is greater than the radius of Sphere A (5). Wait, no, Sphere A's radius is 5, so the plane x=4 is inside Sphere A. Wait, the distance from the center of Sphere A to the plane x=4 is 4 units. So the cap height for Sphere A is from x=4 to the "top" of Sphere A along the x-axis. The maximum x-coordinate in Sphere A is 5 (at the point (5,0,0)). So the height of the cap for Sphere A is 5 -4=1.Similarly, for Sphere B, the plane x=4 is at its center. So the cap height for Sphere B is from x=4 to the "top" of Sphere B, which is at x=4 +3=7. Wait, but Sphere B is centered at (4,0,0), so its maximum x-coordinate is 7, but Sphere A only extends to x=5. So actually, the portion of Sphere B that is inside Sphere A is from x=1 to x=5 (since Sphere A is up to x=5). Wait, no, maybe this is where the confusion comes.Wait, the intersection is the plane x=4. So for Sphere A, the cap is the part of Sphere A where x >=4, which is a spherical cap of height h_A=5 -4=1.For Sphere B, the cap is the part of Sphere B where x <=4, but since the Sphere B is centered at x=4, the cap is actually the entire hemisphere of Sphere B where x <=4. But the radius of Sphere B is 3, so the distance from its center to the plane is zero, so the height of the cap would be 3. Wait, but if the center is at the plane, then the cap height is equal to the radius? Let me check the formula again.The formula for the volume of a spherical cap is ( V = frac{pi h^2}{3}(3r - h) ). If the plane is passing through the center of the sphere, then the cap height h would be equal to the radius. Wait, no. If the plane cuts through the center, then the cap is a hemisphere. The height of a hemisphere is equal to the radius. So in this case, for Sphere B, the height h_B is 3, since it's a hemisphere. Let's check:If h=3, then the volume would be ( frac{pi (3)^2}{3}(9 -3) = frac{9pi}{3}(6) =3pi *6=18pi ). But the volume of a hemisphere is ( frac{2}{3}pi r^3 = frac{2}{3}pi 27=18pi ). So yes, that matches. So the cap for Sphere B is a hemisphere with volume 18π.For Sphere A, the cap has h=1, so the volume is ( frac{pi (1)^2}{3}(15 -1) = frac{pi}{3}(14) = frac{14pi}{3} ).Therefore, the total volume of the intersection is the sum of these two caps: ( 18pi + frac{14pi}{3} = frac{54pi +14pi}{3} = frac{68pi}{3} ). Hmm, but I need to verify this because sometimes the formula for the spherical cap can be tricky depending on how the cap is oriented.Wait, let me think again. The formula for the spherical cap is when you have a sphere of radius r, and a cap of height h. The height is the distance from the base of the cap (the intersection plane) to the top of the sphere. For Sphere A, the intersection plane is at x=4, which is 1 unit away from the top of Sphere A at x=5. So h=1. For Sphere B, the intersection plane is at x=4, which is the center of Sphere B, so the cap is a hemisphere, which has h=3 (the radius). But actually, the part of Sphere B that lies inside Sphere A is the hemisphere where x <=4, but since Sphere B is centered at x=4, the hemisphere where x <=4 is a hemisphere, but how much of that hemisphere is inside Sphere A?Wait, hold on. Sphere B's hemisphere from x=4 to x=1 (since it's radius 3). But Sphere A only extends to x=5. So actually, the entire hemisphere of Sphere B where x <=4 is inside Sphere A? Wait, no. Let me check coordinates.Sphere B is centered at (4,0,0) with radius 3. So the part of Sphere B that is inside Sphere A must satisfy both Sphere B's equation and Sphere A's equation. The intersection is the plane x=4, so the part of Sphere B that is inside Sphere A is where x <=4 (since beyond x=4 is outside Sphere A). Wait, no. If you take a point in Sphere B where x >4, then its distance from Sphere A's center (origin) is more than 4 units. But Sphere A's radius is 5, so points in Sphere B with x >4 can still be inside Sphere A as long as their distance from the origin is <=5.Wait, this is getting confusing. Maybe it's better to use the formula for the volume of intersection of two spheres. Let me recall that formula.The volume of the intersection of two spheres can be calculated using the formula involving the heights of the spherical caps. The formula is:( V = frac{pi (h_1^2)(3r_1 - h_1)}{3} + frac{pi (h_2^2)(3r_2 - h_2)}{3} )where ( h_1 ) and ( h_2 ) are the heights of the caps from each sphere.To find ( h_1 ) and ( h_2 ), we can use the following relations:If the distance between the centers is ( d ), then:( h_1 = r_1 - frac{d^2 + r_1^2 - r_2^2}{2d} )( h_2 = r_2 - frac{d^2 + r_2^2 - r_1^2}{2d} )Wait, let me check that. I think there is a formula for the height of the cap in terms of the radii and the distance between centers.Alternatively, we can use the formula:Given two spheres with radii ( r ) and ( R ), separated by distance ( d ). The height of the cap for the first sphere is ( h = r - frac{d^2 + r^2 - R^2}{2d} )Similarly for the second sphere.Let me try that.For Sphere A (radius 5) and Sphere B (radius 3), distance d=4.So, ( h_1 = 5 - frac{4^2 +5^2 -3^2}{2*4} )Compute numerator: 16 +25 -9 =32So ( h_1 =5 -32/(8)=5 -4=1 ). That matches the previous calculation.For Sphere B:( h_2=3 - frac{4^2 +3^2 -5^2}{2*4} )Compute numerator:16 +9 -25=0Thus ( h_2=3 -0/8=3 ). So h_2=3. Wait, but that seems to suggest that the height of the cap for Sphere B is 3, which is the entire radius, meaning the cap is a hemisphere. But as we saw earlier, Sphere B's center is at 4 units from Sphere A's center, and Sphere B has radius 3, so the portion of Sphere B inside Sphere A is not a hemisphere. Wait, but according to the formula, h_2=3. So perhaps the formula is correct, but we need to consider that the cap here is not a "height" in the same direction. Let me think.Since the distance between centers is 4, and Sphere B has radius 3, the overlap of Sphere B inside Sphere A is a spherical cap of height h_2=3. But since Sphere B is centered 4 units away from Sphere A, and has radius 3, the portion of Sphere B that lies within Sphere A is all points of Sphere B that are within 5 units from Sphere A's center. But Sphere B's center is 4 units away, so the overlap is all points of Sphere B that are within 5 units of the origin. The set of points in Sphere B that are also in Sphere A is the region of Sphere B where the distance from the origin is <=5. The boundary of this region is the intersection circle we found earlier at x=4, radius 3.But how does that translate to the cap height? If we consider Sphere B, the cap height h_2 is the distance from the intersection plane (x=4) to the farthest point of Sphere B from its own center (which is at (4,0,0)). But the farthest point of Sphere B from its own center is in the direction away from Sphere A, which would be at (7,0,0). However, the portion of Sphere B inside Sphere A is from (1,0,0) to (4,0,0). Wait, this is confusing.Wait, perhaps the cap height h_2 is the distance from the intersection plane (x=4) to the "bottom" of Sphere B's cap. Wait, no. The formula for the spherical cap height is the distance from the base of the cap (the intersection plane) to the top of the sphere. But in Sphere B's case, the intersection plane is at x=4, which is exactly the center of Sphere B. So the cap height would be the distance from x=4 to x=7 (the top of Sphere B) which is 3 units, but that cap is outside Sphere A. However, the part of Sphere B that is inside Sphere A is the hemisphere from x=4 to x=1. Wait, but x=1 is 3 units away from Sphere B's center in the opposite direction.Wait, this is getting me confused. Let me approach it differently.The formula gives h_2=3, so according to the formula, the volume contributed by Sphere B's cap is ( frac{pi (3)^2}{3}(9 -3) = frac{9pi}{3}*6=18pi ), which is the volume of a hemisphere. But the hemisphere of Sphere B that is inside Sphere A is not the entire hemisphere. Because Sphere B's center is at x=4, and it has radius 3, the points of Sphere B range from x=1 to x=7. Sphere A, with radius 5, extends from x=-5 to x=5. Therefore, the overlap of Sphere B with Sphere A is from x=1 to x=5. The intersection circle is at x=4, radius 3. Therefore, the part of Sphere B inside Sphere A is a spherical cap from x=1 to x=4? Wait, no. Let me parametrize Sphere B.Sphere B is centered at (4,0,0) with radius 3. So the equation is ( (x -4)^2 + y^2 + z^2 <=9 ). The overlap with Sphere A (( x^2 + y^2 + z^2 <=25 )) occurs where both inequalities hold. The intersection is the set of points where ( (x -4)^2 + y^2 + z^2 <=9 ) and ( x^2 + y^2 + z^2 <=25 ).Subtracting the two equations as before, we found that the plane of intersection is x=4. Therefore, the overlap region for Sphere B is all points in Sphere B with x <=4. Since Sphere B is centered at x=4, x <=4 is a hemisphere of Sphere B. So the volume of this hemisphere is ( frac{2}{3}pi r^3 = frac{2}{3}pi 27=18pi ). But wait, is this hemisphere entirely inside Sphere A?Sphere B's hemisphere where x <=4 is centered at (4,0,0) and extends from x=1 to x=4. The points from x=1 to x=4. Now, in Sphere A, any point with x<=5 is inside. So x=1 to x=4 is entirely within Sphere A. Therefore, the hemisphere of Sphere B where x <=4 is entirely inside Sphere A. Therefore, the volume contributed by Sphere B is indeed 18π.Similarly, the volume contributed by Sphere A is the cap from x=4 to x=5, which is a height of 1. So the volume is ( frac{pi h^2}{3}(3r - h) = frac{pi (1)^2}{3}(15 -1)= frac{14pi}{3} ).Therefore, the total volume of the intersection is ( 18pi + frac{14pi}{3}= frac{54pi +14pi}{3}= frac{68pi}{3} ).But wait, let me verify this with another method to make sure.Alternatively, the volume of the intersection can be calculated using the formula for two intersecting spheres:( V = frac{pi (r_1 + r_2 - d)^2 [d^2 + 2d(r_1 + r_2) - 3(r_1 - r_2)^2]}{12d} )But I'm not sure if this formula is correct. Let me check.Alternatively, another formula for the volume of intersection is:( V = frac{pi (4 + r_1 + r_2 - d)(d + r_1 - r_2)(d + r_2 - r_1)(d + r_1 + r_2)}{12d} )Wait, actually, that formula is for the lens volume. Let me check.Wait, actually, I think the formula for the volume of the intersection (lens) of two spheres is:( V = frac{pi (h_1^2)(3r_1 - h_1) + pi (h_2^2)(3r_2 - h_2)}{3} )Which is exactly what we computed earlier.Given that h_1 =1 and h_2=3, then:( V = frac{pi (1)^2(15 -1) + pi (3)^2(9 -3)}{3} = frac{pi (14) + pi (54)}{3} = frac{68pi}{3} )So that's 68π/3.But let me cross-validate this with an example. Suppose two spheres of equal radius intersecting. For example, if two spheres of radius r with centers separated by d, the volume of intersection should be ( frac{4pi r^3}{3} - 2*frac{pi h^2}{3}(3r -h) ), where h is the height of the cap. But maybe that's overcomplicating.Alternatively, let's take the values here and see if 68π/3 makes sense.Sphere A has volume (4/3)π*5³ = 500π/3 ≈166.666πSphere B has volume (4/3)π*3³=36π≈113.097πThe intersection is 68π/3≈22.666π. That seems reasonable, as it's smaller than both spheres.Alternatively, if the two spheres were completely overlapping, the intersection would be the volume of the smaller sphere, but here the distance is 4, so they're partially overlapping.Alternatively, suppose the distance was 0, the intersection would be the smaller sphere's volume, 36π. If the distance was 5 +3=8, the intersection would be zero. Here, distance 4, so the intersection should be between 0 and 36π, but in this case, the answer is ~22.666π, which seems plausible.But wait, another sanity check. If the centers are 4 units apart, Sphere A radius 5, Sphere B radius 3.The overlap on Sphere A is a cap of height 1, volume 14π/3≈4.666πThe overlap on Sphere B is a hemisphere, volume 18π≈56.548π. Wait, that can't be. Wait, no, the total intersection is the sum, 4.666π +56.548≈61.214, but 68π/3≈71.209. Wait, no, 18π≈56.548 and 14π/3≈14.66, so total is≈71.209.Wait, but Sphere B's hemisphere is 18π, which is part of the intersection. But Sphere B's hemisphere from x=4 to x=1 is entirely inside Sphere A. But Sphere A's cap from x=4 to x=5 is a small region. So the total intersection is 18π +14π/3≈22.666π. Wait, no, wait 18π is about 56.548, and 14π/3≈4.666π. So total≈61.214π. Wait, but 68/3≈22.666, but that's in terms of multiples of π. Wait, 68π/3≈22.666π. Wait, no, 68 divided by 3 is≈22.666, so 68π/3≈22.666π. But according to the previous, it's 18π +14π/3≈(54/3 +14/3)π=68/3π≈22.666π. Yes, that's correct. So the total intersection volume is≈22.666π. Which is about 22.666*3.1416≈71.2. Which is less than both spheres' volumes.But if I imagine, the larger sphere has a small cap (volume≈4.666π) and the smaller sphere contributes a hemisphere (18π≈56.548). But when you add them, you get≈22.666π, which is the total intersection volume. But how can the intersection be the sum of the two caps? Wait, actually, no. The intersection is the region common to both spheres, which can be seen as the sum of the two caps, but actually, each cap is the part of each sphere that is inside the other sphere. So the total intersection is the union of these two caps, but since the intersection is a single lens-shaped region, the volume is the sum of the two caps. Wait, but that would double-count the overlapping region. Wait, no, because each cap is from a different sphere. Wait, no, in reality, the intersection region is exactly the part that is inside both spheres, which can be calculated by considering the portion of each sphere that is inside the other. However, when you take the union, it's the same as the intersection volume. But in reality, the intersection is a lens which is the union of two spherical caps. But since they are overlapping, the total volume is the sum of the two caps. Wait, but in reality, the intersection is the same region, so it should be equal to one cap plus the other cap. But maybe they don't overlap. Wait, but how?Wait, actually, no. The intersection of the two spheres is a lens shape, which is formed by the two spherical caps. However, each cap is part of one sphere, and together they form the lens. So the total volume of the intersection is the sum of the two caps. Therefore, the formula is correct.But here's the problem: when we calculated the cap for Sphere A, it's the part of Sphere A that lies within Sphere B, and the cap for Sphere B is the part of Sphere B that lies within Sphere A. Since the intersection is the region common to both spheres, it's equivalent to the union of the two caps? Wait, no, actually, the intersection is exactly the region that is inside both spheres, which is the same as the cap from Sphere A inside Sphere B and the cap from Sphere B inside Sphere A, which are the same region. Therefore, adding them would double count. Wait, no, that can't be. Each cap is the portion of one sphere inside the other, but the intersection region is the same. Therefore, adding them is not correct.Wait, now I'm confused. Maybe I made a mistake here.Wait, the intersection of the two spheres is the set of points that are in both spheres. Therefore, it's the same as the part of Sphere A that is inside Sphere B, which is the cap of Sphere A with volume 14π/3, and it's also the same as the part of Sphere B that is inside Sphere A, which is the hemisphere of Sphere B with volume 18π. But these are two different ways of describing the same region. So the actual volume should be the same whether calculated from Sphere A's perspective or Sphere B's perspective. But according to the calculations, they are different. That's a problem.Wait, this inconsistency shows that there is a mistake in my reasoning. So if the intersection volume is the same regardless of which sphere's cap we consider, but according to the calculations, they are different, then something is wrong.Wait, this is a critical point. The volume of the intersection must be unique. Therefore, if calculating from Sphere A's cap gives 14π/3 and from Sphere B's cap gives 18π, but these are supposed to represent the same volume, then clearly there's an error. Therefore, my previous approach is flawed.So where is the mistake?Ah, I think the confusion arises from misunderstanding what the caps represent. The intersection volume is a lens-shaped region that is part of both Sphere A and Sphere B. To compute its volume, you can consider it as the sum of the two spherical caps from each sphere that make up the lens. However, these caps are not separate regions; they are the same region. Therefore, adding them would be incorrect. Instead, the intersection volume is just the volume of one of the caps, but since the caps are parts of different spheres, you need to compute them correctly.Wait, no, actually, each cap is part of one sphere, but the intersection is the combination. Let me refer to the formula for the volume of intersection of two spheres.According to the formula on mathworld: http://mathworld.wolfram.com/Sphere-SphereIntersection.htmlThe volume of intersection is:( V = frac{pi (4 + r_1 + r_2 - d)(d + r_1 - r_2)(d + r_2 - r_1)(d + r_1 + r_2)}{12d} )But let's plug in the values:r1=5, r2=3, d=4.First compute each term:4 + r1 + r2 -d =4 +5+3 -4=8d + r1 -r2=4+5-3=6d + r2 -r1=4+3-5=2d + r1 + r2=4+5+3=12Multiply them all together:8*6*2*12=8*6=48; 48*2=96; 96*12=1152Then divide by 12d=12*4=48So V= (1152 /48 )π=24π.Wait, that's 24π. But according to our previous calculation, we had 68π/3≈22.666π. These are different. Therefore, my previous calculation was wrong, and the correct answer is 24π.So where did I go wrong?Looking back, when I calculated the volume of the intersection as the sum of the two spherical caps, I was wrong. The correct formula for the intersection volume is not the sum of the two caps, but actually the sum of the two caps minus the overlapping part, but since the caps are the same region, the intersection volume is the same as either one of the caps. Wait, no. The formula from mathworld gives 24π, so that must be correct.Alternatively, maybe the formula for the intersection is the sum of the two spherical caps. Wait, let me verify.Let me recompute using the formula for spherical caps.For Sphere A:h1=1, so cap volume is (π h1² /3)(3r1 - h1)= (π*1/3)(15 -1)=14π/3≈4.666πFor Sphere B:h2=3, cap volume is (π h2² /3)(3r2 - h2)= (π*9/3)(9 -3)=3π*6=18π≈56.548πBut 4.666π +56.548π≈61.214π, which is different from mathworld's 24π. Therefore, this approach is wrong.Thus, my mistake was thinking that the intersection volume is the sum of the two caps. Actually, the intersection is a lens shape which is formed by each sphere contributing a cap. However, the volume of the intersection is the sum of the individual caps minus the overlapping region. But since the caps are the same region, adding them would count the intersection twice. Therefore, the correct formula must be different.Wait, no. The intersection is one region, so it should be calculated either as the cap of Sphere A inside Sphere B or the cap of Sphere B inside Sphere A. But since the lens is entirely within both spheres, why would the volumes be different? That suggests a problem.Wait, according to mathworld, the formula is 24π. Let's see:24π≈75.398, which is higher than our previous sum. Wait, no, 24π≈75.398? Wait, no, 24*3.1416≈75.398. But in our calculation, we had 68π/3≈22.666*3.1416≈71.2. But mathworld says 24π≈75.398. So different answer.Wait, but plugging into mathworld's formula:V = [π (4 + r1 + r2 - d)(d + r1 - r2)(d + r2 - r1)(d + r1 + r2)] / (12d)So with r1=5, r2=3, d=4:4 +5+3 -4=84 +5 -3=64 +3 -5=24 +5+3=12Multiply:8*6*2*12=1152Divide by 12*4=48:1152/48=24Thus, V=24π.Alternatively, another formula from Wikipedia:The volume of intersection of two spheres is:( V = frac{pi (r_1 + r_2 - d)^2 [d^2 + 2d(r_1 + r_2) - 3(r_1^2 + r_2^2)]}{12d} )Wait, let's plug in:(r1 + r2 -d)^2=(5+3-4)^2=4²=16[d² +2d(r1 +r2) -3(r1² +r2²)]= [16 +8*8 -3*(25 +9)]=16 +64 -3*34=80 -102= -22Thus V= π*16*(-22)/(12*4)= π*(-352)/48= -22π/3. Which is negative, which can't be. So this formula must be incorrect or I misapplied it.Therefore, probably the formula from Wolfram is correct, giving 24π.So where did the earlier approach go wrong?The key mistake was assuming that the intersection volume is the sum of the two caps. Instead, the formula from Wolfram indicates it's 24π. To reconcile this, let's see:If the intersection volume is 24π, how does that relate to the caps?Alternatively, maybe the formula using the heights is also valid but requires a different approach. Let me check the derivation.The volume of the intersection can be calculated as the sum of the two spherical caps. However, each cap is a portion of one sphere, but they together form the lens. But according to the formula, the answer is 24π, so perhaps the caps are calculated differently.Wait, perhaps I made a mistake in the direction of the caps. For Sphere A, the cap is the part beyond the plane x=4 (towards Sphere B), and for Sphere B, the cap is the part beyond the plane x=4 (towards Sphere A). Wait, but Sphere B is centered at x=4, so beyond x=4 is outside of Sphere A. Wait, no.Wait, the intersection of the two spheres is the set of points that are in both. So for Sphere A, the part inside Sphere B is the region of Sphere A that is on the side of the plane x=4 towards Sphere B (i.e., x >=4). For Sphere B, the part inside Sphere A is the region of Sphere B that is on the side of the plane x=4 towards Sphere A (i.e., x <=4). Therefore, these are two different caps, but they form the same lens-shaped intersection. But since the lens is the overlap, its volume is the same as each of these caps. However, according to the calculations, these caps have different volumes, which is a contradiction. Therefore, there must be a misunderstanding in how h1 and h2 are defined.Wait, let me recast the problem.The intersection is a lens which is the union of two spherical caps: one from Sphere A and one from Sphere B. But the volume of the lens is the sum of these two caps. But according to mathworld, the volume is 24π. However, according to the separate cap calculations:Sphere A's cap volume:14π/3≈4.666πSphere B's cap volume:18π≈56.548πSum:≈61.214πBut this is larger than mathworld's 24π. Hence, clearly, this approach is wrong.Thus, the error is in assuming that the intersection volume is the sum of the two caps. In reality, the intersection is one single region, which is the same as each individual cap. Therefore, only one cap should be calculated, but which one?But according to mathworld, the correct volume is 24π. How can this be?Let me use the method of integration to compute the volume.The intersection volume can be calculated by integrating over the region where both spheres overlap.The spheres are:Sphere A: x² + y² + z² <=25Sphere B: (x -4)² + y² + z² <=9The intersection is all points satisfying both inequalities. Let's find the bounds for x. From Sphere B's equation:(x -4)^2 + y^2 + z^2 <=9This implies that y² + z² <=9 - (x -4)^2But also from Sphere A's equation:x² + y² + z² <=25Which implies y² + z² <=25 -x²Therefore, combining these:y² + z² <= min(9 - (x -4)^2, 25 -x²)The intersection region is thus bounded in x where both inequalities hold. Let's find the range of x where both are possible.First, find the x-values where 9 - (x -4)^2 >=0 and 25 -x² >=0.For Sphere B: 9 - (x -4)^2 >=0 => (x -4)^2 <=9 => x ∈ [1,7]For Sphere A:25 -x² >=0 => x ∈ [-5,5]So the overlap x-range is [1,5]Therefore, the intersection region is x from 1 to5.For each x in [1,5], the cross-sectional area is a circle with radius squared equal to the minimum of 9 - (x -4)^2 and 25 -x².But where is 9 - (x -4)^2 <=25 -x²?Set 9 - (x -4)^2 <=25 -x²Expand:9 - (x² -8x +16) <=25 -x²Simplify:9 -x² +8x -16 <=25 -x²=> -x² +8x -7 <=25 -x²Add x² to both sides:8x -7 <=258x <=32x <=4Therefore, for x <=4, 9 - (x -4)^2 <=25 -x²So the cross-sectional radius squared is 9 - (x -4)^2For x >=4, the radius squared is25 -x²Therefore, the intersection volume can be calculated as the integral from x=1 to x=4 of π(9 - (x -4)^2) dx plus the integral from x=4 to x=5 of π(25 -x²) dxCompute these integrals.First integral, x from1 to4:π ∫ [9 - (x -4)^2] dxLet u =x -4, then du=dx, when x=1, u=-3; x=4, u=0But expanding:9 - (x² -8x +16)=9 -x² +8x -16= -x² +8x -7Wait, but this is not necessary. Alternatively, integrate as is:∫ [9 - (x -4)^2] dx from1 to4= ∫9 dx - ∫(x -4)^2 dx from1 to4=9x|1 to4 - [ ( (x -4)^3 ) /3 ] from1 to4=9*(4 -1) - [ (0 - (-3)^3 ) /3 ]=27 - [ (0 - (-27))/3 ]=27 - [27/3]=27 -9=18Multiply by π:18πSecond integral, x from4 to5:π ∫(25 -x²) dx=π [25x -x³/3] from4 to5=π [ (25*5 -125/3) - (25*4 -64/3) ]=π [ (125 -125/3) - (100 -64/3) ]=π [ (375/3 -125/3) - (300/3 -64/3) ]=π [250/3 -236/3]=π [14/3]=14π/3Therefore, total intersection volume is18π +14π/3= (54π +14π)/3=68π/3≈22.666πBut this contradicts the Wolfram formula's result of24π. So what's going on here?Wait, the integration method gives68π/3≈22.666π, while the formula from Wolfram gives24π≈75.398/3≈24. But actually, wait, 24π is approximately75.398, which is different from22.666π≈71.2. But our integration result is68π/3≈22.666π, which is approximately71.2, whereas24π is approximately75.398. These are different results.There must be a mistake in either the integration or the Wolfram formula. Let me check the integration steps again.First integral from1 to4:∫1 to4 of [9 - (x -4)^2] dxExpand the integrand:9 - (x² -8x +16)=9 -x² +8x -16= -x² +8x -7Integrate term by term:∫(-x² +8x -7)dx from1 to4= [ -x³/3 +4x² -7x ] from1 to4At x=4:-64/3 +4*16 -7*4= -64/3 +64 -28= (-64/3) +36= (-64 +108)/3=44/3At x=1:-1/3 +4 -7= (-1/3) -3= (-10/3)Subtract:44/3 - (-10/3)=54/3=18Multiply by π:18π. Correct.Second integral from4 to5:∫(25 -x²)dx=25x -x³/3 evaluated from4 to5At x=5:125 -125/3= (375 -125)/3=250/3At x=4:100 -64/3= (300 -64)/3=236/3Subtract:250/3 -236/3=14/3Multiply by π:14π/3. Correct.Total volume:18π +14π/3=68π/3≈22.666π. So the integration is correct. Therefore, the formula from Wolfram must be wrong or I applied it incorrectly.Wait, let's check Wolfram's formula again.Formula: ( V = frac{pi (4 + r_1 + r_2 - d)(d + r_1 - r_2)(d + r_2 - r_1)(d + r_1 + r_2)}{12d} )Given r1=5, r2=3, d=4.First term:4 + r1 + r2 -d=4+5+3-4=8Second term:d + r1 -r2=4+5-3=6Third term:d + r2 -r1=4+3-5=2Fourth term:d + r1 + r2=4+5+3=12Multiply all terms:8*6*2*12=1152Divide by12d=12*4=481152/48=24Thus V=24π. But integration gives68π/3≈22.666π. Therefore, there is a discrepancy.This suggests that either the Wolfram formula is incorrect, or my integration is wrong. However, the integration step-by-step seems correct. Let me verify with another source.According to the formula for the volume of intersection of two spheres on Wikipedia: yes, there's a formula here:The volume of the intersection of two spheres of radii ( r ) and ( R ) with centers separated by distance ( d ) is:( V = frac{pi (r + R - d)^2 [d^2 + 2d(r + R) - 3(r - R)^2]}{12d} )Let's apply this formula with r=5, R=3, d=4.First, compute ( r + R -d =5 +3 -4=4 )Next, compute ( d^2 +2d(r + R) -3(r - R)^2=16 +8*8 -3*(2)^2=16 +64 -12=68Then, ( V= frac{pi *4^2 *68}{12*4}= frac{pi*16*68}{48}= frac{1088π}{48}= frac{68π}{3}≈22.666π )This matches the result from the integration method. So the correct formula is indeed Wikipedia's formula, giving68π/3, and Wolfram's formula must have a different parametrization or a mistake.Therefore, the correct volume is68π/3, which aligns with the integration result. Therefore, my initial calculation was correct, and Wolfram's formula might have a typo or different notation.Therefore, the answer is68π/3.But now, to reconcile with the spherical cap approach:The volume contributed by Sphere A's cap is14π/3, and Sphere B's cap is18π. But why does the integration method give the sum of these two? But according to the integration, the volume is the sum of the two caps. However, the spherical cap volumes are from each sphere's perspective, so the intersection is indeed the sum of these two caps. But according to the spherical cap formula, if the intersection is the sum of the two caps, then it's correct. However, according to the formula from Wolfram, which might be incorrect, it was different, but Wikipedia's formula gives the same result as integration, which is the sum of the two caps.Therefore, the correct answer is68π/3.But let me check another example to confirm.Suppose two spheres of equal radius r, separated by distance d. The volume of intersection should be:( 2*frac{pi h^2}{3}(3r - h) ), where h =r -d/2For example, if r=5, d=4, but spheres are equal. Wait, no, for equal spheres.Wait, for two spheres of radius r, separated by distance d, the height of each cap is h =r -d/2Therefore, the volume of intersection is ( 2*frac{pi h^2}{3}(3r - h) )In our original problem, the spheres are not equal, so this formula doesn't apply, but in the case of equal spheres, integrating should give the same result as the formula.But in our case, unequal spheres, the integration gives the sum of the two different caps, which is68π/3, and Wikipedia's formula gives the same. So I think the correct answer is68π/3.Therefore, the volume of the common part of the two spheres is68π/3.Final AnswerThe volume of the common part of the two spheres is boxed{dfrac{68}{3}pi}.</think>To find the volume of the common part of two spheres with radii 5 and 3 units, and a distance of 4 units between their centers, we use the formula for the volume of intersection of two spheres. The key steps are as follows:1. Identify the heights of the spherical caps: - For the larger sphere (radius 5), the height ( h_1 ) of the cap is calculated as: [ h_1 = 5 - frac{4^2 + 5^2 - 3^2}{2 cdot 4} = 5 - frac{16 + 25 - 9}{8} = 5 - 4 = 1 ] - For the smaller sphere (radius 3), the height ( h_2 ) of the cap is calculated as: [ h_2 = 3 - frac{4^2 + 3^2 - 5^2}{2 cdot 4} = 3 - frac{16 + 9 - 25}{8} = 3 - 0 = 3 ]2. Calculate the volume of each spherical cap: - Volume of the cap from the larger sphere: [ V_1 = frac{pi h_1^2}{3} (3 cdot 5 - h_1) = frac{pi (1)^2}{3} (15 - 1) = frac{14pi}{3} ] - Volume of the cap from the smaller sphere: [ V_2 = frac{pi h_2^2}{3} (3 cdot 3 - h_2) = frac{pi (3)^2}{3} (9 - 3) = frac{9pi cdot 6}{3} = 18pi ]3. Sum the volumes of the two caps: - The total volume of the intersection is the sum of the volumes of the two caps: [ V = V_1 + V_2 = frac{14pi}{3} + 18pi = frac{14pi + 54pi}{3} = frac{68pi}{3} ]Thus, the volume of the common part of the two spheres is boxed{dfrac{68}{3}pi}.,<think>Okay, so I need to find the volume of the common part of two spheres. The radii are 5 and 3 units, and the distance between their centers is 4 units. Hmm, this sounds like a problem involving the intersection of two spheres. I remember that when two spheres intersect, the common part is called a lens-shaped volume, and to calculate it, you need to use the formula for the volume of intersection of two spheres. But wait, how exactly does that formula work?Let me recall. I think it involves calculating the volume of each spherical cap that is formed by the intersection and then adding them together. Each sphere contributes a cap to the intersection. The formula for the volume of a spherical cap is (πh²(3r - h))/3, where h is the height of the cap and r is the radius of the sphere. So, if I can find the heights of the caps on each sphere, I can compute their volumes and sum them up.First, let me visualize the problem. There are two spheres: one with radius 5 and the other with radius 3. The distance between their centers is 4 units. Since the sum of the radii is 8, which is more than the distance between centers (4), they must intersect each other. The lens-shaped common volume is formed where the two spheres overlap.To find the height of each cap, I need to determine how much of each sphere is inside the other. Let me denote the two spheres as Sphere A (radius 5) and Sphere B (radius 3). The distance between their centers is 4 units. Let's set up a coordinate system where the center of Sphere A is at the origin, and the center of Sphere B is along the positive x-axis at (4, 0, 0). This way, the line connecting the centers is the x-axis, simplifying calculations.Now, the equation of Sphere A is x² + y² + z² = 25, and the equation of Sphere B is (x - 4)² + y² + z² = 9. To find the intersection, I need to solve these two equations simultaneously. Subtract the equation of Sphere B from Sphere A:x² + y² + z² - [(x - 4)² + y² + z²] = 25 - 9 x² - (x² - 8x + 16) = 16 x² - x² + 8x - 16 = 16 8x - 16 = 16 8x = 32 x = 4.Wait, that can't be right. If x = 4, that's the center of Sphere B. Plugging back into Sphere A's equation: 4² + y² + z² = 25 => 16 + y² + z² = 25 => y² + z² = 9. So the intersection is a circle of radius 3 in the plane x = 4, which is the center of Sphere B. But Sphere B has radius 3, so its surface at x = 4 is just a single point. Hmm, this seems contradictory.Wait, maybe I made a mistake in the algebra. Let me check again.Starting over: Sphere A: x² + y² + z² = 25 Sphere B: (x - 4)² + y² + z² = 9 Subtracting Sphere B from Sphere A: x² - (x - 4)² = 16 Expanding (x - 4)²: x² - 8x + 16 So, x² - (x² - 8x + 16) = 16 Simplify: x² - x² + 8x - 16 = 16 8x - 16 = 16 8x = 32 x = 4.Hmm, same result. So the intersection is at x = 4, which is the center of Sphere B. But Sphere B's equation at x = 4 would be 0 + y² + z² = 9, which is indeed a circle with radius 3. However, Sphere A at x = 4 has y² + z² = 25 - 16 = 9, which is the same circle. So, the intersection is a circle of radius 3 lying on the plane x = 4, which is the surface of Sphere B. But Sphere A extends beyond Sphere B here. Wait, but Sphere B has radius 3, so the entire intersection circle is on the surface of Sphere B. That seems odd. Let me visualize again.If the centers are 4 units apart, Sphere A (radius 5) is larger, so it should encompass part of Sphere B. Sphere B (radius 3) is smaller, so the intersection should be a lens where both spheres overlap. But according to the equations, the intersection is the circle x = 4, y² + z² = 9, which is exactly the set of points on Sphere B's surface. But Sphere A's surface at x = 4 is the same circle. So, actually, Sphere A and Sphere B intersect exactly at the surface of Sphere B. Wait, that would mean that Sphere B is just touching the inside of Sphere A? But the distance between centers is 4, which is less than the radius of Sphere A (5), so the center of Sphere B is inside Sphere A. So Sphere B is entirely within Sphere A? Wait, if the distance between centers is 4, and Sphere B has radius 3, then the farthest point of Sphere B from the center of Sphere A is 4 + 3 = 7, which is less than 5? No, wait, Sphere A has radius 5, so the farthest any point in Sphere A can be from its center is 5. But Sphere B's center is 4 units away from Sphere A's center, and Sphere B has radius 3, so the points on Sphere B extend from 4 - 3 = 1 to 4 + 3 = 7 units from Sphere A's center. But Sphere A only goes up to 5 units. Therefore, the part of Sphere B that is beyond 5 units from Sphere A's center is outside Sphere A, but the part from 1 to 5 units is inside Sphere A. So the intersection would be the part of Sphere B that is within Sphere A, which is a spherical cap on Sphere B. But also, Sphere A's portion of the intersection is a spherical cap as well. Wait, but in this case, since Sphere B's center is inside Sphere A (distance between centers is 4, which is less than Sphere A's radius 5), then the intersection is just the portion of Sphere B that is inside Sphere A. But is that the case?Wait, no. Because Sphere A is larger, so part of Sphere A is inside Sphere B? Wait, Sphere B has radius 3, and the distance from Sphere A's center to Sphere B's center is 4. So, the distance from Sphere B's center to Sphere A's center is 4. The radius of Sphere B is 3, so the part of Sphere A that is within 4 - 3 = 1 unit from Sphere B's center is inside Sphere B. So, actually, both spheres have parts inside each other. Wait, no. Let me think again.If two spheres intersect, the intersection is the set of points that are in both spheres. If Sphere A's center is at (0,0,0) and Sphere B's center is at (4,0,0), with radii 5 and 3 respectively, then:A point (x,y,z) is in both spheres if it satisfies both x² + y² + z² ≤ 25 and (x - 4)² + y² + z² ≤ 9.So, solving these inequalities, the intersection is a lens shape. To compute the volume, we can compute the volume of the spherical caps that each sphere contributes to the intersection.To compute the heights of these caps, we can use the formula from the geometry of sphere-sphere intersection.I recall that when two spheres intersect, the height of the cap on each sphere can be calculated using the formula:h₁ = r₁ - d₁,where d₁ is the distance from the center of the first sphere to the plane of intersection, and similarly for h₂ = r₂ - d₂.But how do we find d₁ and d₂?Alternatively, there's a formula involving the distance between centers (denoted as d), radii r₁ and r₂.The formula for the volume of intersection is:V = π(h₁²(3r₁ - h₁) + h₂²(3r₂ - h₂))/3,where h₁ and h₂ are the heights of the spherical caps on each sphere.To find h₁ and h₂, we can use the relations:h₁ = r₁ - (d² + r₁² - r₂²)/(2d),h₂ = r₂ - (d² + r₂² - r₁²)/(2d).Wait, let me verify this.From the geometry of two intersecting spheres, the distance from the center of each sphere to the plane of intersection (the plane where the two spheres intersect) can be found using the formula:Let’s denote the distance between centers as d. Then, for Sphere 1 with radius r₁, the distance from its center to the plane of intersection is (d² + r₁² - r₂²)/(2d). Similarly, for Sphere 2, it's (d² + r₂² - r₁²)/(2d).Then, the height of the cap on Sphere 1 is h₁ = r₁ - that distance, and similarly for h₂.Yes, I think that's correct.So, let's compute h₁ and h₂.Given:r₁ = 5,r₂ = 3,d = 4.First, compute the distance from Sphere A's center to the plane of intersection:d₁ = (d² + r₁² - r₂²)/(2d) = (16 + 25 - 9)/(8) = (32)/8 = 4.Wait, that's 32/8 = 4. So h₁ = r₁ - d₁ = 5 - 4 = 1.Similarly, compute d₂ = (d² + r₂² - r₁²)/(2d) = (16 + 9 - 25)/8 = (0)/8 = 0.Thus, h₂ = r₂ - d₂ = 3 - 0 = 3.Wait, so the height of the cap on Sphere A (radius 5) is 1 unit, and the height of the cap on Sphere B (radius 3) is 3 units.But wait, if d₁ is 4, which is the distance from Sphere A's center to the plane of intersection. But the distance between centers is 4, so if Sphere A's center is at 0, and the plane of intersection is at 4 units from Sphere A's center, then that plane is at the center of Sphere B. So the cap on Sphere A is from x = 4 to x = 5 (since h₁ = 1), but Sphere A's radius is 5, so the cap height is 1. While the cap on Sphere B is from x = 4 to x = 4 + 3 = 7, but since the plane of intersection is at x = 4, and h₂ = 3, does that mean the entire Sphere B is a cap? Wait, but Sphere B's center is at x = 4, so if the distance from Sphere B's center to the plane of intersection is d₂ = 0, then the cap height is h₂ = 3 - 0 = 3. But since Sphere B's radius is 3, that would mean the entire Sphere B is the cap? That can't be right. Wait, no. If the plane of intersection is at the center of Sphere B, then the cap on Sphere B would be a hemisphere. Wait, no. The cap height h is the distance from the plane of intersection to the top of the cap. If the plane of intersection is at the center of Sphere B, then the cap height would be equal to the radius, so h₂ = 3, which would make the cap the entire Sphere B. But that can't be, because Sphere B is not entirely inside Sphere A. Wait, but earlier calculation suggested that the plane of intersection is at x = 4, which is Sphere B's center, so the part of Sphere B that is inside Sphere A is the hemisphere of Sphere B that is on the side towards Sphere A's center. Wait, but if the plane is at x = 4, then the part of Sphere B that is within Sphere A is the half where x ≤ 4. But Sphere B is centered at x = 4, so x ≤ 4 would be a hemisphere. However, Sphere A's equation is x² + y² + z² ≤ 25. At x = 4, the cross-section is y² + z² ≤ 9, which is exactly the same as Sphere B's cross-section. So the intersection is the set of points in both spheres, which is the hemisphere of Sphere B (since Sphere B's points with x ≤ 4 are inside Sphere A). But Sphere A's contribution is a small cap near x = 4.Wait, maybe the volume of the intersection is the sum of two spherical caps: one from Sphere A with height 1 and one from Sphere B with height 3. But how does that work?Wait, let's recast the problem. The intersection of the two spheres is a lens shape. To compute the volume, you can compute the volume of each spherical cap (the part of each sphere that is inside the other) and add them together.So, for Sphere A (radius 5), the cap that is inside Sphere B has height h₁ = 1. The volume of this cap is (πh₁²(3r₁ - h₁))/3 = (π*1²*(15 - 1))/3 = (π*14)/3 ≈ 14.666π.For Sphere B (radius 3), the cap that is inside Sphere A has height h₂ = 3. Wait, but if h₂ is 3, then the cap volume is (π*3²*(9 - 3))/3 = (π*9*6)/3 = 18π. But Sphere B's total volume is (4/3)π*3³ = 36π. If the cap is 18π, that's exactly half of Sphere B. But earlier, we saw that the intersection is a hemisphere of Sphere B. So this makes sense.Therefore, the total volume of the intersection is 14.666π + 18π = 32.666π. But 14.666 is 44/3, so 44/3 π + 18π = (44/3 + 54/3)π = 98/3 π ≈ 32.666π.But let me check the formulas again. The formula for the volume of a spherical cap is indeed (πh²(3r - h))/3. Let me verify with h₁ = 1 and r₁ = 5:Volume for cap on Sphere A: (π*1²*(15 - 1))/3 = (14π)/3.For h₂ = 3 and r₂ = 3:Volume for cap on Sphere B: (π*3²*(9 - 3))/3 = (π*9*6)/3 = (54π)/3 = 18π.So total intersection volume is 14π/3 + 18π = (14π + 54π)/3 = 68π/3 ≈ 22.666π. Wait, hold on, 14/3 + 18 = 14/3 + 54/3 = 68/3. So 68π/3. Wait, but 68 divided by 3 is approximately 22.666. But earlier, when I thought of Sphere B's hemisphere, that's 18π, and the cap on Sphere A is 14π/3, which is about 4.666π. So total is approximately 22.666π. But is that correct?Wait, but let's think again. If Sphere B is centered at x = 4 with radius 3, and the intersection is all points of Sphere B that are within Sphere A. Since Sphere A is centered at x = 0 with radius 5, points on Sphere B must satisfy x² + y² + z² ≤ 25. But Sphere B's points satisfy (x - 4)² + y² + z² ≤ 9. So substituting x from Sphere B into Sphere A's equation: let's express x in terms of Sphere B.Wait, actually, solving the two inequalities:From Sphere B: (x - 4)^2 + y^2 + z^2 <= 9 => y^2 + z^2 <= 9 - (x - 4)^2.From Sphere A: x^2 + y^2 + z^2 <= 25.Substituting y^2 + z^2 from Sphere B into Sphere A:x^2 + [9 - (x - 4)^2] <= 25.Simplify:x^2 + 9 - (x^2 - 8x + 16) <= 25x^2 + 9 - x^2 + 8x - 16 <= 258x -7 <= 258x <= 32x <= 4.So, the x-coordinate in the intersection must be less than or equal to 4. But Sphere B's center is at x=4, so the intersection region in Sphere B is all points of Sphere B with x <=4. But since Sphere B is centered at x=4 with radius 3, the set of points in Sphere B with x <=4 is a hemisphere (since it's cut through the center). Therefore, the volume contributed by Sphere B is half of its total volume, which is (4/3)π*3³ /2 = (36π)/2 = 18π. That matches the calculation above.For Sphere A, the intersection is the part of Sphere A that is also in Sphere B. So, points in Sphere A must satisfy (x - 4)^2 + y^2 + z^2 <=9. Let's find the range of x in Sphere A that satisfies this.From Sphere A: x² + y² + z² <=25.From Sphere B's condition: (x -4)^2 + y² + z² <=9.Subtracting the two equations as before gives x <=4. So in Sphere A, the intersection is the set of points with x <=4. But Sphere A is centered at x=0, so x <=4 in Sphere A is all points from x=-5 to x=4. The volume of Sphere A with x <=4 is the volume of Sphere A minus the volume of the spherical cap from x=4 to x=5.Wait, yes. The cap on Sphere A that is outside the intersection is from x=4 to x=5, which has height h = 5 -4 =1. Therefore, the volume of this cap is (πh²(3r - h))/3 = (π*1²*(15 -1))/3 =14π/3. Therefore, the volume of Sphere A that is inside the intersection is the total volume of Sphere A minus the volume of this cap: (4/3)π*5³ -14π/3 = (500π/3 -14π/3) = 486π/3 = 162π. Wait, but that can't be, because the intersection is supposed to be the common part. Wait, maybe I confused something.Wait, no. If we have Sphere A, and we are looking for the part that is inside Sphere B, then that part is a spherical cap on Sphere A. To find its volume, we can use the formula for the spherical cap. However, the height of this cap is not 1, but is determined by how much of Sphere A is inside Sphere B.Wait, let's re-examine. The distance between the centers is 4. The radius of Sphere A is 5, so the portion of Sphere A that lies within Sphere B is a spherical cap whose height is h₁ = 5 - (distance from center of Sphere A to the plane of intersection). Wait, the plane of intersection is x=4. The distance from Sphere A's center (x=0) to the plane x=4 is 4 units. So the height of the cap on Sphere A is h₁ = r₁ - d = 5 -4=1, which is the same as before. Therefore, the volume of the cap on Sphere A is 14π/3. So the part of Sphere A inside the intersection is 14π/3. But earlier, when considering Sphere B, the part inside the intersection is 18π. Therefore, the total intersection volume is 14π/3 +18π= 68π/3 ≈22.666π.Wait, but this seems to be the case, but I thought the intersection is a lens shape. However, given the positions and radii, maybe it's correct.Wait, Sphere B is centered at x=4, radius 3. The part of Sphere B that's inside Sphere A is a hemisphere (since the plane x=4 cuts through Sphere B, and the hemisphere x<=4 is inside Sphere A). The volume of that hemisphere is 18π. Sphere A is centered at x=0, radius 5. The part of Sphere A that's inside Sphere B is a small cap near x=4, with height 1, volume 14π/3. So the total intersection is the union of these two regions? Wait, no. The intersection is the set of points that are in both spheres, so it's the overlap. But in this case, since the plane of intersection is x=4, the overlap region is the set of points that are in both spheres. So for Sphere B, the overlap is the hemisphere x<=4, and for Sphere A, the overlap is the cap x>=4 (but how? Wait, no. Wait, Sphere A's points with x<=4 are part of Sphere A that are closer to the center, but Sphere B's points with x<=4 are the ones inside Sphere A. Wait, this is confusing.Wait, let's think in terms of coordinates. Take a point at x=4, which is on the plane of intersection. For Sphere A, x=4 is part of its surface (since 4² + y² + z² =16 + ...=25 when y² + z²=9). For Sphere B, x=4 is its center.A point in the intersection must satisfy both Sphere A and Sphere B's inequalities. So, in Sphere A, points are from x=0 to x=5, but to be in Sphere B, they must satisfy (x -4)^2 + y² + z² <=9. Let's solve for x in Sphere A such that (x -4)^2 + y² + z² <=9.But in Sphere A, x² + y² + z² <=25. Let me express y² + z² from Sphere A's equation: y² + z² =25 -x². Substitute into Sphere B's inequality:(x -4)^2 + (25 -x²) <=9Expand (x -4)^2: x² -8x +16So:x² -8x +16 +25 -x² <=9Simplify:(-8x +41) <=9-8x <= -32x >=4So, in Sphere A, the points that are also in Sphere B must have x >=4. But in Sphere A, x can go up to 5. So the intersection region in Sphere A is the set of points with x from 4 to 5, and in Sphere B, the set of points with x from 1 to4 (since Sphere B is centered at 4 with radius 3, so x can go from 4 -3=1 to 4 +3=7. But to be in Sphere A, x must be <=5. Wait, but the previous calculation said x >=4. Wait, maybe there's confusion here.Wait, solving (x -4)^2 + y² + z² <=9 and x² + y² + z² <=25.From substituting y² + z² from Sphere A's equation into Sphere B's equation:(x -4)^2 + (25 -x²) <=9Which simplified to x >=4. Therefore, in Sphere A, the overlapping region is x >=4, which is a spherical cap from x=4 to x=5 with height 1. Similarly, in Sphere B, the overlapping region is x <=4. But Sphere B's center is at x=4, so x <=4 in Sphere B is a hemisphere. Therefore, the total intersection is the union of these two regions? Wait, no. The intersection is the set of points that are in both spheres. So a point in the intersection must satisfy both Sphere A and Sphere B's inequalities. So such points must satisfy both x² + y² + z² <=25 and (x -4)^2 + y² + z² <=9. As we saw earlier, this implies x >=4. Therefore, the intersection is only the region x >=4, which is the spherical cap on Sphere A and the hemisphere on Sphere B. Wait, but how can it be both?Wait, actually, points in the intersection must satisfy both inequalities. So if x >=4, then in Sphere A, x ranges from 4 to5, but in Sphere B, x ranges from 4 to7. However, the intersection requires that they also are in Sphere A, so x cannot exceed5. Therefore, the intersection in terms of x is 4 <=x <=5. But in Sphere B, x ranges from1 to7. But to be in Sphere A, x must be <=5. Therefore, the intersection is x from4 to5, which is part of both spheres.But how does that relate to the previous calculations? If the intersection is x from4 to5, then this is a lens shape that is a part of both spheres. Wait, but according to the earlier solutions, the intersection is x >=4. So perhaps the overlapping region is the set of points where x >=4 and also in both spheres. But in Sphere B, points with x >=4 are from4 to7, but limited by Sphere A to x <=5. So the overlapping region is a segment from x=4 tox=5. So how does this relate to the spherical caps?Wait, maybe the confusion arises from the coordinate system. Let me consider the line connecting the centers of the two spheres (the x-axis). The intersection is symmetric around this line, so we can model this in 2D and then revolve it around the x-axis to get the 3D volume.In 2D, we have two circles: one centered at (0,0) with radius5, and another at (4,0) with radius3. The intersection points can be found by solving the equations:x² + y² =25 (x -4)² + y² =9Subtracting the second equation from the first:x² - (x -4)² =16 x² - (x² -8x +16)=16 8x -16=16 8x=32 x=4. Then y²=25 -16=9 => y=±3.So the circles intersect at (4,3) and (4,-3). So in 2D, the intersection is the lens between x=4 and the two circles. Wait, but in this case, both circles intersect at (4,3) and (4,-3). For the larger circle (radius5), at x=4, y=±3, which is inside the smaller circle (radius3) centered at (4,0). Wait, the smaller circle has radius3, so at (4,3) and (4,-3) are points on its perimeter. So in 2D, the overlapping region is the lens between the two circles. However, in this case, since the smaller circle is centered at (4,0) with radius3, its entire left half (x<=4) is inside the larger circle. But the larger circle's rightmost point is at x=5, so the overlapping region is the part of the smaller circle that is to the left of x=4 and the part of the larger circle that is to the right of x=4 up to x=5.But in reality, solving the equations, the overlapping region is the set of points that satisfy both circle equations, which are only the two points (4,3) and (4,-3). Wait, that can't be right. Wait, no. In 2D, if two circles intersect, they either intersect at two points, one point (tangent), or no points. Here, the two circles intersect at two points: (4,3) and (4,-3). Therefore, the overlapping region is the lens-shaped area between the two circles. However, since the smaller circle is centered at (4,0) with radius3, and the larger circle is centered at (0,0) with radius5, the overlapping region is the part of each circle that lies inside the other.But in this specific case, the two intersection points are on the vertical line x=4. So in 2D, the overlapping region is the area that is inside both circles. For the smaller circle (radius3), the overlapping region is the part from x=1 to x=4, y between the two arcs. For the larger circle, the overlapping region is the part from x=4 to x=5, y between the two arcs.Wait, but according to the equations, substituting x=4 into both circles gives y=±3. So the overlapping region is two separate lens shapes? Wait, no. Wait, in 2D, the overlapping region is the set of points that are inside both circles. For the smaller circle (radius3, center (4,0)), all points inside it are from x=1 to x=7. For the larger circle (radius5, center (0,0)), all points inside it are from x=-5 to x=5. Therefore, the overlapping region in terms of x is x=1 to x=5. However, when we solved the equations, we found that the circles intersect at x=4, y=±3, which suggests that between x=1 and x=4, the overlapping region is bounded by the smaller circle's upper and lower arcs, and between x=4 and x=5, it's bounded by the larger circle's upper and lower arcs.But actually, that might not be the case. Let's take a point at x=3. For the smaller circle (radius3, center (4,0)), at x=3, y² + (3-4)^2 <=9 => y² +1 <=9 => y² <=8 => |y| <=2√2 ≈2.828. For the larger circle (radius5, center (0,0)), at x=3, y² <=25 -9=16 => |y| <=4. So at x=3, the overlapping region is |y| <=2√2. Similarly, at x=4, |y| <=3 for both circles. At x=5, for the larger circle, y=0, but for the smaller circle, x=5 is outside since the smaller circle goes up to x=7. So in the overlapping region, from x=1 to x=5, the upper and lower boundaries are determined by which circle has the smaller y value at each x.Therefore, the overlapping region in 2D is a lens that from x=1 to x=4 is bounded by the smaller circle, and from x=4 to x=5 is bounded by the larger circle. When revolved around the x-axis, this forms a 3D shape that is the intersection of the two spheres.Therefore, the volume of the intersection can be calculated by integrating the area of the overlapping region as it revolves around the x-axis. Alternatively, using the formula for the volume of the intersection of two spheres, which is the sum of the two spherical caps.Earlier, we calculated the volume as 14π/3 (from Sphere A) + 18π (from Sphere B) = 68π/3. But this seems to suggest that the intersection is the union of these two caps. However, in reality, the intersection is a single lens-shaped volume, not the union of two separate caps. But perhaps the formula correctly accounts for the combined volume by summing the two caps. Let me check with the standard formula for the volume of intersection of two spheres.The formula for the volume of intersection of two spheres with radii r and R, separated by distance d is:V = π(h₁²(3r - h₁) + h₂²(3R - h₂))/3,where h₁ = [r² - R² + d²]/(2d),Wait, no, I think the heights h₁ and h₂ are calculated as:h₁ = r - (r² - R² + d²)/(2d),h₂ = R - (R² - r² + d²)/(2d).Wait, let me verify.From the article on sphere-sphere intersection, the volume of the intersection is the sum of the two spherical caps.The heights of the caps are:h₁ = r₁ - (r₁² - r₂² + d²)/(2d),h₂ = r₂ - (r₂² - r₁² + d²)/(2d).Plugging in our values:For h₁ (cap on Sphere A):h₁ = 5 - (25 - 9 + 16)/(8) =5 - (32)/8=5 -4=1,For h₂ (cap on Sphere B):h₂ = 3 - (9 -25 +16)/8=3 - (0)/8=3 -0=3.So the heights are h₁=1 and h₂=3. Therefore, the volume of the intersection is:V = [πh₁²(3r₁ - h₁) + πh₂²(3r₂ - h₂)] /3= [π(1²)(15 -1) + π(3²)(9 -3)] /3= [π(14) + π(54)] /3= (68π)/3.So the answer is 68π/3. Let me confirm this with an alternative method.Alternatively, we can compute the volume using integration. For a sphere centered at the origin with radius r, the volume from x=a to x=b is:π∫[a to b] (r² - x²) dx.For the intersection, the limits are from x=1 to x=5 (as established earlier). Wait, no. Wait, in 3D, the overlapping region in terms of x is from x=1 to x=5? Wait, no. The overlapping region is the set of points that are in both spheres. For the smaller sphere (Sphere B), x ranges from1 to7, but to be in Sphere A, x must be <=5. Therefore, x ranges from1 to5. However, the actual overlap is more nuanced. Wait, but according to the equations, solving for points in both spheres gives x >=4. Wait, previously, solving for points in both spheres gave x >=4.Wait, this is conflicting. Let's re-examine the equations.We have:Sphere A: x² + y² + z² <=25,Sphere B: (x -4)^2 + y² + z² <=9.A point (x,y,z) is in both spheres if both inequalities are satisfied.Let's solve for x in terms of y and z.From Sphere A: y² + z² <=25 -x².From Sphere B: y² + z² <=9 - (x -4)^2.Therefore, to satisfy both, we need 25 -x² >=9 - (x -4)^2.So:25 -x² >=9 - (x² -8x +16)25 -x² >=9 -x² +8x -1625 >= -7 +8x25 +7 >=8x32 >=8xx <=4.Therefore, the x-coordinate must be <=4 for a point to satisfy both inequalities. Wait, this contradicts my previous conclusion where solving the equations gave x=4. What's happening here?Wait, substituting y² + z² from Sphere A into Sphere B's equation:(x -4)^2 + (25 -x²) <=9Which simplifies to x >=4.But here, solving the inequality 25 -x² >=9 - (x -4)^2 gives x <=4.There's a contradiction here. Which one is correct?Wait, let's go through the steps again.From Sphere A: y² + z² <=25 -x².From Sphere B: y² + z² <=9 - (x -4)^2.To satisfy both, 25 -x² >=9 - (x -4)^2, otherwise, the upper bound on y² + z² from Sphere A is less than that from Sphere B, which would mean there are no solutions for y² + z².Wait, no. To satisfy both inequalities, y² + z² must be <= the minimum of 25 -x² and9 - (x -4)^2.Therefore, for there to exist y and z such that both inequalities hold, we need that 25 -x² >=0 and9 - (x -4)^2 >=0.Which gives for Sphere A: x² <=25 => -5 <=x <=5,For Sphere B: (x -4)^2 <=9 => -3 <=x -4 <=3 =>1 <=x <=7.Therefore, the x-coordinates in the intersection must satisfy both -5 <=x <=5 and1 <=x <=7. Therefore, overlapping x is1 <=x <=5.So, the intersection is the set of points (x,y,z) such that1 <=x <=5 and y² + z² <=min(25 -x²,9 - (x -4)^2).Therefore, the volume of the intersection can be calculated by integrating over x from1 to5 the area of the circle with radius squared equal to the minimum of (25 -x²) and (9 - (x -4)^2).To find where 25 -x² =9 - (x -4)^2, set them equal:25 -x² =9 - (x² -8x +16)25 -x² =9 -x² +8x -1625 = -7 +8x8x =32x=4.So, for x from1 to4, the limiting radius squared is9 - (x -4)^2,and for x from4 to5, the limiting radius squared is25 -x².Therefore, the volume is:π∫[1 to4] [9 - (x -4)^2] dx + π∫[4 to5] [25 -x²] dx.Compute each integral separately.First integral: from1 to4, 9 - (x -4)^2 dx.Let’s expand (x -4)^2 =x² -8x +16.So 9 - (x² -8x +16)= -x² +8x -7.Therefore, the integral becomes ∫[1 to4] (-x² +8x -7) dx.Integrate term by term:∫(-x²)dx = -x³/3,∫8x dx=4x²,∫-7 dx=-7x.Evaluated from1 to4:At x=4: -64/3 +4*(16) -7*4= -64/3 +64 -28= (-64/3) +36= (-64 +108)/3=44/3.At x=1: -1/3 +4 -7= -1/3 -3= (-10)/3.Subtract:44/3 - (-10/3)=54/3=18.So the first integral is π*18.Second integral: from4 to5,25 -x² dx.Integrate term by term:∫25 dx=25x,∫-x² dx= -x³/3.Evaluated from4 to5:At x=5:25*5 -125/3=125 -125/3= (375 -125)/3=250/3.At x=4:25*4 -64/3=100 -64/3= (300 -64)/3=236/3.Subtract:250/3 -236/3=14/3.So the second integral is π*(14/3).Total volume=18π +14π/3= (54π +14π)/3=68π/3≈22.666π.This matches the earlier result using the spherical cap formula. Therefore, the volume of the intersection is 68π/3.So, after going through the integration process, confirming with the spherical cap formula, and checking the steps multiple times, the final answer is 68π/3.Final AnswerThe volume of the common part of the two spheres is boxed{dfrac{68}{3}pi}.</think>To find the volume of the common part of two spheres with radii 5 and 3 units, and a distance of 4 units between their centers, we use the formula for the volume of intersection of two spheres. This involves calculating the volumes of the spherical caps formed on each sphere and summing them.1. Identify the heights of the spherical caps: - For Sphere A (radius 5), the height ( h_1 ) is calculated as: [ h_1 = 5 - frac{5^2 - 3^2 + 4^2}{2 cdot 4} = 5 - frac{25 - 9 + 16}{8} = 5 - 4 = 1 ] - For Sphere B (radius 3), the height ( h_2 ) is calculated as: [ h_2 = 3 - frac{3^2 - 5^2 + 4^2}{2 cdot 4} = 3 - frac{9 - 25 + 16}{8} = 3 - 0 = 3 ]2. Calculate the volumes of the spherical caps: - Volume of the cap on Sphere A: [ V_1 = frac{pi h_1^2 (3 cdot 5 - h_1)}{3} = frac{pi cdot 1^2 (15 - 1)}{3} = frac{14pi}{3} ] - Volume of the cap on Sphere B: [ V_2 = frac{pi h_2^2 (3 cdot 3 - h_2)}{3} = frac{pi cdot 3^2 (9 - 3)}{3} = frac{54pi}{3} = 18pi ]3. Sum the volumes of the caps: [ V = V_1 + V_2 = frac{14pi}{3} + 18pi = frac{14pi + 54pi}{3} = frac{68pi}{3} ]Thus, the volume of the common part of the two spheres is (boxed{dfrac{68}{3}pi}).]

🔑:I. Solution. Consider an arbitrary plane section passing through the centers of the two spheres. Let the centers of the two circles visible in the plane section be O_{1} and O_{2}, and the intersection points be P_{1} and P_{2}.angle P_{1} O_{1} O_{2} = 90^{circ}, since the sides of the triangle P_{1} O_{1} O_{2} form a Pythagorean triple, and the angle in question is opposite the hypotenuse of the triangle.Let the intersection point of O_{1} O_{2} and P_{1} P_{2} be M. We know that the line connecting the intersection points of two circles is perpendicular to the line segment connecting the centers, so angle P_{1} M O_{2} = 90^{circ}.From the above, it follows that M = O_{1}, i.e., O_{1} lies on the segment P_{1} P_{2}.Returning to the space: the common part of the two spheres consists of two spherical caps (which are separated by a circle in a plane perpendicular to O_{1} O_{2} and centered at O_{1}), one of which is equal to half of the smaller sphere. The volume of this spherical cap is V_{1}, and the volume of the other is V_{2}. Let the height of the spherical cap be m_{1}, with its value being: m_{1} = 5 - 4 = 1. Using the formula for the volume of a sphere and a spherical cap:begin{aligned}& V_{1} = frac{1}{2} cdot frac{4}{3} cdot 3^{3} pi = 18 pi & V_{2} = 5 cdot 1^{2} cdot pi - frac{1^{3} pi}{3} = frac{14}{3} piend{aligned}The volume of the common part of the two spheres is V_{1} + V_{2} = frac{68}{3} pi approx 71.21.II. Solution. We cut the two spheres with a plane passing through their centers. The plane section will be a circle with radius 3 and a circle with radius 5, and the distance between the centers of the two circles is 4.Place the two circles in a coordinate system such that their centers lie on the x-axis, and the larger circle touches the y-axis. Then the equations of the two circles are:begin{array}{r}(x-9)^{2} + y^{2} - 9 = 0 (x-5)^{2} + y^{2} - 25 = 0end{array}We can calculate the coordinates of the intersection points of the two circles by equating the left-hand sides of the two equations: x^{2} - 18x + 72 = x^{2} - 10x, from which x = 9 and y = pm 3.We divide the desired solid into two parts and calculate the volume using integration. One part is formed by the rotation of the smaller circle around the x-axis, and the other part is formed by the rotation of the larger circle around the x-axis. For the smaller circle, f_{1}^{2}(x) = -x^{2} + 18x - 72, and for the larger circle, f_{2}^{2}(x) = -x^{2} + 10. Therefore:begin{aligned}V_{1} & = pi int_{6}^{9} f_{1}^{2}(x) , dx = pi int_{6}^{9} left(-x^{2} + 18x - 72right) , dx = pi left[ frac{-x^{3}}{3} + frac{18x^{2}}{2} - 72x right]_{6}^{9} = & = pi left( frac{-9^{3}}{3} + frac{18 cdot 9^{2}}{2} - 72 cdot 9 - frac{-6^{3}}{3} - frac{18 cdot 6^{2}}{2} + 72 cdot 6 right) = 18 pi V_{2} & = pi int_{9}^{10} f_{2}^{2}(x) , dx = pi int_{9}^{10} left(-x^{2} + 10right) , dx = pi left[ frac{-x^{3}}{3} + frac{10x^{2}}{2} right]_{9}^{10} = & = pi left( frac{-10^{3}}{3} + frac{10 cdot 10^{2}}{2} - frac{-9^{3}}{3} - frac{10 cdot 9^{2}}{2} right) = 4 frac{2}{3} piend{aligned}The sum of the two volumes is: V_{1} + V_{2} = frac{68}{3} pi.

🧠:[<think>Okay, so I have this problem about an ellipse. The equation given is x²/16 + y²/9 = 1. They mention a point P on the ellipse, and the distance from P to one of the foci is 3. I need to find the distance from P to the other focus. The options are A: 2, B: 3, C: 4, D: 5. Let me try to work through this step by step.First, I remember that an ellipse has some standard properties. The general equation of an ellipse centered at the origin is x²/a² + y²/b² = 1, where a is the semi-major axis and b is the semi-minor axis. Depending on whether the major axis is along the x-axis or y-axis, but in this case, since 16 is under x² and 9 is under y², and 16 > 9, so the major axis is along the x-axis. That means a² = 16, so a = 4, and b² = 9, so b = 3.Now, the foci of an ellipse are located at a distance of c from the center along the major axis, where c² = a² - b². Let me calculate that. Here, a² is 16, b² is 9, so c² = 16 - 9 = 7, so c = sqrt(7). Therefore, the foci are at (±sqrt(7), 0).Now, the problem states that the distance from point P to one of the foci is 3. Let's denote the two foci as F1 and F2, located at (sqrt(7), 0) and (-sqrt(7), 0) respectively. The question is asking for the distance from P to the other focus.I recall that one of the defining properties of an ellipse is that the sum of the distances from any point on the ellipse to the two foci is constant and equal to 2a, which in this case is 2*4 = 8. So, if the distance from P to one focus is 3, then the distance to the other focus should be 8 - 3 = 5. Hmm, but 5 is option D. Let me make sure I'm not making a mistake here.Wait, let's verify this. The sum of the distances from any point on the ellipse to both foci is always 2a. Since a is 4, then 2a is 8. Therefore, if one distance is 3, the other should be 8 - 3 = 5. So, the answer would be D: 5.But hold on, maybe I should check if there's any trick here. Let me think again. The ellipse equation x²/16 + y²/9 =1. The foci are at (±sqrt(7),0). Let's pick a point P on the ellipse. Suppose P is at (4,0), which is the vertex on the right. Then, the distance to the right focus (sqrt(7),0) would be |4 - sqrt(7)|, which is approximately 4 - 2.6458 ≈ 1.354, which is less than 3. Similarly, the distance to the left focus would be 4 + sqrt(7) ≈ 6.645. So the sum is 8, which checks out. But in this case, the distance to the right focus is not 3. So, there must be a point where the distance to one focus is 3. Let me try another point.Take the top of the ellipse at (0,3). The distance to each focus would be sqrt((0 - sqrt(7))² + (3 - 0)²) = sqrt(7 + 9) = sqrt(16) = 4. So, the distance from (0,3) to either focus is 4. Therefore, the sum is 4 + 4 = 8, which is still 2a. So, in this case, both distances are 4. So, maybe the point P is somewhere else.Wait, the problem says the distance to one of the foci is 3. So, if one distance is 3, then the other should be 5, as per the property. Therefore, the answer is 5. So, D.But let me check another example. Suppose the point P is such that the distance to the right focus is 3. Then, the sum is 8, so distance to left focus is 5. Similarly, if the distance to left focus is 3, then distance to right is 5. So, regardless, it's 5. Therefore, the answer is D:5.Alternatively, maybe I can approach this algebraically. Let the foci be F1 and F2 at (c,0) and (-c,0), where c = sqrt(a² - b²) = sqrt(16 -9)=sqrt(7). Let P be (x,y) on the ellipse. Then, the distance from P to F1 is sqrt((x - c)^2 + y²) and to F2 is sqrt((x + c)^2 + y²). Given that one of these is 3, need to find the other.But we know that sqrt((x - c)^2 + y²) + sqrt((x + c)^2 + y²) = 2a = 8. So, if one of them is 3, then the other is 5. Therefore, regardless of which focus it is, the other distance is 5. So, the answer is D.Hence, the correct answer is D:5.Final Answerboxed{D}</think>Given the ellipse (frac{x^2}{16} + frac{y^2}{9} = 1), we know that the semi-major axis (a = 4) and the semi-minor axis (b = 3). The distance (c) from the center to each focus is calculated as (c = sqrt{a^2 - b^2} = sqrt{16 - 9} = sqrt{7}). Therefore, the foci are located at ((pm sqrt{7}, 0)).A key property of an ellipse is that the sum of the distances from any point on the ellipse to the two foci is constant and equal to (2a). Here, (2a = 8).Given that the distance from a point (P) on the ellipse to one of the foci is 3, we use the property that the sum of the distances to both foci is 8. Therefore, the distance to the other focus is (8 - 3 = 5).Thus, the distance from point (P) to the other focus is (boxed{D}).,<think>Okay, let's see. I need to find the distance from a point P on the ellipse to the other focus, given that the distance from P to one of the foci is 3. The ellipse equation is x²/16 + y²/9 = 1. The options are A:2, B:3, C:4, D:5. Hmm, let me recall what I know about ellipses.First, the standard equation of an ellipse is x²/a² + y²/b² = 1, where a is the semi-major axis and b is the semi-minor axis. The foci are located at (±c, 0) where c is the distance from the center to each focus, and c² = a² - b². So in this case, a² is 16, so a = 4, and b² is 9, so b = 3. Therefore, c² = 16 - 9 = 7, so c = √7. That means the foci are at (√7, 0) and (-√7, 0). Now, one of the key properties of an ellipse is that the sum of the distances from any point on the ellipse to the two foci is constant and equal to 2a. So, for this ellipse, that sum should be 2*4 = 8. The problem says that the distance from P to one of the foci is 3. Let's call the two foci F1 and F2. If the distance from P to F1 is 3, then the distance from P to F2 must be 8 - 3 = 5. Similarly, if the distance from P to F2 is 3, then the distance to F1 is 5. So regardless of which focus we're talking about, the other distance should be 5. But wait, the options given are 2, 3, 4, 5. So 5 is an option, which is D. But let me double-check to make sure I didn't make a mistake here.Wait, the problem is phrased as "the distance from P to one of the foci is 3". So the other distance would be 8 - 3 = 5. Therefore, the answer should be D:5. But let me think again. Maybe I made an error in calculating the sum?Wait, the sum is 2a. For this ellipse, a² = 16, so a = 4. Therefore, 2a = 8. So sum of distances is 8. If one distance is 3, the other is 8 - 3 = 5. That seems correct. So the answer should be D:5. But let me check if there's another way this could be approached.Alternatively, maybe the problem is referring to something else? Let me think. Suppose the point P is located at a particular position. For example, if P is at a vertex, then the distance to the nearest focus would be a - c. Let's compute a - c. a is 4, c is √7 ≈ 2.645. So a - c ≈ 4 - 2.645 ≈ 1.355. But the given distance is 3, which is bigger than that. Similarly, the distance to the farthest focus would be a + c ≈ 4 + 2.645 ≈ 6.645. But 3 is between these two values. So it's possible for a point on the ellipse to be 3 units away from a focus. Alternatively, maybe using parametric equations. Let's parametrize the ellipse. The standard parametric equations are x = a cosθ, y = b sinθ. So here, x = 4 cosθ, y = 3 sinθ. Then, the distance from P to a focus (c, 0) is sqrt[(4 cosθ - c)^2 + (3 sinθ)^2]. Given that this distance is 3. Let's set up the equation:sqrt[(4 cosθ - √7)^2 + (3 sinθ)^2] = 3Square both sides:(4 cosθ - √7)^2 + 9 sin²θ = 9Expand the first term:16 cos²θ - 8√7 cosθ + 7 + 9 sin²θ = 9Combine like terms. Note that 16 cos²θ + 9 sin²θ can be written as 9 (cos²θ + sin²θ) + 7 cos²θ = 9 + 7 cos²θ. So:9 + 7 cos²θ - 8√7 cosθ + 7 = 9Combine constants:9 + 7 = 16, so:16 + 7 cos²θ - 8√7 cosθ = 9Subtract 9:7 cos²θ - 8√7 cosθ + 7 = 0This is a quadratic in cosθ. Let me denote u = cosθ:7u² - 8√7 u + 7 = 0Solve for u:Discriminant D = (8√7)^2 - 4*7*7 = 64*7 - 196 = 448 - 196 = 252Square root of discriminant: √252 = √(36*7) = 6√7So solutions:u = [8√7 ± 6√7]/(2*7) = [ (8√7 ± 6√7) ] /14So two solutions:(8√7 + 6√7)/14 = 14√7/14 = √7and (8√7 - 6√7)/14 = 2√7 /14 = √7/7But cosθ must be between -1 and 1. Let's check √7 ≈ 2.645, which is more than 1, so that solution is invalid. The other solution is √7/7 ≈ 2.645/7 ≈ 0.377, which is valid.So cosθ = √7/7 ≈ 0.377. Then sinθ can be found via sin²θ = 1 - cos²θ = 1 - 7/49 = 42/49 = 6/7, so sinθ = ±√(6/7).Therefore, the point P is (4*(√7/7), 3*(±√(6/7))) ≈ (4*0.377, ±3*0.926) ≈ (1.508, ±2.778).Now, let's compute the distance from this point to the other focus. The other focus is at (-√7, 0). Let's compute the distance between (4*(√7/7), 3*√(6/7)) and (-√7, 0).Distance squared is [4*(√7/7) + √7]^2 + [3*√(6/7) - 0]^2First term: 4*(√7/7) + √7 = (4√7/7 + 7√7/7) = (11√7/7)So first term squared: (121*7)/49 = 121/7 ≈ 17.2857Second term squared: [9*(6/7)] = 54/7 ≈ 7.7143Total distance squared: 121/7 + 54/7 = 175/7 = 25Therefore, distance is 5. So that checks out. Therefore, the distance to the other focus is 5. So answer D.Alternatively, since I already knew the property that the sum is 2a = 8, so 8 - 3 = 5, which is D. So either way, seems like the answer is D. Therefore, I think the correct answer is D:5.Final Answerboxed{D}</think>Given the ellipse (frac{x^2}{16} + frac{y^2}{9} = 1), we know that (a^2 = 16) (so (a = 4)) and (b^2 = 9) (so (b = 3)). The distance to the foci is calculated using (c^2 = a^2 - b^2), which gives (c = sqrt{16 - 9} = sqrt{7}). The foci are located at ((pm sqrt{7}, 0)).A key property of an ellipse is that the sum of the distances from any point on the ellipse to the two foci is constant and equal to (2a). For this ellipse, this sum is (2 times 4 = 8).Given that the distance from a point (P) on the ellipse to one of the foci is 3, the distance to the other focus must be (8 - 3 = 5).Thus, the distance from (P) to the other focus is (boxed{D}).]

🔑:Since the equation of the ellipse is frac{x^{2}}{16} + frac{y^{2}}{9} = 1, we have a = 4 and b = 3. The distance between the foci is given by c = sqrt{a^2 - b^2} = sqrt{7}. Therefore, the foci are F_1(-sqrt{7}, 0) and F_2(sqrt{7}, 0).By definition of an ellipse, the sum of the distances from any point P on the ellipse to the two foci is constant and equals to 2a. In this case, it is 2 cdot 4 = 8.If the distance from point P to one focus is 3, then the distance from P to the other focus would be 8 - 3 = 5.Therefore, the answer is: boxed{D}.

🧠:[<think>Okay, let's try to figure out this problem. The question is asking for the largest possible prime value of the expression ( n^{2} - 12n + 27 ) where ( n ) is a positive integer. The options given are 91, 37, 23, 17, and 7. Hmm, let me start by understanding the problem step by step.First, I know that a prime number is a number greater than 1 that has no positive divisors other than 1 and itself. So, our goal is to find the largest prime number from the given options that can be generated by plugging in some positive integer ( n ) into the quadratic expression ( n^{2} - 12n + 27 ).Let me write down the expression again: ( n^2 - 12n + 27 ). Maybe factoring this quadratic could help simplify things or give me some insights into possible values. Let me try to factor it.Looking at ( n^2 - 12n + 27 ), I need two numbers that multiply to 27 and add up to -12. Let's see... The factors of 27 are 1 and 27, 3 and 9. Since the middle term is negative, both numbers should be negative. But 3 and 9 add up to 12. So, if I factor it, it should be ( (n - 3)(n - 9) ). Let me check: ( (n - 3)(n - 9) = n^2 - 9n - 3n + 27 = n^2 - 12n + 27 ). Yes, that works. So the expression factors into ( (n - 3)(n - 9) ).Wait a second, so the quadratic expression simplifies to the product of ( (n - 3) ) and ( (n - 9) ). For this product to be a prime number, one of the factors must be 1 or -1 because primes have only two distinct positive divisors. But since ( n ) is a positive integer, let's analyze the possible cases.Let me recall that if ( a times b ) is a prime, then one of ( a ) or ( b ) must be ±1, and the other must be ± the prime itself. However, since we are dealing with positive integers ( n ), both ( (n - 3) ) and ( (n - 9) ) are integers. But since ( n ) is positive, let's see the possible values of ( n ):Case 1: Suppose ( n - 3 = 1 ). Then ( n = 4 ). Plugging into the other factor: ( 4 - 9 = -5 ). Then the product would be ( 1 times (-5) = -5 ). But we're looking for a prime value, and primes are positive by definition, so this gives -5, which is not prime. So that's not valid.Case 2: Suppose ( n - 9 = 1 ). Then ( n = 10 ). Plugging into the other factor: ( 10 - 3 = 7 ). The product is ( 7 times 1 = 7 ), which is prime. So when ( n = 10 ), the expression equals 7. That's option (E). But maybe there are higher primes?Case 3: Maybe the other factor is -1. Let's check. If ( n - 3 = -1 ), then ( n = 2 ). Then ( 2 - 9 = -7 ). The product is ( (-1) times (-7) = 7 ), which is prime. So ( n = 2 ) also gives 7. Similarly, if ( n - 9 = -1 ), then ( n = 8 ). Then ( 8 - 3 = 5 ). The product is ( 5 times (-1) = -5 ), which again isn't prime. So in these cases, we only get 7 or -5, but since primes are positive, 7 is the prime here.But wait, so far, both ( n = 2 ) and ( n = 10 ) give 7. But 7 is an option here (E), but maybe there are other primes. The question is asking for the largest possible prime value from the options given, so 91 is the highest there. But maybe 91 can be achieved?Alternatively, maybe there's a case where one of the factors is a prime and the other is 1 or -1. But we tried that. But perhaps for other primes, maybe we can set one of the factors equal to a prime and the other factor equal to 1 or -1. But since the expression is the product of two integers, the only way it's prime is if one is ±1 and the other is ±prime. So, unless the factors themselves are primes but the product would be composite unless one is 1 or -1.Alternatively, maybe I made a mistake here. Let me think again. If the product ( (n - 3)(n - 9) ) is a prime number, then one of the factors must be ±1, and the other must be ±prime. Because a prime number can only be factored as 1 times itself or -1 times -itself. Therefore, to get a prime, we need one of the factors to be 1 or -1, and the other to be the prime or -prime.But since ( n ) is a positive integer, let's consider possible positive values of ( n ):If ( n - 3 = 1 ), then ( n = 4 ), which gives ( (1)(-5) = -5 ), not prime.If ( n - 3 = -1 ), then ( n = 2 ), which gives ( (-1)(-7) = 7 ), prime.If ( n - 9 = 1 ), then ( n = 10 ), which gives ( (7)(1) = 7 ), prime.If ( n - 9 = -1 ), then ( n = 8 ), which gives ( (5)(-1) = -5 ), not prime.So these are the only cases where the product is ±prime. So the only primes achievable here are 7 and -5 (but negative primes aren't considered). So according to this, the only prime value is 7. But wait, the answer choices include 17, 23, 37, 91, and 7. So perhaps this reasoning is incomplete?Wait, hold on. Maybe my initial approach is wrong. Because if we factor the quadratic as ( (n - 3)(n - 9) ), then unless one of the factors is 1 or -1, the product would be composite. But perhaps for some ( n ), even if neither factor is 1 or -1, the product might still be prime. Wait, no. If both factors are integers with absolute value greater than 1, then the product would have more divisors and thus be composite. So primes can only be formed when one of the factors is ±1. So in that case, the only possible primes are 7 and -5, but since primes are positive, only 7. But that contradicts the answer choices, since 7 is the smallest option. But the answer choices include 17, 23, 37, and 91. So maybe my initial factoring is wrong?Wait, let me check again. Let me compute ( n^2 -12n +27 ). Let me plug in n = 3: 9 - 36 +27 = 0. So 3 is a root. Similarly, n = 9: 81 - 108 +27 = 0. So the quadratic factors correctly as (n -3)(n -9). So that's correct. So if the expression factors as (n -3)(n -9), then indeed, for the product to be prime, one of the factors must be ±1.But perhaps the problem is considering n as a positive integer, but maybe for some n, the value of the quadratic is prime even if the factors aren't 1 or -1. Wait, but that would require that both factors are primes and their product is a prime, but the product of two primes is semi-prime (i.e., composite), unless one of them is 1. So unless one of the factors is 1, the product can't be prime. Therefore, the only way the expression is prime is when one of the factors is 1 or -1. Therefore, the only possible primes are 7 and -5, but only 7 is prime. But then why are the answer choices including larger primes?Wait a second, maybe I need to check if the quadratic can actually result in these primes for some n. Maybe the factoring approach is correct, but perhaps there's a case where even if both factors are greater than 1 in absolute value, but one of them is a prime and the other is 1, but in such a way that their product is prime.Wait, but let me think again. Let's take n=10: (10-3)(10-9)=7*1=7. That's prime. n=2: (2-3)(2-9)=(-1)(-7)=7. Prime. But if I take n=4: (4-3)(4-9)=1*(-5)=-5. Not prime. n=8: (8-3)(8-9)=5*(-1)=-5. Not prime. So the only primes are 7. But according to the answer choices, 7 is the smallest. However, the problem says "the largest possible prime value". So maybe the answer is 7. But that seems counterintuitive because 7 is the smallest option here. But maybe my reasoning is flawed.Alternatively, perhaps the expression can sometimes be prime even if neither factor is 1 or -1. Let's test some values of n and compute the expression. Let me make a table.Let's start plugging in values of n and compute ( n^2 -12n +27 ):n=1: 1 -12 +27=16. 16 is not prime.n=2: 4 -24 +27=7. Prime. (As before)n=3: 9 -36 +27=0. Not prime.n=4: 16 -48 +27= -5. Not prime.n=5: 25 -60 +27= -8. Not prime.n=6: 36 -72 +27= -9. Not prime.n=7: 49 -84 +27= -8. Not prime.n=8: 64 -96 +27= -5. Not prime.n=9: 81 -108 +27=0. Not prime.n=10: 100 -120 +27=7. Prime.n=11: 121 -132 +27=16. Not prime.n=12: 144 -144 +27=27. Not prime.n=13: 169 -156 +27=40. Not prime.n=14: 196 -168 +27=55. Not prime.n=15: 225 -180 +27=72. Not prime.n=16: 256 -192 +27=91. 91 is 7*13, so composite. Not prime.n=17: 289 -204 +27=112. Not prime.n=18: 324 -216 +27=135. Not prime.n=19: 361 -228 +27=160. Not prime.n=20: 400 -240 +27=187. 187 is 11*17, composite.n=21: 441 -252 +27=216. Not prime.n=22: 484 -264 +27=247. 247 is 13*19, composite.n=23: 529 -276 +27=280. Not prime.n=24: 576 -288 +27=315. Not prime.n=25: 625 -300 +27=352. Not prime.Hmm, so according to this, the only n values that give a prime are n=2, 10, which give 7, and then n=16 gives 91, which is composite. Wait, but 91 is 7*13. So that's not prime. So according to this, 7 is the only prime value obtained from this expression. But that's conflicting with the answer choices given, which include 17, 23, 37, 91, and 7. The problem states "the largest possible prime value", so if 7 is the only prime, then the answer is 7, which is option E. But maybe I made a mistake here.Wait, let me check n=0. Wait, n is a positive integer, so n=0 is excluded. What about n=5? Let me check again. n=5: 25 -60 +27= -8. Not prime. n=6: -9. n=7: -8. n=8: -5. n=9:0. So all negative or zero. Starting from n=10, it's 7. Then n=11:16, which is 4 squared. n=12:27. So up to n=16, which is 91. Wait, 91 is composite. So why is 91 even an option here? Maybe there's a mistake in the problem? Or maybe my approach is wrong.Wait, let me check n=16 again. 16^2 is 256. 256 -12*16 is 256 -192 = 64. 64 +27=91. Yes, 91. Which is 7*13. So composite. So 91 is not prime. So why is 91 an option? The problem might be trying to trick us. Similarly, 37 is an option. Let me check if 37 can be obtained.Is there any n where ( n^2 -12n +27 = 37 ). Let's solve the equation:n^2 -12n +27 = 37n^2 -12n -10 =0Using quadratic formula:n = [12 ± sqrt(144 + 40)] / 2= [12 ± sqrt(184)] / 2sqrt(184) is approximately 13.564, so n would be (12 +13.564)/2 ≈25.564/2≈12.782, and (12 -13.564)/2≈-1.564/2≈-0.782. Since n has to be a positive integer, 12.782 is not an integer, so no solution. Therefore, 37 is not attainable.Similarly, check for 23:Set n^2 -12n +27 =23n^2 -12n +4 =0Discriminant: 144 -16=128sqrt(128)=8*sqrt(2)≈11.313n=(12 ±11.313)/2Positive solution: (12+11.313)/2≈23.313/2≈11.656. Not integer. So 23 is not achievable.Check 17:n^2 -12n +27=17n^2 -12n +10=0Discriminant:144 -40=104sqrt(104)=~10.198n=(12±10.198)/2Positive solution≈22.198/2≈11.099. Not integer. So no solution.Check 7:n^2 -12n +27=7n^2 -12n +20=0Factor: (n-10)(n-2)=0. So n=10 or n=2. Which matches our earlier results. So 7 is attainable.So according to these calculations, the only prime number achievable is 7, which is option E. But the answer choices include 7, 17, 23, 37, 91, with 91 being the largest. But 91 is composite, so it's not prime. So the answer must be 7, but the problem is presented as if there's a larger prime. Maybe the problem is written incorrectly?Wait, maybe I made a mistake in factoring the quadratic. Let me check again.Original expression: ( n^2 -12n +27 ). Let's compute discriminant: ( (-12)^2 -4*1*27 =144 -108=36. sqrt(36)=6. So roots are [12 ±6]/2= (18)/2=9 and (6)/2=3. So yes, factors are (n-3)(n-9). So factoring is correct.Alternatively, maybe the problem is not about factoring but considering different forms. Let's try completing the square.( n^2 -12n +27 )Complete the square:Take n^2 -12n, add and subtract (12/2)^2=36.So, ( n^2 -12n +36 -9 )Which is ( (n -6)^2 -9 ). So the expression becomes ( (n -6)^2 -9 ).Hmm, so we have a square minus 9. So, primes would be when ( (n -6)^2 -9 ) is prime. That is, ( (n -6)^2 = prime +9 ). So, ( prime +9 ) must be a perfect square.So, for example, if prime +9 is a square, then prime must be a square minus 9. So primes are of the form ( k^2 -9 ), which factors as ( (k-3)(k+3) ). For this to be prime, one of the factors must be 1. So ( k -3 =1 implies k=4 implies prime= (4)^2 -9=16-9=7. Alternatively, ( k +3=1 implies k=-2 implies prime= (-2)^2 -9=4-9=-5 ). So again, same conclusion, the only prime is 7. Therefore, the answer is 7.But the problem gives 37, 23, 17, which are primes, but according to this, they can't be expressed as ( k^2 -9 ), since ( k^2 ) would have to be 46, 32, 26, which are not perfect squares.Alternatively, perhaps I made a mistake here.Wait, but in the expression ( (n -6)^2 -9 ), primes would be achieved when this expression is a prime. So, primes p such that p = (n -6)^2 -9. Therefore, (n -6)^2 = p +9. Therefore, p +9 must be a perfect square. So for p to be prime, p +9 must be a perfect square. So possible primes are when p = k^2 -9, where k is an integer greater than 3 (since p must be positive). Then, k^2 -9 must be prime. Let's check k=4: 16-9=7, prime. k=5:25-9=16, not prime. k=6:36-9=27, composite. k=7:49-9=40, composite. k=8:64-9=55, composite. k=10:100-9=91, composite. k=11:121-9=112, composite. So, only k=4 gives prime. So again, only prime possible is 7.Therefore, despite the answer choices, the only possible prime is 7, which is option E. So why are there larger primes as options? Maybe it's a trick question. Wait, but 91 is an option. Wait, 91 is composite, but maybe the problem is written incorrectly. Let me check once again by plugging in n=16. n=16, expression is 256 -192 +27=91. But 91 is 7*13, so composite. So 91 is not prime. Therefore, it's not a valid answer.Alternatively, maybe the problem is not factoring correctly? Wait, original expression is ( n^2 -12n +27 ). Let's check if when n= something else, we can get 37. Wait, earlier solving for 37 gave non-integer n. Let's check n=11: ( 121 - 132 +27=16 ). n=12: 27. n=13:40. n=14:55. n=15:72. n=16:91. n=17:112. n=18:135. n=19:160. n=20:187. 187 is 11*17. So composite. n=21:216. n=22:247. 247 is 13*19. n=23:280. n=24:315. n=25:352. n=26: 26²=676 -12*26=312 +27=676-312=364+27=391. 391 is 17*23. So composite. So none of these give primes. So seems like only n=2 and n=10 give 7, which is prime. All other n give composite numbers or non-primes.Therefore, despite the answer choices including higher primes, the only possible prime is 7. So the answer should be E)7. But wait, the options given are (A)91, (B)37, (C)23, (D)17, (E)7. So E is 7. So unless there's a mistake in my calculations, the answer is E. However, the problem is presented in a way that suggests a larger prime is possible, given the options. Maybe the problem is written incorrectly?Alternatively, maybe I need to check negative n, but n is specified as a positive integer. So n=2 and n=10 are the only positive integers giving 7. So 7 is the maximum prime value. Therefore, the correct answer is E)7.But let me check once again if maybe there's an n beyond 25 that might yield a prime. Let's try n=28:n=28: 784 -336 +27=784-336=448+27=475. 475 is 5*95=5*5*19. Composite.n=29:841 -348 +27=841-348=493+27=520. 520 is divisible by 10. Composite.n=30:900-360+27=567. 567 is divisible by 3. 5+6+7=18, which is divisible by 3.n=31:961 -372 +27=961-372=589+27=616. 616 is even, composite.n=32:1024-384+27=1024-384=640+27=667. 667: Let's check divisibility. 667 divided by 23 is 29. 23*29=667. So composite.n=33: 1089 -396 +27=1089-396=693+27=720. Composite.n=34:1156 -408 +27=1156-408=748+27=775. Ends with 5, divisible by 5.So even as n increases, the expression ( n^2 -12n +27 ) grows quadratically, and primes become less frequent. Moreover, since the expression can be written as (n-3)(n-9), unless one of those terms is 1 or -1, the product will be composite. As n increases beyond 10, both (n-3) and (n-9) become larger positive integers, so their product is definitely composite. Therefore, the only primes achievable are 7.Therefore, the answer must be E)7. But the problem includes higher primes as options. Maybe there's a typo in the problem, or maybe I missed something. But according to all calculations and logical reasoning, 7 is the only prime obtainable from that quadratic for positive integers n. Hence, the largest possible prime is 7.Final Answerboxed{E}</think>To determine the largest possible prime value of ( n^2 - 12n + 27 ) where ( n ) is a positive integer, we start by factoring the quadratic expression:[n^2 - 12n + 27 = (n - 3)(n - 9)]For the product ((n - 3)(n - 9)) to be a prime number, one of the factors must be (pm 1) because a prime number has only two distinct positive divisors: 1 and itself. We analyze the possible values of ( n ):1. Case 1: ( n - 3 = 1 ) - ( n = 4 ) - The other factor: ( 4 - 9 = -5 ) - Product: ( 1 times (-5) = -5 ) (not a prime)2. Case 2: ( n - 9 = 1 ) - ( n = 10 ) - The other factor: ( 10 - 3 = 7 ) - Product: ( 7 times 1 = 7 ) (prime)3. Case 3: ( n - 3 = -1 ) - ( n = 2 ) - The other factor: ( 2 - 9 = -7 ) - Product: ( (-1) times (-7) = 7 ) (prime)4. Case 4: ( n - 9 = -1 ) - ( n = 8 ) - The other factor: ( 8 - 3 = 5 ) - Product: ( 5 times (-1) = -5 ) (not a prime)From these cases, the only prime values obtained are 7 when ( n = 2 ) or ( n = 10 ).We also checked other values of ( n ) by solving the quadratic equation for the given options (17, 23, 37, 91) and found that none of these primes can be achieved with a positive integer ( n ). Additionally, verifying higher values of ( n ) up to 34 confirmed that the expression results in composite numbers.Thus, the largest possible prime value of ( n^2 - 12n + 27 ) is (boxed{E}).,<think>Okay, let's see. I need to find the largest possible prime value of the quadratic expression n² - 12n + 27 when n is a positive integer. The answer choices are given, so maybe I can work through this step by step.First, let me write down the expression again to make sure I have it right: n² - 12n + 27. Hmm, maybe I can factor this quadratic to see if that helps. Let me try factoring. The quadratic is in the form ax² + bx + c, so here a=1, b=-12, c=27. To factor it, I need two numbers that multiply to 27 and add up to -12. Let's see, factors of 27 are 1 and 27, 3 and 9. Adding 3 and 9 gives 12. Since the middle term is -12n, both numbers should be negative. So, yes, it factors as (n - 3)(n - 9). Let me check that: (n - 3)(n - 9) = n² -9n -3n +27 = n² -12n +27. Perfect, that's correct.So the expression factors into (n - 3)(n - 9). Now, since we want this product to be a prime number, and primes have exactly two distinct positive divisors: 1 and themselves. But here, (n - 3) and (n - 9) are two integers (since n is a positive integer), and their product is prime. That can only happen if one of the factors is 1 or -1. However, since n is a positive integer, let's see if the factors can be 1 or -1.Wait, let's think carefully. If (n - 3)(n - 9) is prime, then one of the factors must be ±1, and the other must be ±prime. But since n is a positive integer, let's check possible cases.Case 1: (n - 3) = 1. Then n = 4. Then (n - 9) = 4 - 9 = -5. Then the product is 1 * (-5) = -5. But primes are positive integers greater than 1, so -5 is not considered here. So maybe we need to take absolute values? Wait, the problem says "prime value," so maybe negative primes aren't considered. Typically primes are positive, so I think we can ignore the negative product here. So even though 1*(-5) gives -5, which is a prime in some contexts, but not in the standard sense. So perhaps that's not acceptable.Case 2: (n - 3) = -1. Then n = 2. Then (n - 9) = 2 - 9 = -7. The product is (-1)*(-7) = 7. 7 is a prime. So that works. So when n=2, we get 7, which is prime.Case 3: (n - 9) = 1. Then n = 10. Then (n - 3) = 10 - 3 = 7. So the product is 7*1 = 7, which is prime. So n=10 also gives 7.Case 4: (n - 9) = -1. Then n=8. Then (n - 3)=8-3=5. The product is 5*(-1)= -5. Again, negative, so not considered.So from these cases, we get prime values of 7 (from n=2 and n=10) and -5 and -7 (from other cases). But since we want the prime to be positive, only 7 is valid here.But wait, the answer choices include 7 as an option (E), but there are higher primes in the options like 17, 23, 37, 91. So maybe there's a different way to get a prime value where the factors are not 1 or -1. But if the product of two integers is prime, then one of them must be ±1 and the other must be ±prime, right? Because primes have only two divisors. So unless one of the factors is 1 or -1, the product would be composite.But maybe I made a mistake here. Let's check for n=1: 1 -12 +27 = 16, which is 16, not prime. n=2: 4 -24 +27=7. n=3: 9 -36 +27=0, not prime. n=4: 16 -48 +27= -5. n=5: 25 -60 +27= -8. n=6: 36 -72 +27= -9. n=7: 49 -84 +27= -8. n=8: 64 -96 +27= -5. n=9: 81 -108 +27=0. n=10: 100 -120 +27=7. So n=2 and n=10 give 7, which is prime, and n=3 and n=9 give 0. All other values between 1 and 10 give negative numbers or composites. Wait, but maybe n is larger than 10? Let's check n=11: 121 -132 +27=16. Not prime. n=12: 144 -144 +27=27. Composite. n=13: 169 -156 +27=40. Composite. n=14: 196 -168 +27=55. Composite. n=15: 225 -180 +27=72. Composite. n=16: 256 -192 +27=91. 91 is 7*13, so composite. Wait, but 91 is one of the options (A). But it's composite. So maybe they included it to trick us. Then n=17: 289 -204 +27=112. Composite. n=18: 324 -216 +27=135. Composite. Hmm, so as n increases beyond 10, the value of n² -12n +27 starts increasing again, but for n=16, it's 91, which is composite. Let's check n=5: we had n=5 gives -8. Negative. So is there a way to get a prime higher than 7? The options include 17,23,37, which are primes. Maybe I need to check if the quadratic can somehow produce those primes without factoring into 1*prime.Wait, perhaps the quadratic can take a prime value even when the factors are not 1 or -1. But that would mean that (n-3)(n-9) is prime. But if (n-3) and (n-9) are integers, then unless one is 1 or -1, their product would have more than two divisors. For example, if (n-3) is 2 and (n-9) is 3, their product is 6, which is composite. So maybe there's no way for (n-3)(n-9) to be prime unless one of the factors is ±1. Therefore, the only primes obtainable are 7 and maybe 5 if negatives are allowed, but since they are not, only 7. But according to the options, there are higher primes. So perhaps my initial approach is wrong.Alternatively, maybe I need to compute n² -12n +27 for different n and check if any of them result in primes from the options. Let's check each answer choice:(A) 91: Let's see if 91 can be achieved. n² -12n +27=91. Then n² -12n -64=0. Solve this quadratic equation. Discriminant D=144 + 256=400. sqrt(400)=20. So n=(12 ±20)/2. So n=(32)/2=16 or n=(-8)/2=-4. Since n must be positive, n=16. Then check if 16² -12*16 +27=256-192+27=91. Yes, that's correct. But 91 is 7*13, which is composite. So even though the equation equals 91 at n=16, 91 isn't prime, so that's a trap.(B)37: Let's see if 37 is achievable. So set n² -12n +27=37. Then n² -12n -10=0. Discriminant D=144 +40=184. sqrt(184)≈13.56. Not a perfect square, so n would not be integer. So no integer n gives 37? Wait, but maybe I made a mistake. Let's compute for some n:Let me check n=7: 49 -84 +27= -8. n=8: 64 -96 +27= -5. n=9:0. n=10:7. n=11:16. n=12:27. n=13:169-156+27=40. n=14:196-168+27=55. n=15:225-180+27=72. n=16:91. n=17:112. n=18:135. n=19:160. n=20:187. Wait, 187 is 11*17, composite. n=5:25-60+27=-8. Wait, maybe n=13 gives 40, which is composite. Hmm, so where does 37 come? Maybe there's a value beyond n=20? Let's check n=1:16, n=2:7, n=3:0, n=4:-5, n=5:-8, n=6:-9, n=7:-8, n=8:-5, n=9:0, n=10:7, n=11:16, n=12:27, n=13:40, n=14:55, n=15:72, n=16:91, n=17:112, n=18:135, n=19:160, n=20:187, n=21:216, n=22:247 (which is 13*19), n=23: 23² -12*23 +27. Let's compute that: 529 -276 +27=529-276=253; 253+27=280. 280 is composite. Hmm, n=24: 576 -288 +27=315. Composite. Wait, maybe n=7 gives -8. So perhaps the expression is only prime at n=2 and n=10, giving 7. But 7 is an option (E), but the answer choices have (B)37 and (C)23. So why are these options here?Wait, perhaps the question is asking for the largest possible prime value, not necessarily requiring that the factors have to be 1 or -1. Maybe my initial assumption was wrong. Let me check n=5: expression value is -8, which is not prime. n=6: -9. n=7: -8. n=8: -5. So negative numbers. But maybe if n is less than 3, n-3 is negative, so let's check n=1:1 -12 +27=16. 16 is composite. n=0: Not allowed since n is positive integer. Wait, n ranges over positive integers, so starting from 1.Wait, maybe I need to check primes in the answer choices and see if they can be represented as n² -12n +27. Let's check 37. If 37 is a possible value, then n² -12n +27=37 => n² -12n -10=0. The discriminant is 144 +40=184. Which is not a perfect square, so n would not be integer. Similarly, 23: n² -12n +27=23 => n² -12n +4=0. Discriminant 144 -16=128, which is not a perfect square. 17: n² -12n +27=17 => n² -12n +10=0. Discriminant 144 -40=104, not a perfect square. 7 we already know is possible. So the only primes achievable with integer n are 7. But the answer choices have higher primes. That seems contradictory. Wait, maybe there is a mistake in my reasoning.Wait, hold on. Maybe I didn't check all possible n values. Let me check higher n. For example, n=13: 40, composite. n=14:55, composite. n=15:72, composite. n=16:91, composite. n=17:112, composite. n=18:135, composite. n=19:160, composite. n=20:187, composite. n=21:216, composite. n=22:247, composite. n=23:280, composite. n=24:315, composite. n=25: 625 -300 +27=352, composite. n=26:676 -312 +27=391, which is 17*23. Composite. n=27:729 -324 +27=432, composite. Hmm. So maybe there is no n where the expression is prime except n=2 and n=10, which gives 7. Then why are the answer choices including higher primes?Wait, but wait a second. Let me check n=5 again. Wait, n=5 gives 25 -60 +27= -8. Not prime. n=4 gives 16 -48 +27= -5. Wait, -5 is prime in some contexts but not positive. The problem says "prime value", so maybe they consider absolute value? If so, then n=4 gives |-5|=5, which is prime, and n=8 gives |-5|=5. Similarly, n=7 gives |-8|=8, which is composite. But 5 is prime. But 5 is not one of the answer choices. The options are 7,17,23,37,91. So 5 isn't there. So perhaps the question is only considering positive primes. Then even n=4 and n=8 would result in -5, which is not positive, so 5 is excluded.Alternatively, maybe I made a mistake in factoring. Let me re-express the quadratic. Maybe completing the square would help. Let's try that.n² -12n +27. To complete the square, take half of 12, which is 6, square it to get 36. So,n² -12n +36 -9 = (n -6)² -9. So the expression can be written as (n -6)^2 -9. So that is equal to (n -6 -3)(n -6 +3) = (n -9)(n -3), which matches the factoring I did earlier. So the expression is (n -6)^2 -9. So the minimum value of this quadratic is when (n -6)^2 is minimized, which is 0 at n=6. Then the expression becomes 0 -9= -9. So the minimum value is -9. Then as n increases or decreases from 6, the value of the expression increases. So for n>6, as n increases beyond 6, (n -6)^2 grows, so the expression becomes (n -6)^2 -9. For example, at n=10, (10-6)^2 -9=16 -9=7. At n=16, (16-6)^2 -9=100 -9=91. Wait, but 91 is composite. So even though the quadratic can take prime values like 7, but when it increases, it can reach 91 which is composite. But according to the answer choices, 37 is there. Let me check if when n is such that (n -6)^2 -9=37. Then (n -6)^2=46. So n -6=±√46. But √46 is not an integer, so n is not an integer. Similarly, for 23: (n -6)^2=32. Not a square. For 17: (n -6)^2=26. Not a square. For 7: (n -6)^2=16. Then n -6=±4, so n=10 or n=2. Which gives us 7. So indeed, the only prime value achievable is 7, and that occurs at n=2 and n=10. So the answer should be 7, option (E). But the problem says "the largest possible prime value," and 7 is the only prime in the answer choices that is achievable. But the options include higher primes, which is confusing.Wait, perhaps I made a mistake in calculating the discriminant for 37. Let me double-check that. If n² -12n +27=37, then n² -12n -10=0. The discriminant is b² -4ac = (-12)^2 -4*1*(-10)=144 +40=184. 184 is not a perfect square. So there's no integer n here. Similarly, for 23, setting n² -12n +27=23 gives n² -12n +4=0. Discriminant:144 -16=128. Not a square. For 17: n² -12n +27=17 =>n² -12n +10=0. Discriminant 144 -40=104. Not a square. So none of these primes can be achieved with integer n.Therefore, the only prime achievable is 7. But the answer choices have (B) 37 and (C)23. So perhaps I made a mistake in my initial approach.Wait, wait. Maybe there's a different interpretation. The problem says "where n ranges over all positive integers." So n must be a positive integer, but perhaps the quadratic can take prime values even if the factors are both primes? For example, if (n -3) is prime and (n -9)=1, then the product is prime. Wait, but (n -9)=1 implies n=10, then (n -3)=7, which is prime. So 7*1=7, which is prime. Similarly, (n -3)=1 and (n -9)=prime. But that would require n=4, then (n -9)= -5. So the product is 1*(-5)= -5. Not prime in positive. So only when one factor is 1 or -1, but leading to positive primes only when n=2 or n=10.Alternatively, perhaps if both (n -3) and (n -9) are primes but one is negative. For example, (n -3)= -p and (n -9)= -q, so their product is pq. If both p and q are primes, then pq is composite. So that doesn't help. Alternatively, if one is 1 and the other is prime. So that only gives us 7.Alternatively, perhaps the quadratic can be prime even if the factors are not 1 or -1, but that would require that one factor is ±prime and the other is ±1. So unless that's the case, the product can't be prime. Hence, the only way the product is prime is if one factor is ±1. So in positive integers, only n=2 and n=10 give 7. Therefore, 7 is the only prime achievable. So the answer is (E)7.But the problem gives higher primes as options, so I'm confused. Let me check again n=11: expression value is 16. Not prime. n=12:27. Not prime. n=13:40. Not prime. n=14:55. Not prime. n=15:72. Not prime. n=16:91. Composite. n=17:112. Composite. n=18:135. Composite. n=19:160. Composite. n=20:187. Composite. n=21:216. Composite. n=22:247. Composite. n=23:280. Composite. n=24:315. Composite. n=25:352. Composite. n=26:391. Composite. n=27:432. Composite. n=28: 28² -12*28 +27=784 -336 +27=475. Composite. So no primes beyond 7.Therefore, the answer must be 7, option (E). But the options include (B)37, (C)23, etc. Maybe the problem is a trick question where the largest prime possible is 7, even though higher primes are listed. Alternatively, maybe I miscalculated something.Wait, let's check n=7: 49 -84 +27= -8. Negative. n=8:64 -96 +27= -5. Negative. n=9:81 -108 +27=0. Not prime. n=10:7. Prime. n=11:16. Composite. n=12:27. Composite. So yes, only 7. So the answer is (E). But why are there higher primes in the options? Maybe the test makers expect students to factor the quadratic and see that only 7 is possible, but if someone carelessly computes for n=16, gets 91 (which is composite), but 91 is an option. So perhaps they included distractors.Therefore, after thorough checking, the largest possible prime value is 7. So the answer is E.Final Answerboxed{E}</think>To find the largest possible prime value of ( n^2 - 12n + 27 ) where ( n ) is a positive integer, we start by factoring the quadratic expression:[n^2 - 12n + 27 = (n - 3)(n - 9)]For the product ((n - 3)(n - 9)) to be a prime number, one of the factors must be ( pm 1 ) and the other must be ( pm ) a prime number. We consider the cases where the factors are ( 1 ) or ( -1 ):1. Case 1: ( n - 3 = 1 ) - ( n = 4 ) - ( (n - 9) = 4 - 9 = -5 ) - Product: ( 1 times (-5) = -5 ) (not a prime)2. Case 2: ( n - 3 = -1 ) - ( n = 2 ) - ( (n - 9) = 2 - 9 = -7 ) - Product: ( -1 times (-7) = 7 ) (prime)3. Case 3: ( n - 9 = 1 ) - ( n = 10 ) - ( (n - 3) = 10 - 3 = 7 ) - Product: ( 7 times 1 = 7 ) (prime)4. Case 4: ( n - 9 = -1 ) - ( n = 8 ) - ( (n - 3) = 8 - 3 = 5 ) - Product: ( 5 times (-1) = -5 ) (not a prime)Thus, the only prime values obtained are ( 7 ) from ( n = 2 ) and ( n = 10 ).We also checked other values of ( n ) from 1 to 28 and found that the expression ( n^2 - 12n + 27 ) does not yield any prime numbers other than 7. The quadratic expression was evaluated for higher values of ( n ), but all results were composite numbers.Therefore, the largest possible prime value of ( n^2 - 12n + 27 ) is ( boxed{E} ).]

🔑:10. (E)Note that n^{2}-12 n+27=(n-9)(n-3). For this number to be a prime, either n= 10 , in which case n^{2}-12 n+27=7; or n=2, in which case n^{2}-12 n+27=7. Since a prime number cannot be factorized in other ways, we know that 7 is the only answer.

🧠:[<think>Okay, let's try to tackle this problem step by step. First, part (1) asks us to find the expressions for the arithmetic sequence (a_n) and the geometric sequence (b_n). Let me start with the arithmetic sequence.We know that in an arithmetic sequence, each term is the previous term plus a common difference (d). The general formula for the (n)-th term is (a_n = a_1 + (n-1)d). We are given that (a_2 = 6), so substituting (n=2) into the formula:(a_2 = a_1 + d = 6). So that's one equation: (a_1 + d = 6).We also have information involving the sum of the first (n) terms, (S_n). The sum of an arithmetic sequence is given by (S_n = frac{n}{2}(2a_1 + (n-1)d)) or (S_n = frac{n}{2}(a_1 + a_n)). Then, the problem mentions a geometric sequence (b_n) where each term is positive, (b_1 = 1), and two equations: (b_2 + S_4 = 33) and (b_3 = S_2). For the geometric sequence, the general term is (b_n = b_1 cdot r^{n-1}), where (r) is the common ratio. Since (b_1 = 1), this simplifies to (b_n = r^{n-1}).Let me write down the given equations:1. (b_2 + S_4 = 33)2. (b_3 = S_2)We need to express (b_2), (b_3), (S_4), and (S_2) in terms of (a_1), (d), and (r).Starting with the geometric sequence:- (b_2 = r^{2-1} = r)- (b_3 = r^{3-1} = r^2)For the arithmetic sequence sums:- (S_4 = frac{4}{2}(2a_1 + (4-1)d) = 2(2a_1 + 3d) = 4a_1 + 6d)- (S_2 = frac{2}{2}(2a_1 + (2-1)d) = 1(2a_1 + d) = 2a_1 + d)Now plug these into the equations:1. (r + (4a_1 + 6d) = 33) → (r + 4a_1 + 6d = 33)2. (r^2 = 2a_1 + d)We already have from the arithmetic sequence that (a_1 + d = 6). So now we have three equations:1. (a_1 + d = 6)2. (r + 4a_1 + 6d = 33)3. (r^2 = 2a_1 + d)Let me try to solve these equations step by step.First, from equation 1: (a_1 = 6 - d). Let's substitute (a_1) into equations 2 and 3.Substituting into equation 2:(r + 4(6 - d) + 6d = 33)Simplify:(r + 24 - 4d + 6d = 33)Combine like terms:(r + 24 + 2d = 33)Subtract 24 from both sides:(r + 2d = 9) → (r = 9 - 2d) ...(4)Now substitute (a_1 = 6 - d) into equation 3:(r^2 = 2(6 - d) + d)Simplify:(r^2 = 12 - 2d + d = 12 - d) ...(5)Now, substitute equation (4) into equation (5):((9 - 2d)^2 = 12 - d)Expand the left side:(81 - 36d + 4d^2 = 12 - d)Bring all terms to the left side:(81 - 36d + 4d^2 - 12 + d = 0)Simplify:(69 - 35d + 4d^2 = 0)Let me write it as:(4d^2 - 35d + 69 = 0)Now, solving this quadratic equation for (d). Let's use the quadratic formula:(d = [35 pm sqrt{(-35)^2 - 4 cdot 4 cdot 69}]/(2 cdot 4))Calculate discriminant:(D = 1225 - 4 cdot 4 cdot 69 = 1225 - 1104 = 121)Square root of discriminant:(sqrt{121} = 11)Therefore:(d = [35 pm 11]/8)So two possibilities:1. (d = (35 + 11)/8 = 46/8 = 23/4 = 5.75)2. (d = (35 - 11)/8 = 24/8 = 3)So (d = 3) or (d = 23/4). Let's check which one is valid.First, if (d = 3), then from equation (1):(a_1 = 6 - 3 = 3)From equation (4):(r = 9 - 2(3) = 9 - 6 = 3)Check equation (5):(r^2 = 3^2 = 9), and (12 - d = 12 - 3 = 9). So that works.Now check (d = 23/4 = 5.75):From equation (1):(a_1 = 6 - 23/4 = 24/4 - 23/4 = 1/4)From equation (4):(r = 9 - 2*(23/4) = 9 - 23/2 = (18 - 23)/2 = -5/2)But the geometric sequence terms are all positive, so the common ratio (r) must be positive. Therefore, (r = -5/2) is invalid. So we discard this solution.Therefore, the valid solution is (d = 3), (a_1 = 3), (r = 3).Thus, the arithmetic sequence (a_n) is:(a_n = a_1 + (n-1)d = 3 + (n-1)*3 = 3 + 3n - 3 = 3n). So (a_n = 3n).Wait, that simplifies nicely. Let me verify:If (a_n = 3n), then (a_1 = 3*1 = 3), (a_2 = 3*2 = 6), which matches the given (a_2 = 6). Then the common difference (d = a_2 - a_1 = 6 - 3 = 3), which matches our earlier calculation. Good.Now the geometric sequence (b_n):(b_n = r^{n-1} = 3^{n-1}). Let's check the given conditions.First term (b_1 = 3^{0} = 1), correct.(b_2 = 3^{1} = 3). Then (S_4 = 4a_1 + 6d = 4*3 + 6*3 = 12 + 18 = 30). So (b_2 + S_4 = 3 + 30 = 33), which matches.(b_3 = 3^{2} = 9). (S_2 = 2a_1 + d = 2*3 + 3 = 6 + 3 = 9). So (b_3 = S_2 = 9), which is correct. All conditions are satisfied.So part (1) answers:(a_n = 3n) and (b_n = 3^{n-1}).Now part (2): We need to find the smallest positive integer (n) such that (T_n > S_6), where (T_n) is the sum of the first (n) terms of sequence (c_n), defined by (c_n = 4b_n - a_5).First, let's compute (a_5). Since (a_n = 3n), (a_5 = 3*5 = 15).Then (c_n = 4b_n - 15). Since (b_n = 3^{n-1}), this becomes:(c_n = 4*3^{n-1} - 15).Now, (T_n) is the sum of the first (n) terms of (c_n), so:(T_n = sum_{k=1}^n c_k = sum_{k=1}^n (4*3^{k-1} - 15) = 4sum_{k=1}^n 3^{k-1} - 15sum_{k=1}^n 1).The sum (sum_{k=1}^n 3^{k-1}) is a geometric series with first term 1, ratio 3, so sum is ((3^n - 1)/(3 - 1) = (3^n - 1)/2).Therefore, (T_n = 4*(3^n -1)/2 - 15n = 2*(3^n -1) -15n = 2*3^n - 2 -15n).Simplify: (T_n = 2*3^n -15n -2).Now, we need to compute (S_6), the sum of the first 6 terms of the arithmetic sequence (a_n). Since (a_n = 3n), the sum (S_n) is ( frac{n}{2}(a_1 + a_n) = frac{n}{2}(3 + 3n) = frac{n}{2}*3(n +1) = frac{3n(n +1)}{2}).Therefore, (S_6 = frac{3*6*7}{2} = frac{126}{2} = 63).So we need to find the smallest positive integer (n) such that (T_n > 63). That is:(2*3^n -15n -2 > 63)Simplify:(2*3^n -15n -2 -63 > 0)(2*3^n -15n -65 > 0)Now, let's compute (T_n) for successive values of (n) until we find the smallest (n) where this inequality holds.Starting with (n=1):(T_1 = 2*3^1 -15*1 -65 = 6 -15 -65 = -74 < 0)n=2:(T_2 = 2*9 -30 -65 = 18 -30 -65 = -77 < 0)n=3:(T_3 = 2*27 -45 -65 = 54 -45 -65 = -56 <0)n=4:(T_4 = 2*81 -60 -65 = 162 -60 -65 = 37 >0)Wait, 162 -60 is 102, 102 -65 is 37. So 37 >0. So at n=4, T_n=37 which is greater than 0? Wait, but our inequality is (2*3^n -15n -65 >0). So yes, 37 >0.But wait, the original inequality is (T_n > 63). Wait, maybe I made a mistake here.Wait, no: Wait, the problem says "the inequality (T_n > S_6)", which is (T_n > 63). But when I simplified, I wrote:(2*3^n -15n -65 > 0). Wait, that's correct because (T_n = 2*3^n -15n -2), so (T_n > 63) implies (2*3^n -15n -2 >63), which is (2*3^n -15n -65 >0). So yes.But when I calculated for n=4:(2*81 -15*4 -65 = 162 -60 -65 = 162 -125 = 37). So 37 >0. But 37 is less than 63? Wait no, hold on. Wait, T_n = 2*3^n -15n -2. For n=4:2*81 -15*4 -2 = 162 -60 -2 = 100. Wait, but that's conflicting with previous calculation.Wait, maybe I messed up in the calculation steps. Let me redo.First, (T_n = 2*3^n -15n -2). Therefore:For n=1: 2*3 -15 -2 = 6 -15 -2 = -11.Wait, but earlier I thought it was -74. Wait, where did that come from? Wait, no, the correct formula is (T_n = 2*3^n -15n -2). So for n=1: 2*3^1 -15*1 -2 = 6 -15 -2 = -11.Similarly, n=2: 2*9 -30 -2 = 18 -30 -2 = -14.n=3: 2*27 -45 -2 =54 -45 -2=7.n=4: 2*81 -60 -2=162 -60 -2=100.Wait, so my previous calculation was wrong because I did 2*3^n -15n -65 instead of the correct T_n = 2*3^n -15n -2. So when I set (T_n >63), that's 2*3^n -15n -2 >63, so 2*3^n -15n >65. Therefore, we need to find the smallest n where 2*3^n -15n >65.So recalculating:n=1: 2*3 -15 =6-15= -9. Not >65.n=2: 18 -30= -12. No.n=3:54 -45=9. No.n=4:162 -60=102. 102 >65? Yes. So T_4=102 -2=100>63. So n=4 gives T_n=100>63.Wait, but according to the formula T_n=2*3^n -15n -2.So for n=4, 2*81=162, 15*4=60, so 162-60=102, then subtract 2: 100. So T_4=100>63. So n=4 is the answer?But let me check for n=3: T_3=2*27 -15*3 -2=54 -45 -2=7. 7>63? No.n=4:100>63. So n=4 is the first n where T_n>63. But let me check if n=4 is indeed the smallest. Wait, but according to the formula:Wait, maybe the question is about positive integers. So n must be at least 1. So the answer would be n=4.But just to be thorough, let's verify:Calculate T_n for n=1 to n=4:n=1: T_1=2*3 -15 -2=6-15-2=-11n=2: T_2=2*9 -30 -2=18-30-2=-14n=3: T_3=2*27 -45 -2=54-45-2=7n=4: T_4=2*81 -60 -2=162-60-2=100So S_6=63. So T_3=7 <63, T_4=100>63. Therefore, the smallest positive integer n is 4.But wait, the problem says "the smallest positive integer n for which the inequality T_n > S_6 holds". So the answer is 4.Wait, but before finalizing, let me make sure that my formula for T_n is correct.Original definition: c_n =4b_n -a_5. Then T_n is the sum from k=1 to n of c_k.Given that a_5=15, and b_n=3^{n-1}, so c_n=4*3^{n-1}-15.Therefore, sum T_n= sum_{k=1}^n [4*3^{k-1}-15] =4*sum_{k=1}^n3^{k-1} -15*sum_{k=1}^n1.Sum_{k=1}^n3^{k-1}= (3^n -1)/2, as a geometric series. So 4*(3^n -1)/2=2*(3^n -1). Then subtract 15n. Therefore, T_n=2*(3^n -1)-15n=2*3^n -2 -15n. Correct. Then T_n=2*3^n -15n -2.So for n=4: 2*81 -15*4 -2=162 -60 -2=100. 100>63. For n=3:2*27 -15*3 -2=54 -45 -2=7. 7<63. So yes, n=4 is the smallest positive integer. Therefore, the answer to part (2) is 4.Therefore, summarizing:1) (a_n = 3n), (b_n = 3^{n-1})2) The smallest positive integer n is 4.Final Answer(1) (a_n = boxed{3n}) and (b_n = boxed{3^{n-1}}).(2) The smallest positive integer (n) is (boxed{4}).</think>In the arithmetic sequence ({a_n}), we know (a_2 = 6), which gives the equation (a_1 + d = 6). For the geometric sequence ({b_n}), we have (b_1 = 1), (b_2 + S_4 = 33), and (b_3 = S_2). First, we express the terms and sums of the sequences:- The sum of the first 4 terms of the arithmetic sequence: (S_4 = 4a_1 + 6d)- The sum of the first 2 terms of the arithmetic sequence: (S_2 = 2a_1 + d)- The second term of the geometric sequence: (b_2 = r)- The third term of the geometric sequence: (b_3 = r^2)Using the given conditions:1. (r + 4a_1 + 6d = 33)2. (r^2 = 2a_1 + d)3. (a_1 + d = 6)Solving these equations, we substitute (a_1 = 6 - d) into the other equations. This leads to solving the quadratic equation (4d^2 - 35d + 69 = 0), which gives (d = 3) (since (d = 23/4) results in a negative common ratio for the geometric sequence). Thus, (a_1 = 3) and the common ratio (r = 3).The arithmetic sequence is (a_n = 3n) and the geometric sequence is (b_n = 3^{n-1}).For part (2), we need to find the smallest (n) such that (T_n > S_6), where (T_n) is the sum of the first (n) terms of ({c_n}) with (c_n = 4b_n - a_5). Calculating (a_5 = 15) and substituting (b_n = 3^{n-1}), we get (c_n = 4 cdot 3^{n-1} - 15).The sum (T_n) is:[ T_n = 2 cdot 3^n - 15n - 2 ]We need (T_n > 63) (since (S_6 = 63)). Checking values of (n), we find that (T_4 = 100) is the first to exceed 63. Thus, the smallest (n) is 4. Final Answer(1) (a_n = boxed{3n}) and (b_n = boxed{3^{n-1}}).(2) The smallest positive integer (n) is (boxed{4}).,<think>Okay, so I have this problem with two parts. Let me start with part (1). It involves an arithmetic sequence {a_n} and a geometric sequence {b_n}. First, let's recall what an arithmetic sequence is. In an arithmetic sequence, each term is obtained by adding a common difference to the previous term. So, the general formula for the nth term is a_n = a_1 + (n-1)d, where a_1 is the first term and d is the common difference. The sum of the first n terms, S_n, can be calculated using the formula S_n = n/2 * [2a_1 + (n-1)d] or S_n = n*(a_1 + a_n)/2. For the arithmetic sequence {a_n}, we're given that a_2 = 6. Since a_2 = a_1 + d, that means a_1 + d = 6. So, equation (1): a_1 + d = 6.Now, moving on to the geometric sequence {b_n}. In a geometric sequence, each term is obtained by multiplying the previous term by a common ratio r. The general term is b_n = b_1 * r^{n-1}. Here, it's given that b_1 = 1, so the first term is 1. Also, all terms are positive. We have two conditions given for the geometric sequence:1. b_2 + S_4 = 332. b_3 = S_2So, let's unpack these. First, let's express b_2 and b_3. Since b_1 = 1, then b_2 = b_1 * r = r, and b_3 = b_1 * r^2 = r^2.Next, S_4 is the sum of the first 4 terms of the arithmetic sequence. Using the sum formula for arithmetic sequences: S_4 = 4/2 [2a_1 + 3d] = 2*(2a_1 + 3d) = 4a_1 + 6d.Similarly, S_2 is the sum of the first 2 terms: S_2 = 2/2 [2a_1 + d] = (2a_1 + d).So, substituting these into the given conditions:1. b_2 + S_4 = r + (4a_1 + 6d) = 332. b_3 = r^2 = S_2 = 2a_1 + dSo now we have two equations:Equation (2): r + 4a_1 + 6d = 33Equation (3): r^2 = 2a_1 + dBut we also have equation (1): a_1 + d = 6.So, we have three equations here with three variables: a_1, d, and r. Let's solve them step by step.First, from equation (1): a_1 = 6 - d.Let's substitute a_1 = 6 - d into equation (3):r^2 = 2*(6 - d) + d = 12 - 2d + d = 12 - d.So, equation (3a): r^2 = 12 - d.Similarly, substitute a_1 = 6 - d into equation (2):r + 4*(6 - d) + 6d = 33Let's compute that:r + 24 - 4d + 6d = 33Simplify:r + 24 + 2d = 33Subtract 24 from both sides:r + 2d = 9So, equation (2a): r = 9 - 2d.Now, substitute equation (2a) into equation (3a):(9 - 2d)^2 = 12 - dLet's expand the left side:81 - 36d + 4d^2 = 12 - dBring all terms to the left side:81 - 36d + 4d^2 - 12 + d = 0Simplify:4d^2 - 35d + 69 = 0So, we have a quadratic equation: 4d^2 - 35d + 69 = 0.Let's solve for d using the quadratic formula.The quadratic formula says d = [35 ± sqrt(35^2 - 4*4*69)] / (2*4)Compute discriminant:35^2 = 12254*4*69 = 16*69 = 1104Discriminant: 1225 - 1104 = 121sqrt(121) = 11Therefore, d = [35 ± 11]/8So, two solutions:d = (35 + 11)/8 = 46/8 = 23/4 = 5.75d = (35 - 11)/8 = 24/8 = 3So, possible d values are 3 and 23/4. Let's check each.First, d = 3:From equation (1): a_1 = 6 - d = 6 - 3 = 3From equation (2a): r = 9 - 2d = 9 - 6 = 3From equation (3a): r^2 = 9, which should equal 12 - d = 12 - 3 = 9. That works.Now check d = 23/4:d = 5.75From equation (1): a_1 = 6 - 23/4 = 24/4 - 23/4 = 1/4 = 0.25From equation (2a): r = 9 - 2*(23/4) = 9 - 23/2 = 18/2 -23/2 = -5/2 = -2.5But the problem states that each term of the geometric sequence {b_n} is positive. However, if r = -2.5, then b_2 = r = -2.5, which is negative. That's not allowed. So, we discard this solution.Therefore, the only valid solution is d = 3, a_1 = 3, and r = 3.Therefore, the arithmetic sequence {a_n} has a_1 = 3 and common difference d = 3. So the nth term is a_n = 3 + (n-1)*3 = 3 + 3n - 3 = 3n. So, a_n = 3n.Check: a_2 = 3*2 = 6, which matches the given. Good.For the geometric sequence {b_n}, b_1 = 1, common ratio r = 3. Therefore, b_n = 1*3^{n-1} = 3^{n-1}.Check the conditions:First condition: b_2 + S_4 = 3 + S_4.Compute S_4 for arithmetic sequence: 4/2*(2*3 + 3*3) = 2*(6 + 9) = 2*15 = 30. So, 3 + 30 = 33. Correct.Second condition: b_3 = 3^2 = 9. S_2 for arithmetic sequence: 2/2*(2*3 + 3) = 1*(6 + 3) = 9. Correct.So, part (1) seems solved. a_n = 3n, b_n = 3^{n-1}.Now, moving to part (2). We need to find the smallest positive integer n such that T_n > S_6, where T_n is the sum of the first n terms of sequence {c_n}, and c_n = 4b_n - a_5.First, let's compute a_5. Since a_n = 3n, a_5 = 3*5 = 15. So, c_n = 4b_n - 15. But b_n = 3^{n-1}. Therefore, c_n = 4*3^{n-1} - 15.So, the sequence {c_n} is defined by c_n = 4*3^{n-1} - 15. Then, T_n is the sum of the first n terms of this sequence.First, let's compute S_6, which is the sum of the first 6 terms of the arithmetic sequence {a_n}. Since a_n = 3n, the sum S_6 can be calculated as:S_6 = 6/2*(a_1 + a_6) = 3*(3 + 18) = 3*21 = 63. Alternatively, since it's an arithmetic series with a_1=3, d=3, S_6 = 6/2 [2*3 + 5*3] = 3*(6 +15)=3*21=63. Either way, S_6=63.So, we need to find the smallest n where T_n > 63.Now, T_n is the sum of c_1 + c_2 + ... + c_n. Let's express this sum.First, write T_n = sum_{k=1}^n [4*3^{k-1} - 15] = 4*sum_{k=1}^n 3^{k-1} - 15*sum_{k=1}^n 1.So, this is T_n = 4*(sum of geometric series) - 15n.The sum of the geometric series sum_{k=1}^n 3^{k-1} is (3^n - 1)/(3 - 1) = (3^n -1)/2.Therefore, T_n = 4*( (3^n -1)/2 ) -15n = 2*(3^n -1) -15n = 2*3^n - 2 -15n.So, T_n = 2*3^n -15n -2.We need to find the smallest positive integer n such that T_n > 63.So, we need to solve 2*3^n -15n -2 > 63.Let's simplify:2*3^n -15n -2 > 63Add 2 to both sides:2*3^n -15n > 65Divide both sides by 1 (no change):2*3^n -15n > 65Now, let's compute T_n for n=1,2,3,... until we find the smallest n where this inequality holds.Compute step by step:For n=1:T_1 = 2*3^1 -15*1 -2 = 6 -15 -2 = -11. Not >63.n=2:T_2=2*3^2 -15*2 -2=2*9 -30 -2=18 -30 -2= -14. Still no.n=3:2*27 -45 -2=54 -45 -2=7. Still 7 <63.n=4:2*81 -60 -2=162 -60 -2=100. 100>63. So, n=4 gives T_n=100>63.Wait, but let's check between n=3 and n=4.Wait n=3 gives 7, n=4 gives 100. So, the jump is big. Let's verify that.Wait, is that correct? Let's compute T_n step by step by summing c_n.Alternatively, compute T_n as the sum of c_1 to c_n.Compute c_1 = 4*3^{0} -15=4*1 -15=4 -15=-11c_2=4*3^{1} -15=12 -15=-3c_3=4*9 -15=36 -15=21c_4=4*27 -15=108 -15=93c_5=4*81 -15=324 -15=309So, sum T_n:T_1 = -11T_2 = -11 + (-3) = -14T_3 = -14 +21=7T_4=7 +93=100T_5=100 +309=409So, T_4=100>63. Therefore, the smallest n is 4.Wait, but before confirming, let me check if perhaps there's an error in the formula for T_n.Earlier, we derived T_n =2*3^n -15n -2.Let's verify for n=4:2*3^4 -15*4 -2=2*81 -60 -2=162 -60 -2=100. Correct.Similarly, n=3:2*27 -45 -2=54 -45 -2=7. Correct.Therefore, indeed, starting from n=4, T_n=100>63= S_6. So the minimal positive integer n is 4.But let me check if maybe T_3 is 7, which is less than 63, and T_4 is 100, which is greater. So yes, n=4 is the smallest.Therefore, the answer to part (2) is 4.But let me check once again to make sure.Alternatively, perhaps solving the inequality 2*3^n -15n -2 >63.Which is 2*3^n -15n >65.We can test n=4: 2*81 -60=162-60=102>65. Yes, 102>65, so 102-2=100>63.But wait, the original inequality is 2*3^n -15n -2>63.Yes, 2*3^4 -15*4 -2=162-60-2=100>63. Correct.So, n=4 is indeed the smallest.Therefore, the answers are:(1) a_n = 3n, b_n = 3^{n-1}(2) The smallest positive integer n is 4.Final Answer(1) The general terms are boxed{a_n = 3n} and boxed{b_n = 3^{n-1}}.(2) The smallest positive integer n is boxed{4}.</think>Given the arithmetic sequence ({a_n}) with (a_2 = 6) and the geometric sequence ({b_n}) with (b_1 = 1), we need to find the general terms (a_n) and (b_n), and the smallest positive integer (n) such that (T_n > S_6). Part (1)1. Arithmetic Sequence ({a_n}): - Given (a_2 = 6), we have (a_1 + d = 6). - The sum of the first (n) terms (S_n) is given by (S_n = frac{n}{2} [2a_1 + (n-1)d]).2. Geometric Sequence ({b_n}): - Given (b_1 = 1), each term is positive. - Conditions: (b_2 + S_4 = 33) and (b_3 = S_2).Solving the equations:- From (a_1 + d = 6), we get (a_1 = 6 - d).- For the geometric sequence: (b_2 = r), (b_3 = r^2).- Using the conditions: 1. (r + 4a_1 + 6d = 33) 2. (r^2 = 2a_1 + d)Substituting (a_1 = 6 - d) into the conditions:1. (r + 4(6 - d) + 6d = 33 implies r + 24 + 2d = 33 implies r = 9 - 2d)2. (r^2 = 2(6 - d) + d implies r^2 = 12 - d)Substituting (r = 9 - 2d) into (r^2 = 12 - d):[(9 - 2d)^2 = 12 - d implies 81 - 36d + 4d^2 = 12 - d implies 4d^2 - 35d + 69 = 0]Solving the quadratic equation, we get (d = 3) (since (d = 23/4) leads to a negative common ratio (r) which is invalid).Thus, (a_1 = 3), (d = 3), and (r = 3). Therefore:- (a_n = 3 + (n-1) cdot 3 = 3n)- (b_n = 3^{n-1}) Part (2)Given (c_n = 4b_n - a_5) and (T_n) is the sum of the first (n) terms of ({c_n}):- (a_5 = 15), so (c_n = 4 cdot 3^{n-1} - 15)- The sum (T_n = sum_{k=1}^n (4 cdot 3^{k-1} - 15) = 4 cdot frac{3^n - 1}{3 - 1} - 15n = 2 cdot 3^n - 2 - 15n)We need (T_n > S_6) where (S_6 = 63):[2 cdot 3^n - 15n - 2 > 63 implies 2 cdot 3^n - 15n > 65]Checking values of (n):- (n = 4): (2 cdot 81 - 60 - 2 = 100 > 63)Thus, the smallest positive integer (n) is 4. Final Answers(1) The general terms are (boxed{a_n = 3n}) and (boxed{b_n = 3^{n-1}}).(2) The smallest positive integer (n) is (boxed{4}).]

🔑:(1) Let d be the common difference of the arithmetic sequence {a_n} and let q > 0 be the common ratio of the geometric sequence {b_n}.Given a_2 = 6, b_1 = 1, and also b_2 + S_4 = 33, b_3 = S_2, we have:a_1+ d = 6q + 6a_1+ 6d = 33q^2 = 2a_1 + dSolving these equations concurrently, we find q=3, a_1=3, and d=3. Thus, the nth-term of the arithmetic sequence is:a_n = 3 + 3(n-1) = 3nThe nth-term of the geometric sequence is:b_n = 3^{(n-1)}(2) Since c_n = 4b_n - a_5, we have:c_n = 4 cdot 3^{(n-1)} - 15The sum of the first n terms of the sequence {c_n} is T_n:T_n = 4 times frac{3^n - 1}{3 - 1} - 15n = 2 cdot 3^n - 2 - 15nThe sum of the first 6 terms of {a_n}, S_6, is:S_6 = 6 cdot 3 + frac{6 cdot 5}{2} cdot 3 = 63To solve the inequality T_n > S_6, we express it as:2 cdot 3^n - 2 - 15n > 63Let's define the function f(n) = 2 cdot 3^n - 15n - 65.Since:f(n+1) - f(n) = 2 cdot 3^{(n+1)} - 15(n+1) - 65 - (2 cdot 3^n - 15n - 65) = 4 cdot 3^n - 15For n geq 2, we have that f(n+1) > f(n), which means the function f(n) is monotonically increasing. With:f(1) = -9 65we find that the minimum value of n for which f(n) = 2 cdot 3^n - 15n - 65 > 0 is 4.Thus, the smallest positive integer n for which the inequality T_n > S_6 holds is boxed{4}.