🧠:[<think>Okay, so I have this problem about a triangle ABC where each vertex is on the unit circle. The sides opposite angles A, B, and C are a, b, and c respectively. The given condition is 2a cos A = c cos B + b cos C. I need to find two things: (1) the value of cos A, and (2) the area of the triangle if b² + c² = 4.First, let me digest the problem. The triangle is inscribed in the unit circle, which probably means the circumradius R is 1 because the unit circle is the circumcircle. In the Law of Sines, we know that a/(sin A) = 2R, so since R = 1, a = 2 sin A, similarly b = 2 sin B, c = 2 sin C. That seems useful.Given that, let me note that in any triangle, the sides can be related to the angles through the Law of Sines. Also, maybe using the Law of Cosines could be helpful here. The problem gives an equation involving a, cos A, c, cos B, b, cos C. So maybe I can express all variables in terms of angles or sides and then solve for cos A.Let me write down the given equation: 2a cos A = c cos B + b cos C.Since the triangle is on the unit circle, as I thought, R = 1, so sides are related to angles via a = 2 sin A, b = 2 sin B, c = 2 sin C. Let me substitute these into the equation.So replacing a, b, c with 2 sin A, 2 sin B, 2 sin C:2*(2 sin A)*cos A = (2 sin C)*cos B + (2 sin B)*cos CSimplify both sides:Left side: 4 sin A cos ARight side: 2 sin C cos B + 2 sin B cos CFactor out 2 on the right: 2(sin C cos B + sin B cos C)But wait, sin C cos B + sin B cos C is equal to sin(B + C). Because sin(x + y) = sin x cos y + cos x sin y. So sin(B + C) = sin(π - A) because in a triangle, angles sum to π. So sin(π - A) = sin A. Therefore, the right side becomes 2 sin A.So equation becomes:4 sin A cos A = 2 sin AAssuming sin A ≠ 0 (since A is an angle of a triangle, it can't be 0 or π), we can divide both sides by 2 sin A:2 cos A = 1Therefore, cos A = 1/2. So that's part (1) done. The value of cos A is 1/2.Wait, let me verify that step again. So starting from:Left side: 4 sin A cos ARight side: 2(sin C cos B + sin B cos C) = 2 sin(B + C) = 2 sin(π - A) = 2 sin AHence equation is 4 sin A cos A = 2 sin ADivide both sides by 2 sin A (assuming sin A ≠ 0, which is valid in a triangle):2 cos A = 1 ⇒ cos A = 1/2. Yep, that seems correct. So part (1) is cos A = 1/2. So angle A must be π/3 (60 degrees) since cos π/3 = 1/2.Okay, part (1) seems straightforward once using the sine addition formula and the Law of Sines for circumradius.Now part (2): If b² + c² = 4, find the area of triangle ABC.We need to find the area given that b² + c² = 4. Since we know cos A = 1/2, so angle A is π/3. Let's recall that in a triangle, the area can be calculated by (1/2)ab sin C, but maybe using other formulae.Alternatively, since the triangle is inscribed in the unit circle, the sides are related to the angles by a = 2 sin A, b = 2 sin B, c = 2 sin C. So maybe we can express b and c in terms of angles B and C.Given that angle A is π/3, so angles B + C = π - π/3 = 2π/3.Also, since a = 2 sin A = 2 sin(π/3) = 2*(√3/2) = √3.Given that, but we have the condition b² + c² = 4. Let's write b and c in terms of angles:b = 2 sin B, c = 2 sin C. Then, b² + c² = 4 sin² B + 4 sin² C = 4(sin² B + sin² C) = 4. Therefore, sin² B + sin² C = 1.But angle B + angle C = 2π/3. Let me set angle B = x, then angle C = 2π/3 - x. So sin² x + sin²(2π/3 - x) = 1.Let me compute sin² x + sin²(2π/3 - x). Maybe use trigonometric identities.First, sin(2π/3 - x) can be expanded as sin(2π/3)cos x - cos(2π/3)sin x. sin(2π/3) = √3/2, cos(2π/3) = -1/2. So sin(2π/3 - x) = (√3/2)cos x - (-1/2)sin x = (√3/2)cos x + (1/2)sin x.But maybe squaring that is messy. Alternatively, use the identity sin² θ = (1 - cos 2θ)/2.So sin² x = (1 - cos 2x)/2, sin²(2π/3 - x) = [1 - cos(4π/3 - 2x)]/2.Therefore, sum is [1 - cos 2x + 1 - cos(4π/3 - 2x)]/2 = [2 - cos 2x - cos(4π/3 - 2x)]/2.Simplify cos(4π/3 - 2x). 4π/3 is π + π/3, so cos(4π/3 - 2x) = cos(π + π/3 - 2x) = -cos(π/3 - 2x) because cos(π + θ) = -cos θ.Wait, actually, cos(4π/3 - 2x) can be written as cos(π + π/3 - 2x) = cos(π + (π/3 - 2x)) = -cos(π/3 - 2x) because cos(π + α) = -cos α.Alternatively, let's compute cos(4π/3 - 2x) = cos(4π/3)cos 2x + sin(4π/3)sin 2x. Cos 4π/3 = -1/2, sin 4π/3 = -√3/2. So:cos(4π/3 - 2x) = (-1/2)cos 2x + (-√3/2) sin 2x.Therefore, sin² x + sin²(2π/3 - x) = [2 - cos 2x - (-1/2 cos 2x - √3/2 sin 2x)]/2 = [2 - cos 2x + (1/2 cos 2x + √3/2 sin 2x)]/2 = [2 - (1/2 cos 2x) + (√3/2 sin 2x)]/2.Simplify numerator:2 - (1/2)cos 2x + (√3/2) sin 2x. Let's write this as 2 + [ - (1/2 cos 2x) + (√3/2 sin 2x) ].Note that - (1/2 cos 2x) + (√3/2 sin 2x) can be written as sin(2x - π/6). Because sin(a - b) = sin a cos b - cos a sin b. Let's see:sin(2x - π/6) = sin 2x cos π/6 - cos 2x sin π/6 = sin 2x*(√3/2) - cos 2x*(1/2). Which is (√3/2 sin 2x - 1/2 cos 2x), which is the same as above. So:sin² x + sin²(2π/3 - x) = [2 + sin(2x - π/6)] / 2.But we know this sum is equal to 1, so:[2 + sin(2x - π/6)] / 2 = 1 ⇒ 2 + sin(2x - π/6) = 2 ⇒ sin(2x - π/6) = 0.Thus, sin(2x - π/6) = 0 ⇒ 2x - π/6 = kπ ⇒ 2x = π/6 + kπ ⇒ x = π/12 + kπ/2.But x is angle B of the triangle, which must be between 0 and π, and since angles B and C add up to 2π/3, x must be between 0 and 2π/3. So let's find k such that x is in that interval.For k = 0: x = π/12 ≈ 15°, which is valid.For k = 1: x = π/12 + π/2 = 7π/12 ≈ 105°, but 105° + angle C = 105° + (2π/3 - x) = 105° + 120° - 105° = 120°, which is okay, but 7π/12 is about 105°, which is less than 2π/3 ≈ 120°, so it's still valid.Wait, but if x = π/12, then angle C = 2π/3 - π/12 = 8π/12 - π/12 = 7π/12. Similarly, if x = 7π/12, then angle C = 2π/3 - 7π/12 = 8π/12 - 7π/12 = π/12. So there are two possibilities: angle B is π/12 and angle C is 7π/12, or vice versa. So both solutions are valid.Therefore, the solutions are x = π/12 or x = 7π/12. Hence, angles B and C are π/12 and 7π/12 in some order.Therefore, the sides b and c would be 2 sin B and 2 sin C. Let's compute them:If angle B = π/12, then b = 2 sin(π/12) and c = 2 sin(7π/12). Similarly, if angle B = 7π/12, then b = 2 sin(7π/12) and c = 2 sin(π/12). Since sin(7π/12) = sin(π - 5π/12) = sin(5π/12) = same as sin(π/12 + π/3). Wait, sin(7π/12) is sin(105°) which is sin(60° + 45°) = sin 60° cos 45° + cos 60° sin 45° = (√3/2)(√2/2) + (1/2)(√2/2) = (√6/4 + √2/4) = (√6 + √2)/4. Similarly, sin(π/12) is sin(15°) = sin(45° - 30°) = sin 45° cos 30° - cos 45° sin 30° = (√2/2)(√3/2) - (√2/2)(1/2) = (√6/4 - √2/4) = (√6 - √2)/4.Therefore, sin(π/12) = (√6 - √2)/4, sin(7π/12) = (√6 + √2)/4. Therefore, b and c can be either 2*(√6 - √2)/4 = (√6 - √2)/2 and 2*(√6 + √2)/4 = (√6 + √2)/2, or vice versa. So regardless, the product bc would be [(√6 - √2)/2] * [(√6 + √2)/2] = [(6 - 2)] / 4 = 4/4 = 1. So bc = 1.Wait, but how does that relate to the area? Let me recall that area of triangle ABC can be calculated using (1/2)ab sin C, but maybe with the given data, there's a better formula.Alternatively, since the triangle is inscribed in the unit circle, the area can also be calculated using the formula (a b c)/(4R), where R is the circumradius. Since R = 1, the area would be (a b c)/4. But we know a = √3, and bc = 1, so area would be (√3 * 1)/4 = √3/4. Wait, but let me check if bc = 1.Earlier, we had that when angles B and C are π/12 and 7π/12, then b and c are (√6 - √2)/2 and (√6 + √2)/2. Multiplying these gives [(√6)^2 - (√2)^2]/4 = (6 - 2)/4 = 4/4 = 1. So yes, bc = 1. Therefore, area = (a b c)/4 = (√3 * 1)/4 = √3/4.Alternatively, since we know angle A is π/3, and sides b and c, area can also be calculated using (1/2) b c sin A. Since sin A = sin(π/3) = √3/2, so area = (1/2) * b * c * (√3/2) = (√3/4) * b c. But bc =1, so area = √3/4. Same result. So that's the area.But wait, the problem states that b² + c² = 4. Let me verify if in this case, with b = (√6 - √2)/2 and c = (√6 + √2)/2, does b² + c² equal 4?Compute b² + c²:[(√6 - √2)/2]^2 + [(√6 + √2)/2]^2= [ (6 - 2√12 + 2)/4 + (6 + 2√12 + 2)/4 ]= [ (8 - 4√3)/4 + (8 + 4√3)/4 ]= (8 - 4√3 + 8 + 4√3)/4= 16/4 = 4.Yes, that works. Perfect. So that checks out. Therefore, regardless of the values of angles B and C (whether B is π/12 or 7π/12), bc =1, so the area is √3/4.Hence, the answer to part (2) is √3/4.Wait, let me just recap to make sure I didn't miss any steps. We found that angle A is π/3, so a = √3. Then, using the given b² + c² = 4, we found that angles B and C must be π/12 and 7π/12, leading to bc =1. Then, using the area formula (1/2)bc sin A, which becomes (1/2)(1)(√3/2) = √3/4. Alternatively, via (abc)/4R, which is (√3 *1)/4 = √3/4. So both ways, the area is √3/4.Therefore, the answers are (1) 1/2, and (2) √3/4.Final Answer(1) boxed{dfrac{1}{2}} (2) boxed{dfrac{sqrt{3}}{4}}</think>Given a triangle (ABC) inscribed in the unit circle, with sides opposite angles (A), (B), and (C) being (a), (b), and (c) respectively, and the condition (2a cos A = c cos B + b cos C):1. Value of (cos A): - Using the Law of Sines for a triangle inscribed in the unit circle (circumradius (R = 1)), we have (a = 2 sin A), (b = 2 sin B), and (c = 2 sin C). - Substituting these into the given equation: [ 2(2 sin A) cos A = 2 sin C cos B + 2 sin B cos C ] - Simplifying both sides: [ 4 sin A cos A = 2(sin C cos B + sin B cos C) ] - Using the sine addition formula (sin(B + C) = sin(pi - A) = sin A): [ 4 sin A cos A = 2 sin A ] - Dividing both sides by (2 sin A) (assuming (sin A neq 0)): [ 2 cos A = 1 implies cos A = frac{1}{2} ] Thus, the value of (cos A) is (boxed{dfrac{1}{2}}).2. Area of (triangle ABC) given (b^2 + c^2 = 4): - Since (cos A = frac{1}{2}), angle (A = frac{pi}{3}). - Using the Law of Sines, (a = 2 sin frac{pi}{3} = sqrt{3}). - Given (b^2 + c^2 = 4), we express (b) and (c) in terms of angles (B) and (C): [ b = 2 sin B, quad c = 2 sin C ] - Substituting into the given condition: [ (2 sin B)^2 + (2 sin C)^2 = 4 implies 4(sin^2 B + sin^2 C) = 4 implies sin^2 B + sin^2 C = 1 ] - Using angles (B + C = frac{2pi}{3}) and solving, we find (B = frac{pi}{12}) and (C = frac{7pi}{12}) (or vice versa). - Calculating (b) and (c): [ b = 2 sin frac{pi}{12} = frac{sqrt{6} - sqrt{2}}{2}, quad c = 2 sin frac{7pi}{12} = frac{sqrt{6} + sqrt{2}}{2} ] - Product (bc = 1). - Area using (frac{1}{2} bc sin A): [ text{Area} = frac{1}{2} cdot 1 cdot sin frac{pi}{3} = frac{1}{2} cdot frac{sqrt{3}}{2} = frac{sqrt{3}}{4} ] Thus, the area of (triangle ABC) is (boxed{dfrac{sqrt{3}}{4}}).,<think>Okay, so I have this geometry problem here about triangle ABC. The triangle is inscribed in the unit circle, which means all its vertices lie on the unit circle. The sides opposite angles A, B, and C are a, b, and c respectively. The problem gives an equation: 2a cos A = c cos B + b cos C. Then there are two parts: (1) find cos A, and (2) if b² + c² = 4, find the area of triangle ABC.First, let me focus on part (1). Since the triangle is inscribed in the unit circle, I remember that in such a triangle, the sides are related to the angles via the formula a = 2 sin A, b = 2 sin B, and c = 2 sin C. Wait, is that right? Because in the unit circle, the radius is 1, but the formula for the side length in terms of the central angle is 2R sin θ, where R is the radius. Since R = 1 here, that would make the sides a = 2 sin A, etc. So yes, that seems correct.So, if a = 2 sin A, then maybe I can express everything in terms of sine and cosine of the angles. The given equation is 2a cos A = c cos B + b cos C. Let me substitute a, b, c with 2 sin A, 2 sin B, 2 sin C respectively. So:2*(2 sin A) cos A = (2 sin C) cos B + (2 sin B) cos CSimplify both sides:Left side: 4 sin A cos ARight side: 2 sin C cos B + 2 sin B cos C = 2(sin C cos B + sin B cos C)Hmm, the expression sin C cos B + sin B cos C is familiar. That's the sine addition formula: sin(B + C) = sin B cos C + cos B sin C. So, sin C cos B + sin B cos C = sin(B + C). But in a triangle, angles add up to π, so B + C = π - A. Therefore, sin(B + C) = sin(π - A) = sin A. So the right side becomes 2 sin A.So now, we have:Left side: 4 sin A cos ARight side: 2 sin ASet them equal:4 sin A cos A = 2 sin ADivide both sides by 2 sin A (assuming sin A ≠ 0, which it can't be in a triangle because angles are between 0 and π):2 cos A = 1Therefore, cos A = 1/2.So cos A is 1/2. That would mean angle A is π/3 (60 degrees) since cosine of π/3 is 0.5. So part (1) is done, I think. The answer is 1/2.Now part (2): If b² + c² = 4, find the area of triangle ABC.We already know from part (1) that cos A = 1/2, so angle A is π/3. Since the triangle is inscribed in the unit circle, we can use some properties here.First, since the triangle is on the unit circle, the radius R is 1. But in general, for any triangle inscribed in a circle of radius R, the sides are related to the angles by a = 2R sin A, etc. So here, since R = 1, a = 2 sin A, b = 2 sin B, c = 2 sin C. Wait, so in our case, a = 2 sin A, and we know angle A is π/3, so sin A = sin π/3 = √3/2. Therefore, a = 2*(√3/2) = √3.But the problem gives that b² + c² = 4. Let me write expressions for b and c. Since b = 2 sin B and c = 2 sin C, then b² + c² = 4 sin² B + 4 sin² C = 4(sin² B + sin² C). So 4(sin² B + sin² C) = 4 => sin² B + sin² C = 1.But in triangle ABC, angles B and C are related to angle A. Since A = π/3, then B + C = π - A = π - π/3 = 2π/3. Let me denote angles B and C as x and y, where x + y = 2π/3. Then sin² x + sin² y = 1. Maybe we can express y = 2π/3 - x and substitute.So sin² x + sin² (2π/3 - x) = 1.Let me compute sin² (2π/3 - x):First, sin(2π/3 - x) = sin(2π/3)cos x - cos(2π/3)sin x = (√3/2)cos x - (-1/2)sin x = (√3/2)cos x + (1/2)sin x.Therefore, sin² (2π/3 - x) = [ (√3/2 cos x + 1/2 sin x ) ]² = (3/4 cos² x + √3/2 cos x sin x + 1/4 sin² x).So sin² x + sin² (2π/3 - x) becomes:sin² x + 3/4 cos² x + √3/2 cos x sin x + 1/4 sin² x =( sin² x + 1/4 sin² x ) + 3/4 cos² x + √3/2 cos x sin x =(5/4 sin² x) + 3/4 cos² x + (√3/2 sin x cos x )Set this equal to 1:5/4 sin² x + 3/4 cos² x + √3/2 sin x cos x = 1.Hmm, this seems complicated. Maybe there's a better approach. Let's recall that in triangle ABC, we can use the Law of Cosines. Since the triangle is inscribed in the unit circle, the sides are related to the angles. But perhaps we can relate the sides to the coordinates? Wait, since it's on the unit circle, the coordinates of A, B, C are (cos α, sin α), etc., where α, β, γ are the angles corresponding to each vertex. But maybe that's overcomplicating.Alternatively, since a = √3, as we found, and given that b² + c² = 4, maybe we can use the Law of Cosines on angle A.Law of Cosines says: a² = b² + c² - 2bc cos A.We know a² = (√3)² = 3, b² + c² = 4, and cos A = 1/2. Plugging into the equation:3 = 4 - 2bc*(1/2)Simplify:3 = 4 - bcTherefore, bc = 4 - 3 = 1.So bc = 1. Then, the area of triangle ABC can be found using the formula:Area = (1/2)bc sin A.Since we know bc = 1 and sin A = sin π/3 = √3/2, then:Area = (1/2)*1*(√3/2) = √3/4.Wait, that seems straightforward. Let me verify.Given that we used the Law of Cosines on angle A, which gave us bc = 1. Then, since the area is (1/2)bc sin A, which is (1/2)(1)(√3/2) = √3/4. So that would be the area.But let me cross-check. Alternatively, since the triangle is inscribed in the unit circle, the radius is 1. The area of a triangle inscribed in a circle of radius R is given by (abc)/(4R). But here R = 1, so area = abc/4. Wait, but we have a = √3, and bc = 1. Then area would be (√3 * 1)/4 = √3/4. Which matches the previous result. So that's good.Alternatively, another formula for area is (1/2)ab sin C, but maybe that's more complicated here. Since we already found bc, and angle A, the formula (1/2)bc sin A is appropriate.Therefore, the area is √3/4.So, part (1): cos A = 1/2, part (2): area = √3/4.Wait, but let me make sure about the formula for the area. The formula (abc)/(4R) is correct for a triangle inscribed in a circle of radius R. Since R = 1 here, area = (a*b*c)/4. But if bc = 1, and a = √3, then area is (√3 * 1)/4 = √3/4. Which matches. So that's another way to get the same answer. So I think that's solid.Alternatively, since the triangle is on the unit circle, the coordinates can be represented as (cos α, sin α), etc., but that might complicate things. Let me check.Suppose points A, B, C are on the unit circle. Let’s parametrize them with angles. Let’s say point A is at angle 0, (1, 0). Then points B and C are at some angles θ and φ. The side lengths can be calculated using the chord length formula: the chord between two points on the unit circle at angles α and β is 2 sin( (α - β)/2 ). Wait, but this might complicate things. Alternatively, using the Law of Cosines in terms of the angles.Wait, perhaps I can also compute the area using the sine formula. The area of the triangle is (1/2)ab sin C, but maybe that's not directly helpful here. However, we have bc = 1, a = √3, angle A = π/3, etc.But given that we have two different methods giving the same answer, I think √3/4 is correct.Wait, but let's just recap. From part (1), we found cos A = 1/2, so angle A = 60 degrees. Given that, in part (2), we know b² + c² = 4. Then, using the Law of Cosines:a² = b² + c² - 2bc cos A.We have a = √3, so a² = 3. Then:3 = 4 - 2bc*(1/2) => 3 = 4 - bc => bc = 1.Then area = (1/2)*bc*sin A = (1/2)*1*(√3/2) = √3/4.Yes, that's correct.Alternatively, using the formula for area with sides and angles: (1/2)*b*c*sin A. Which is exactly what we did.Therefore, the area is √3/4.So, summarizing:(1) cos A = 1/2(2) Area = √3/4I think that's the answer. Let me just check if there's any possible mistake here.First, in part (1), using the given equation, substituting a, b, c with 2 sin A, 2 sin B, 2 sin C, then arriving at 4 sin A cos A = 2 sin A, leading to cos A = 1/2. That seems solid.In part (2), using the Law of Cosines with known a, b² + c², and cos A, leading to bc = 1, then using area formula. That also seems correct. So I think the answers are right.Final Answer(1) boxed{dfrac{1}{2}} (2) boxed{dfrac{sqrt{3}}{4}}</think>Given a triangle (ABC) inscribed in the unit circle with sides (a), (b), and (c) opposite to angles (A), (B), and (C) respectively, we know that (2a cos A = c cos B + b cos C).(1) To find (cos A):- Using the relationship for sides in a unit circle triangle, (a = 2 sin A), (b = 2 sin B), and (c = 2 sin C).- Substitute these into the given equation: [ 2(2 sin A) cos A = 2 sin C cos B + 2 sin B cos C ]- Simplify both sides: [ 4 sin A cos A = 2 (sin C cos B + sin B cos C) ]- Recognize the sine addition formula (sin(B + C) = sin pi - A = sin A): [ 4 sin A cos A = 2 sin A ]- Divide both sides by (2 sin A): [ 2 cos A = 1 implies cos A = frac{1}{2} ](2) To find the area of (triangle ABC) given (b^2 + c^2 = 4):- From part (1), (cos A = frac{1}{2}) and (sin A = frac{sqrt{3}}{2}).- Using the Law of Cosines: [ a^2 = b^2 + c^2 - 2bc cos A ]- Substitute (a = sqrt{3}), (b^2 + c^2 = 4), and (cos A = frac{1}{2}): [ 3 = 4 - 2bc left(frac{1}{2}right) implies 3 = 4 - bc implies bc = 1 ]- Calculate the area using (text{Area} = frac{1}{2} bc sin A): [ text{Area} = frac{1}{2} cdot 1 cdot frac{sqrt{3}}{2} = frac{sqrt{3}}{4} ]Final Answers:(1) (boxed{dfrac{1}{2}}) (2) (boxed{dfrac{sqrt{3}}{4}})]

🔑:Solution: (1) Since 2acos A = ccos B + bcos C, by the Law of Sines, we have 2sin A cdot cos A = sin C cos B + sin B cos C Rightarrow 2sin A cdot cos A = sin (B+C) = sin A, and since 0 < A < pi Rightarrow sin A neq 0, therefore 2cos A = 1 Rightarrow cos A = dfrac{1}{2}. (2) Since cos A = dfrac{1}{2} Rightarrow sin A = dfrac{sqrt{3}}{2}, and since the vertex of triangle ABC is on the unit circle, 2R=2, by the Law of Sines, we get: dfrac{a}{sin A} = 2 Rightarrow a = 2sin A = sqrt{3}. By the Law of Cosines, we have: a^{2} = b^{2} + c^{2} - 2bccos A Rightarrow bc = b^{2} + c^{2} - a^{2} = 4 - 3 = 1. therefore S_{triangle ABC} = dfrac{1}{2}bcsin A = dfrac{1}{2} cdot 1 cdot dfrac{sqrt{3}}{2} = boxed{dfrac{sqrt{3}}{4}}.

🧠:[<think>Okay, let me try to figure out this problem. So, Feifei is rowing downstream on a circular river, right? She starts at 8:00 and returns to the starting point at 10:00, so that's 2 hours total. We need to find the length of the river.First, let me note down the given information:- Feifei's rowing speed in still water: 4 km/h- Current speed: 2 km/h- She rests for 5 minutes after every half hour of rowing.- The trip takes exactly 2 hours, ending at the starting point.Hmm. So, since the river is circular, the length of the river is the circumference of the circle. If she starts downstream, that means she's going with the current. Wait, but if it's a circular river, going downstream would just mean going around the circle. But she needs to end up back at the starting point. Since the river is circular, the current is always in the same direction, right? So, she's moving along the circle with the current, but she's also rowing. However, she might have to row against the current at some point to get back? Wait, but the problem says she sets off downstream. So maybe she's just going with the current the whole time? But if she only goes downstream, wouldn't she just keep going around the circle? So how does she end up back at the starting point? Unless she rows upstream part of the time. Wait, maybe I need to think carefully.Wait, the problem says she sets off downstream. So initially, she's going downstream. But the current is 2 km/h, and her rowing speed in still water is 4 km/h. So when she is rowing downstream, her effective speed would be 4 + 2 = 6 km/h. If she rows upstream, her effective speed would be 4 - 2 = 2 km/h. But the problem says she returns to the starting point at 10:00. So she must have gone downstream for some time and upstream for some time? But how does resting factor into this?Wait, the problem says she rests for 5 minutes after every half hour of rowing, during which the boat drifts with the current. So during rest periods, she's moving downstream at the current speed of 2 km/h. So her journey consists of cycles: 30 minutes of rowing followed by 5 minutes of resting. Each cycle is 35 minutes. But she starts at 8:00 and ends at 10:00, which is 2 hours or 120 minutes. Let's see how many cycles she can have in 120 minutes.Let's divide 120 by 35. 35*3 = 105, so 3 cycles would take 105 minutes, leaving 15 minutes. So maybe she does 3 full cycles (each 35 minutes) and then a final 15 minutes. Wait, but in each cycle, it's rowing for 30 minutes and resting for 5. So if she does 3 cycles, that's 90 minutes of rowing and 15 minutes of resting, totaling 105 minutes. Then, she has 15 minutes left. But she can't do another full cycle. Maybe she rows for 15 minutes and then rests? But the problem says she rests for 5 minutes after every half hour of rowing. So if she doesn't complete a half hour of rowing, does she still rest? The problem states "rests for 5 minutes after every half hour of rowing," which suggests that she only rests after completing a half hour of rowing. Therefore, in the last 15 minutes, she might just row for 15 minutes without resting, since she hasn't rowed for a full half hour. So total time would be 3 cycles (30+5, 30+5, 30+5) = 105 minutes, plus 15 minutes of rowing. Total rowing time: 30*3 +15=105 minutes, resting time: 15 minutes. Total time: 105 +15=120 minutes. That adds up.But wait, let's check: 3 cycles of 35 minutes each is 105 minutes. Then 15 minutes left. Since she hasn't completed a half hour of rowing in those 15 minutes, she doesn't rest. So total resting time is 3*5=15 minutes. Total rowing time is 3*30 +15=105 minutes. So yes, 105 +15=120. That works.Now, during each rowing period, she can either be rowing downstream or upstream. But the problem says she sets off downstream. Does she continue downstream the entire time, or does she have to row upstream to get back? This is the key question.Wait, since the river is circular, if she only rows downstream, even with resting, she would just keep going around the circle. But the problem states she returns to the starting point at 10:00. So she must have made an integer number of laps around the river. Therefore, the length of the river must be such that the total distance she traveled downstream (due to rowing and current) minus the distance she might have gone upstream (if she rowed upstream) equals an integer multiple of the river length. But since she started downstream and ended at the same point, maybe she just went around the circle an integer number of times?Wait, but if she only rows downstream, her net displacement would be the total distance rowed downstream plus drifting downstream during rest. But since it's a circular river, every time she completes a full lap, she's back to the start. So if the total distance she traveled is a multiple of the river's length, she would end up back at the start. So perhaps the total distance she traveled downstream is exactly equal to the river length times some integer.But the problem says she returns exactly at 10:00, so it's possible she made exactly one lap? Or multiple laps?Wait, but we don't know the number of laps. The question is asking for the length of the river, so maybe we need to find the length such that the total distance traveled (accounting for rowing and drifting) is exactly an integer number of laps, and the time is 2 hours.But how does the direction come into play? If she only rows downstream, her effective speed while rowing is 6 km/h, and while resting, she drifts at 2 km/h. So in each cycle, she rows for 30 minutes (0.5 hours) at 6 km/h, covering 3 km, then rests for 5 minutes (1/12 of an hour) drifting 2 km/h * 1/12 h = 1/6 km. So each cycle she covers 3 + 1/6 = 19/6 km. Then in the last 15 minutes, she rows downstream for 15 minutes (0.25 hours) at 6 km/h, covering 1.5 km. So total distance would be 3 cycles * 19/6 km + 1.5 km = (19/2) + 1.5 = 9.5 + 1.5 = 11 km. If the river length is 11 km, then she would have done 11 km in total, which is one lap. But 11 km seems a bit arbitrary. Wait, but maybe she goes around multiple times. Wait, but if she goes around multiple times, the total distance would be a multiple of the river length. But since we don't know the number of laps, how can we determine the length?Wait, maybe the problem is that she needs to return to the starting point after 2 hours, so the total displacement must be zero. But in a circular river, displacement is modulo the river length. So if she goes downstream the entire time, her displacement would be the total distance traveled, which must be a multiple of the river length. But if she sometimes rows upstream, her net displacement would be different.Wait, the problem says she "sets off downstream," but doesn't specify that she continues downstream. Maybe she alternates between downstream and upstream? But the problem doesn't mention her changing direction. Hmm, this is a bit confusing.Wait, perhaps the key is that she needs to return to the starting point, so her net displacement must be zero or a multiple of the river's length. But since she is on a circular river, if she goes all the way around once, that's a displacement of zero (back to start). If she goes around twice, also displacement zero. So in that case, the total distance she traveled must be an integer multiple of the river length. But if she only rows downstream, she would have a total distance of 11 km as calculated before. So if 11 km is a multiple of the river length, then the river length could be 11/n km, where n is an integer. But we need more information to determine n.Wait, but maybe I made a mistake in assuming she rows downstream the entire time. Maybe she has to row upstream part of the time to cancel out the downstream drift. For example, if she rows upstream, her effective speed would be 4 - 2 = 2 km/h. Then, perhaps she alternates between rowing downstream and upstream to manage her net displacement. But the problem states she sets off downstream, but doesn't say she changes direction. Hmm. This is unclear.Alternatively, maybe the circular river is such that the current is always in the same direction, so "downstream" is following the current, and "upstream" is against it. Since it's a loop, going downstream continuously would just loop around the river. To return to the starting point, she needs the total distance traveled (with the current) to be an exact multiple of the river's length. But if she only goes downstream, even with rest periods, her total distance would be the sum of rowing downstream and drifting. So perhaps the river length divides evenly into that total distance.But let's re-examine the timeline:From 8:00 to 10:00 is 2 hours = 120 minutes.Her routine is: row for 30 minutes, rest for 5 minutes. So each cycle is 35 minutes. How many full cycles can she do in 120 minutes?120 divided by 35 is 3 with a remainder of 15 minutes. So 3 full cycles (30+5 each) and then 15 minutes remaining. As I thought before.In each cycle:- Rowing for 30 minutes (0.5 hours) at 6 km/h (since downstream): distance = 6 * 0.5 = 3 km- Resting for 5 minutes (1/12 hours) drifting at 2 km/h: distance = 2 * 1/12 = 1/6 km ≈ 0.1667 km- Total per cycle: 3 + 1/6 = 19/6 ≈ 3.1667 kmAfter 3 cycles: 3 * 19/6 = 19/2 = 9.5 kmThen, she has 15 minutes left. Since she can't complete another 30-minute rowing session, she just rows for 15 minutes (0.25 hours) at 6 km/h: distance = 6 * 0.25 = 1.5 kmTotal distance downstream: 9.5 + 1.5 = 11 kmSo, if the total distance she traveled is 11 km, and she ended up back at the starting point, the river must be 11 km in length, or a divisor of 11 km. But 11 is a prime number, so the only divisors are 1 and 11. If the river were 11 km, then 11 km is one lap. If it's 1 km, she would have done 11 laps. But the problem is asking for the length of the river, so it's likely 11 km. Wait, but 11 km seems plausible?Wait, but let me check again. If she goes 11 km downstream in 2 hours, but the river is circular. So if the river is 11 km, then going 11 km brings her back to the start. That makes sense. So 11 km is the answer.But let me check if there's another interpretation where she has to row upstream. Suppose she alternates between rowing upstream and downstream to stay in place? But the problem says she sets off downstream and returns at 10:00. So maybe she only rows downstream, and the length is 11 km.Alternatively, maybe she has to row upstream part of the time. Let's think differently. Suppose that the total displacement must be zero (since she returns to start). But in a circular river, displacement is periodic. So if she travels a distance equal to a multiple of the river's length, she returns. But if the current is always downstream, then even when she rests, she's moving downstream. So if she only rows downstream, she would have to go all the way around the river. But maybe she also rows upstream to counteract the current. Wait, but if she rows upstream, her speed relative to the bank would be 4 - 2 = 2 km/h. So maybe her strategy involves rowing upstream and downstream to manage the total distance.But the problem states she sets off downstream. There's no mention of changing direction. Maybe the problem is designed such that despite only rowing downstream and drifting downstream during rests, the total distance is exactly the circumference. So the answer is 11 km.But let me verify the total time and distance again.First, cycles:3 cycles of 35 minutes each: 3 * 35 = 105 minutesRemaining time: 15 minutesTotal rowing time: 3 * 30 + 15 = 90 + 15 = 105 minutes = 1.75 hoursTotal resting time: 3 * 5 = 15 minutes = 0.25 hoursDistance while rowing: rowing speed downstream is 4 + 2 = 6 km/h. So total rowing distance: 6 km/h * 1.75 h = 10.5 kmDistance while resting: current speed 2 km/h * 0.25 h = 0.5 kmTotal distance: 10.5 + 0.5 = 11 kmTherefore, the total distance traveled is 11 km. Since she ended up at the starting point, the river must be 11 km in length, so that 11 km is exactly one full lap.Therefore, the length of the river is 11 km.Final Answerboxed{11}</think>Feifei sets off downstream in a small boat at 8:00 and returns to the starting point at 10:00, which is a total of 2 hours. Her rowing speed in still water is 4 km/h, and the current speed is 2 km/h. She rests for 5 minutes after every half hour of rowing, during which the boat drifts with the current.First, we calculate the effective speed when rowing downstream:- Rowing speed downstream: (4 , text{km/h} + 2 , text{km/h} = 6 , text{km/h})- Drifting speed during rest: (2 , text{km/h})Each cycle consists of 30 minutes of rowing and 5 minutes of rest, totaling 35 minutes. In 120 minutes (2 hours), there are 3 full cycles (105 minutes) and 15 minutes remaining.Calculations for each cycle:- Rowing distance per 30 minutes: (6 , text{km/h} times 0.5 , text{h} = 3 , text{km})- Drifting distance per 5 minutes: (2 , text{km/h} times frac{1}{12} , text{h} = frac{1}{6} , text{km})- Total distance per cycle: (3 , text{km} + frac{1}{6} , text{km} = frac{19}{6} , text{km})Total distance for 3 cycles:- (3 times frac{19}{6} , text{km} = frac{19}{2} , text{km} = 9.5 , text{km})Remaining 15 minutes:- Rowing distance: (6 , text{km/h} times 0.25 , text{h} = 1.5 , text{km})Total distance traveled:- (9.5 , text{km} + 1.5 , text{km} = 11 , text{km})Since Feifei returns to the starting point, the total distance traveled must be an integer multiple of the river's length. Given that she traveled 11 km in 2 hours and returned to the starting point, the length of the river is 11 km.[boxed{11}],<think>Okay, let's see. So, Feifei starts at 8:00 AM downstream in a circular river. The problem states that her rowing speed in still water is 4 km/h, and the current speed is 2 km/h. She rests for 5 minutes every half hour of rowing, during which the boat drifts with the current. She returns to the starting point exactly at 10:00 AM. We need to find the length of the river.First, let's parse the problem. The river is circular, so the length of the river is the circumference of the circle. Feifei starts rowing downstream at 8:00 AM and returns to the starting point at 10:00 AM, which means the total time taken is 2 hours.Now, her rowing speed in still water is 4 km/h, but when she's going downstream, the current helps her. So her effective speed downstream would be her rowing speed plus the current speed, which is 4 + 2 = 6 km/h. When she rests, she's not rowing, so the boat just drifts with the current at 2 km/h.However, since the river is circular, if she goes all the way around, she would have to counteract the current somehow. Wait, but she's going downstream the entire time. Hmm, maybe the river is circular, so downstream is around the circle. So if she keeps going downstream, she would loop around the river. But she needs to return to the starting point. Wait, but if the river is circular, going downstream continuously would just make her loop around the river. So to return to the starting point, she needs to complete an integer number of loops around the river. Therefore, the length of the river must be such that after 2 hours, considering her rowing and resting periods, she has completed a whole number of loops.Wait, but the problem says she returns to the starting point exactly at 10:00. So, in two hours, with her rowing and resting schedule, she must have gone a distance that is a multiple of the river's circumference. Therefore, the total distance she travels downstream in two hours must equal an integer multiple of the river's length. But since she ends up at the starting point, it must be exactly one full circumference, right? Because if she went around multiple times, she still would have returned to the starting point, but the problem doesn't specify how many loops. Wait, but since it's a circular river, the length is just the circumference. So maybe she goes around once? Or maybe multiple times? But the answer is unique, so it must be once. Wait, but maybe not. Let me think.Alternatively, perhaps the river isn't a closed loop, but a circular path where you can go downstream in a circle. So the starting point is on the circular river, and going downstream means moving along the circle in one direction. To return to the starting point, she must have completed an integer number of laps around the circle. Therefore, the total distance she travels downstream must be an integer multiple of the river's length. Since she ends at the starting point, that integer could be 1, 2, etc., but since the time is 2 hours, we need to find the length such that the total distance equals n*L, where n is an integer, and L is the length. Then, since the problem probably expects a unique answer, we need to compute based on her rowing and resting schedule over two hours.So, let's break down her schedule. She rows for half an hour, then rests for 5 minutes. So each cycle is 35 minutes. Let's see how many cycles she can complete in two hours (120 minutes).Divide 120 by 35: 35*3 = 105 minutes, with 15 minutes remaining. So, three full cycles (each 35 minutes: 30 rowing + 5 resting) and then a remaining 15 minutes. Now, in the remaining 15 minutes, she would row for another 15 minutes? Because after each 30 minutes of rowing, she rests for 5. So the pattern is: row for 30, rest for 5. So in the first 35 minutes: row 30, rest 5; next 35 minutes: row 30, rest 5; third 35 minutes: row 30, rest 5. That's 105 minutes. Then she has 15 minutes left. Since her last rest ended at 105 minutes, she would start rowing again at 105 minutes. She needs to row until 120 minutes, which is 15 minutes. However, does she need to rest after those 15 minutes? Wait, the problem says she rests for 5 minutes after every half hour of rowing. So she rests after every 30 minutes of rowing, not after every 30 minutes of time. Hmm, this is a crucial point.Wait, the problem states: "rests for 5 minutes after every half hour of rowing." So it's after every 30 minutes of rowing, she rests for 5 minutes. So the rest period is triggered by 30 minutes of cumulative rowing.Therefore, in the two hours (120 minutes), we need to calculate how much time she spends rowing and resting, considering that every 30 minutes of rowing is followed by 5 minutes of resting.But she starts at 8:00. Let's try to model her schedule.Starting at 8:00, she begins rowing. She rows for 30 minutes (until 8:30), then rests for 5 minutes (until 8:35). Then she starts rowing again at 8:35 for another 30 minutes (until 9:05), rests until 9:10. Then rows from 9:10 to 9:40, rests until 9:45. Then rows from 9:45 to 10:15? Wait, but she needs to return by 10:00. So this seems conflicting. So perhaps she can't start another 30-minute rowing session at 9:45 because she needs to end at 10:00. Therefore, perhaps in the last segment, she only rows for 15 minutes instead of 30, and doesn't rest because the rest is only after 30 minutes of rowing.Wait, let's step through it.First rowing: 8:00 - 8:30 (30 minutes rowing)Rest: 8:30 - 8:35 (5 minutes)Second rowing: 8:35 - 9:05 (30 minutes rowing)Rest: 9:05 - 9:10 (5 minutes)Third rowing: 9:10 - 9:40 (30 minutes rowing)Rest: 9:40 - 9:45 (5 minutes)Fourth rowing: 9:45 - 10:15 (30 minutes rowing)But she needs to stop at 10:00. So the fourth rowing session would be from 9:45 to 10:00, which is 15 minutes. However, since she only rowed for 15 minutes, which is less than half an hour, she doesn't need to rest afterwards. Therefore, her schedule is:8:00 - 8:30: rowing8:30 - 8:35: rest8:35 - 9:05: rowing9:05 - 9:10: rest9:10 - 9:40: rowing9:40 - 9:45: rest9:45 - 10:00: rowing (15 minutes)So total rowing time: 30 + 30 + 30 + 15 = 105 minutesTotal rest time: 5 + 5 + 5 = 15 minutesTotal time: 105 + 15 = 120 minutes, which is correct.Therefore, her total downstream distance is the sum of the distances during rowing and resting periods.When she is rowing downstream, her speed is 6 km/h (4 km/h rowing + 2 km/h current). When she is resting, her speed is 2 km/h (just drifting with the current).So, let's compute each segment:First rowing: 30 minutes = 0.5 hours. Distance: 6 km/h * 0.5 h = 3 kmRest: 5 minutes = 5/60 hours = 1/12 hours. Distance: 2 km/h * 1/12 h = 1/6 km ≈ 0.1667 kmSecond rowing: 30 minutes = 0.5 h. Distance: 6 * 0.5 = 3 kmRest: 5 minutes. Distance: 1/6 kmThird rowing: 30 minutes. Distance: 3 kmRest: 5 minutes. Distance: 1/6 kmFourth rowing: 15 minutes = 0.25 h. Distance: 6 * 0.25 = 1.5 kmNow, sum all these distances:3 + 1/6 + 3 + 1/6 + 3 + 1/6 + 1.5Let's calculate step by step:First rowing: 3After first rest: 3 + 1/6 ≈ 3.1667Second rowing: +3 → 6.1667After second rest: +1/6 ≈ 6.3333Third rowing: +3 → 9.3333After third rest: +1/6 ≈ 9.5Fourth rowing: +1.5 → 11 kmSo total distance traveled downstream is 11 km.But wait, the problem states that she returns to the starting point. Since the river is circular, this total distance must be an integer multiple of the river's length. Therefore, 11 km = n * L, where n is an integer. Since the problem doesn't specify the number of loops, but asks for the length of the river, which is a unique answer, likely n=1. But 11 km is the total distance, so if she looped around once, L=11 km. However, is there a possibility that she looped multiple times?But if the river's length were 11 km, then in two hours, she would have gone 11 km, which is exactly one loop. But let's check the feasibility.Wait, but the problem says "the length of the river is qquad kilometers." So we need to find L such that 11 km is a multiple of L, but since she ends at the starting point, the total displacement is L * k, where k is an integer. Since the problem probably expects the minimal length, i.e., L=11 km. However, let's check.Alternatively, perhaps my calculation is wrong. Let me recalculate the total distance.Wait, first rowing: 0.5 h * 6 km/h = 3 kmFirst rest: 5 min = 1/12 h * 2 km/h = 2/12 = 1/6 km ≈ 0.1667 kmSecond rowing: 3 kmSecond rest: 1/6 kmThird rowing: 3 kmThird rest: 1/6 kmFourth rowing: 0.25 h * 6 = 1.5 kmTotal: 3 + 1/6 + 3 + 1/6 + 3 + 1/6 + 1.5So that's 3 + 3 + 3 + 1.5 = 10.5 km, and 1/6 + 1/6 + 1/6 = 3/6 = 0.5 km. So total is 10.5 + 0.5 = 11 km. Correct.Therefore, she traveled 11 km in total. Since she ended up at the starting point, the river's length must divide 11 km exactly. The possible lengths are 11 km, 5.5 km, 3.666... km, etc. But which one is the answer?But maybe the problem is implying that she went around once. If she went around once, then the length is 11 km. If she went around twice, then the length is 5.5 km. But we need more information to determine this.Wait, the problem says "the length of the river is qquad kilometers." So likely, the answer is 11 km. But I need to check if that makes sense. If the river is 11 km long, then she went 11 km downstream, which is exactly one loop, bringing her back to the starting point. Alternatively, if the river were shorter, say 5.5 km, then she would have gone around twice. But since the problem doesn't specify the number of loops, we need to assume the minimal length, which is 11 km. But let me think again.Wait, but if the river were 11 km, then her net displacement would be 11 km, which is one loop. But perhaps there's another factor here. Wait, is the current speed affecting her net speed? Wait, no. When she is rowing downstream, her speed relative to the ground is 6 km/h. When resting, it's 2 km/h. So over the entire two hours, her total distance is 11 km. Therefore, if the river is circular, 11 km must be a multiple of the river's length. So the river's length must be a divisor of 11 km. The divisors of 11 are 1 and 11. Since 11 is a prime number. Therefore, the river's length is either 11 km or 1 km. But 1 km seems too short for a river in an amusement park, but maybe.Wait, but 11 km is a very long river for an amusement park as well. So perhaps there is a mistake in my calculation.Wait, let's check the problem again.Feifei sets off downstream in a small boat at 8:00. Her rowing speed in still water is 4 km/h, current speed is 2 km/h. She rests for 5 minutes after every half hour of rowing, during which the boat drifts. She returns to the starting point at 10:00. Find the length of the river.Wait, maybe I didn't consider that when she is rowing, she might be able to go upstream as well? But the problem says she sets off downstream, so perhaps she continues downstream the entire time. However, in a circular river, downstream would loop around, so going downstream continuously would bring her back to the start. But maybe there's a different interpretation.Alternatively, perhaps the river is a loop, so going downstream means moving in the direction of the current, and if she goes all the way around, she returns. However, if she only goes partway, she would have to row upstream to return, but the problem states she returns exactly at 10:00, so maybe she didn't have to row upstream because the current carried her around.Wait, but in my previous calculation, all her movement is downstream, either by rowing or drifting. So if the total distance is 11 km, which is the circumference, then that's the answer. But 11 km seems quite long for an amusement park ride. Maybe I made a mistake in converting time or speed.Wait, let's re-express the total time.From 8:00 to 10:00 is 2 hours, which is 120 minutes.Her schedule is:30 minutes rowing, 5 minutes resting, repeated three times, and then 15 minutes rowing.So:First rowing: 30 min (0.5 h) at 6 km/h: distance = 3 kmFirst rest: 5 min (1/12 h) at 2 km/h: 2*(1/12)=1/6 kmSecond rowing: 30 min: 3 kmSecond rest: 5 min: 1/6 kmThird rowing: 30 min: 3 kmThird rest: 5 min: 1/6 kmFourth rowing: 15 min (0.25 h): 6*0.25=1.5 kmTotal distance:3 + 1/6 + 3 + 1/6 + 3 + 1/6 + 1.5 = 3*3 + 1.5 + (1/6)*3 = 9 + 1.5 + 0.5 = 11 km. Yes, that's correct.So the total distance is 11 km. Since she ended at the starting point, the river must be 11 km in length, as that's the minimal distance that would bring her back after one loop. Alternatively, if the river is shorter, say L km, then 11 km must be a multiple of L. But since 11 is a prime number, the possible lengths are 11 km or 1 km. But 1 km is possible? If the river is 1 km, then she went around 11 times. But 11 loops in 2 hours? That would mean the current is 2 km/h, so drifting alone would take 30 minutes per loop (since 1 km / 2 km/h = 0.5 h). But she's rowing as well, so maybe. But 11 loops seems excessive. But the problem doesn't specify the number of loops, just that she returns to the starting point. So both 11 km and 1 km are possible. But given that amusement parks typically have rides that are not 11 km long, maybe the answer is 11 km, even though it's long. Or perhaps I made a mistake in interpreting the problem.Wait, another angle: when rowing downstream, her speed is 6 km/h, but if the river is circular, then in order to make a full loop, the net distance should be zero? Wait, no. Wait, in a circular river, going downstream means moving along the circumference in one direction. So if she goes all the way around, she would have traveled a distance equal to the circumference, ending up where she started. So, in that case, the total distance she traveled is equal to the circumference. So if she traveled 11 km, then the circumference is 11 km. So the answer should be 11 km.Alternatively, if she traveled multiple loops, but since the problem says "returns to the starting point exactly at 10:00", the first time she returns is at 10:00, which would be after 2 hours, so the circumference is 11 km. Therefore, the length of the river is 11 km. Therefore, the answer is 11, so boxed answer is boxed{11}But just to verify again, let me check the total time:First rowing: 30 minFirst rest: 5 minSecond rowing: 30 minSecond rest: 5 minThird rowing: 30 minThird rest: 5 minFourth rowing: 15 minTotal: 30+5+30+5+30+5+15 = 120 minutes, which is correct. So the total distance is 11 km, and since she ended at the starting point, the river's length is 11 km. So the answer is 11.Final Answerboxed{11}</think>Feifei starts rowing downstream at 8:00 AM and returns to the starting point at 10:00 AM, which is a total of 2 hours. Her rowing speed in still water is 4 km/h, and the current speed is 2 km/h. She rests for 5 minutes after every 30 minutes of rowing.Key Steps:1. Effective Speed Calculation: - Rowing downstream speed: (4 , text{km/h} + 2 , text{km/h} = 6 , text{km/h}) - Resting (drifting) speed: (2 , text{km/h})2. Schedule Breakdown: - Each cycle: 30 minutes rowing + 5 minutes resting (35 minutes total) - In 120 minutes (2 hours), she completes 3 full cycles (105 minutes) and has 15 minutes remaining. - Breakdown: - 8:00 - 8:30: Rowing - 8:30 - 8:35: Resting - 8:35 - 9:05: Rowing - 9:05 - 9:10: Resting - 9:10 - 9:40: Rowing - 9:40 - 9:45: Resting - 9:45 - 10:00: Rowing (15 minutes)3. Distance Calculation: - Rowing segments: (0.5 , text{hours} times 6 , text{km/h} = 3 , text{km}) each (three times) and (0.25 , text{hours} times 6 , text{km/h} = 1.5 , text{km}) - Resting segments: (frac{5}{60} , text{hours} times 2 , text{km/h} = frac{1}{6} , text{km}) each (three times)4. Total Distance: - Rowing: (3 , text{km} times 3 + 1.5 , text{km} = 10.5 , text{km}) - Resting: (frac{1}{6} , text{km} times 3 = 0.5 , text{km}) - Total: (10.5 , text{km} + 0.5 , text{km} = 11 , text{km})Since Feifei returns to the starting point, the total distance traveled (11 km) must be the circumference of the circular river. Therefore, the length of the river is (boxed{11}) kilometers.]

🔑:11

🧠:[<think>Okay, so I need to find the derivative of this function y with respect to x. The function is a natural logarithm of a fraction, and both the numerator and the denominator have similar expressions involving square roots and exponential terms. Let me write down the function first to get a clearer picture.The function is:y = ln left( frac{sqrt{1 + e^x + e^{2x}} - e^x - 1}{sqrt{1 + e^x + e^{2x}} - e^x + 1} right)Alright, so the derivative of y with respect to x will be the derivative of the natural logarithm of a function, which is (1/function) times the derivative of the function. So I'll need to use the chain rule here. Let me recall that if y = ln(u), then y' = u'/u.But here, u is a fraction, so u = numerator/denominator. Therefore, the derivative of u will involve the quotient rule. Alternatively, since we have a logarithm of a quotient, maybe we can simplify that first using logarithm properties before differentiating. Let me think.Yes, logarithm of a quotient is the difference of logarithms. So maybe:y = ln(text{numerator}) - ln(text{denominator})That could make differentiation easier because then I can handle each term separately. Let's try that.So, let me denote:- Let A = sqrt(1 + e^x + e^{2x}) - e^x - 1 (numerator)- Let B = sqrt(1 + e^x + e^{2x}) - e^x + 1 (denominator)Then, y = ln(A) - ln(B). Therefore, dy/dx = (A'/A) - (B'/B). That seems manageable. So I need to compute A' and B', then plug into this expression.But let's check: A and B are similar. Let me write down A and B:A = sqrt(1 + e^x + e^{2x}) - e^x - 1B = sqrt(1 + e^x + e^{2x}) - e^x + 1So the only difference between A and B is the constant term at the end: -1 vs. +1. Therefore, when we take their derivatives A' and B', the only difference will be in the derivative of the constant term, which is 0 in both cases. Wait, actually, the derivative of the constants -1 and +1 with respect to x is 0. So actually, A' and B' are the same? Let's see.Wait, A and B share the same terms except for the constants. Let's confirm:d/dx of sqrt(1 + e^x + e^{2x}) is the same for both A and B.Then, derivative of -e^x is -e^x for A, and derivative of -e^x is also -e^x for B.Then, derivative of -1 (for A) is 0, and derivative of +1 (for B) is 0. So yes, A' and B' are actually equal. Because the only difference between A and B is the constant term, which differentiates to zero. Therefore, A' = B'.Wait, but if A' = B', then dy/dx = (A'/A) - (A'/B) = A' (1/A - 1/B) = A' ( (B - A)/(AB) )Hmm, interesting. Maybe that simplifies things? Let's see.But maybe I should just compute A' and B' step by step. Let me proceed.First, let's compute the derivative of sqrt(1 + e^x + e^{2x}). Let's let C = 1 + e^x + e^{2x}, so sqrt(C). The derivative of sqrt(C) with respect to x is (1/(2*sqrt(C))) * C'Compute C':C = 1 + e^x + e^{2x}C' = 0 + e^x + 2e^{2x}Therefore, derivative of sqrt(C) is (e^x + 2e^{2x})/(2*sqrt(1 + e^x + e^{2x}))Then, the derivative of -e^x is -e^x. So:A' = derivative of sqrt(C) - e^x - 1Which is (e^x + 2e^{2x})/(2*sqrt(C)) - e^xSimilarly, B' = derivative of sqrt(C) - e^x + 1, which is the same as A' because the +1 differentiates to 0. So indeed, A' = B' = (e^x + 2e^{2x})/(2*sqrt(C)) - e^xSo, dy/dx = A'/A - B'/B = A' (1/A - 1/B) = A' ( (B - A)/(AB) )But let's see if B - A can be simplified. Since A and B only differ by the constants -1 and +1, so:B - A = [sqrt(C) - e^x +1] - [sqrt(C) - e^x -1] = (sqrt(C) - e^x +1) - sqrt(C) + e^x +1 = 2So B - A = 2. Therefore, dy/dx = A' * (2)/(AB)But AB is the product of numerator and denominator of the original fraction inside the log. So AB = [sqrt(C) - e^x -1][sqrt(C) - e^x +1]Wait, that looks like (sqrt(C) - e^x)^2 - (1)^2, since (a - b)(a + b) = a^2 - b^2. Let me check:Let me let D = sqrt(C) - e^xThen, (D - 1)(D + 1) = D^2 - 1Therefore, AB = (sqrt(C) - e^x)^2 - 1Compute (sqrt(C) - e^x)^2:= C - 2e^x sqrt(C) + e^{2x}Therefore, AB = C - 2e^x sqrt(C) + e^{2x} -1But C = 1 + e^x + e^{2x}, so substituting:AB = (1 + e^x + e^{2x}) - 2e^x sqrt(C) + e^{2x} -1Simplify:1 -1 cancels, so:AB = e^x + e^{2x} + e^{2x} - 2e^x sqrt(C)= e^x + 2e^{2x} - 2e^x sqrt(C)Hmm, not sure if that helps. Alternatively, maybe it's better to keep AB as [sqrt(C) - e^x -1][sqrt(C) - e^x +1] = (sqrt(C) - e^x)^2 -1But maybe we can find AB in another way.Alternatively, since AB is the product of numerator and denominator of the original fraction, and the original fraction is inside the log, but maybe AB is not needed, because in the derivative expression dy/dx = A' * 2 / (AB)But maybe instead of trying to compute AB directly, we can express AB in terms of C or something else.Alternatively, let's recall that the original argument of the log is A/B. Let me denote that fraction as F = A/B. Then y = ln F, so dy/dx = F'/F. But F is A/B, so F' = (A' B - A B') / B^2. Then dy/dx = (A' B - A B') / (A B). But since A' = B', then:dy/dx = (A' B - A A') / (A B) = A'(B - A)/(A B) = A' * (-2)/(A B), since B - A = - (A - B) = -2. Wait, earlier we had B - A = 2, so A - B = -2. Wait, let's confirm:Original A = sqrt(C) - e^x -1B = sqrt(C) - e^x +1Therefore, B - A = [sqrt(C) - e^x +1] - [sqrt(C) - e^x -1] = 1 - (-1) = 2. Yes, so B - A = 2. Therefore, A - B = -2.So, dy/dx = A' * (2)/(AB). Hmm, but how can we compute AB?Alternatively, maybe we can note that the original expression inside the log is A/B. Let's denote F = A/B. Then, perhaps we can find a simplified form for F, which might help in differentiation.Let me try to simplify F.F = [sqrt(1 + e^x + e^{2x}) - e^x -1] / [sqrt(1 + e^x + e^{2x}) - e^x +1]Let me denote sqrt(1 + e^x + e^{2x}) as S for brevity. Then:F = (S - e^x -1)/(S - e^x +1)Let me see if we can rationalize this fraction or perform some algebraic manipulation.One method could be to multiply numerator and denominator by the conjugate. Let's try multiplying numerator and denominator by (S - e^x +1)/(S - e^x +1), but that's just the denominator. Alternatively, maybe multiply numerator and denominator by (S + e^x +1) or something else.Alternatively, perhaps consider the entire expression under the logarithm. Let me see:Suppose F = [S - e^x -1]/[S - e^x +1]Let me compute F:Let me write S = sqrt(1 + e^x + e^{2x})Notice that 1 + e^x + e^{2x} can be written as e^{2x} + e^x +1, which is a quadratic in e^x. Let me check:Let me set t = e^x, then 1 + e^x + e^{2x} = t^2 + t +1. So sqrt(t^2 + t +1). Hmm, maybe substitution would help here? Let me consider that.Let me try substituting t = e^x. Then, dt/dx = e^x, so dx = dt/t. But since we need dy/dx, perhaps substitution would complicate things. Alternatively, maybe it's better to manipulate F algebraically.Wait, let's try to compute F. Let me write F as:F = [S - (e^x +1)] / [S - (e^x -1)]Wait, that might not help. Alternatively, let's consider S^2 = 1 + e^x + e^{2x}Therefore, S^2 = e^{2x} + e^x +1Therefore, S^2 - e^{2x} - e^x =1Wait, perhaps:(S - e^x -1)(S + e^x +1) = S^2 - (e^x +1)^2But S^2 = 1 + e^x + e^{2x}So, (S - e^x -1)(S + e^x +1) = (1 + e^x + e^{2x}) - (e^{2x} + 2e^x +1) = 1 + e^x + e^{2x} - e^{2x} -2e^x -1 = -e^xSimilarly, (S - e^x +1)(S + e^x -1) = S^2 - (e^x -1)^2 = (1 + e^x + e^{2x}) - (e^{2x} - 2e^x +1) = 1 + e^x + e^{2x} - e^{2x} + 2e^x -1 = 3e^xWait, this seems promising. Let me verify:First, compute (S - e^x -1)(S + e^x +1):= S^2 + S(e^x +1) - S(e^x +1) - (e^x +1)^2Wait, no, that's not correct. Wait, actually, expanding (a - b)(a + b) where a = S and b = e^x +1. So, it's a^2 - b^2. Therefore:(S - (e^x +1))(S + (e^x +1)) = S^2 - (e^x +1)^2Which is equal to:[1 + e^x + e^{2x}] - [e^{2x} + 2e^x +1] = 1 + e^x + e^{2x} - e^{2x} -2e^x -1 = (1 -1) + (e^x -2e^x) + (e^{2x} - e^{2x}) = -e^xSimilarly, compute (S - (e^x -1))(S + (e^x -1)) = S^2 - (e^x -1)^2Which is:[1 + e^x + e^{2x}] - [e^{2x} - 2e^x +1] = 1 + e^x + e^{2x} - e^{2x} + 2e^x -1 = (1 -1) + (e^x + 2e^x) + (e^{2x} - e^{2x}) = 3e^xTherefore, we have:(S - e^x -1)(S + e^x +1) = -e^x(S - e^x +1)(S + e^x -1) = 3e^xBut notice that F = [S - e^x -1]/[S - e^x +1]Let me write F as:F = [S - e^x -1]/[S - e^x +1] = [ (S - e^x -1) ] / [ (S - e^x +1) ]Let me multiply numerator and denominator by (S + e^x +1)/(S + e^x -1). Wait, but maybe use the above products.Wait, we have that (S - e^x -1)(S + e^x +1) = -e^x, so numerator is [S - e^x -1] = -e^x / (S + e^x +1)Similarly, denominator is [S - e^x +1], and from the other product, (S - e^x +1)(S + e^x -1) = 3e^x, so denominator is 3e^x / (S + e^x -1)Therefore, F can be written as:F = [ -e^x / (S + e^x +1) ] / [ 3e^x / (S + e^x -1) ) ] = [ -e^x / (S + e^x +1) ] * [ (S + e^x -1)/3e^x ) ] = [ -1 / (S + e^x +1) ] * [ (S + e^x -1)/3 ] = - (S + e^x -1)/(3(S + e^x +1))So F simplifies to - (S + e^x -1)/(3(S + e^x +1))But then, y = ln F = ln [ - (S + e^x -1)/(3(S + e^x +1)) ]But since the argument of the logarithm must be positive, the negative sign implies that the numerator and denominator have opposite signs. But given the original expressions, perhaps we can check if this is the case. However, maybe I made a miscalculation here. Let me check again.Wait, in the step where I wrote:(S - e^x -1) = -e^x / (S + e^x +1). That comes from (S - e^x -1)(S + e^x +1) = -e^x, so solving for (S - e^x -1) gives -e^x / (S + e^x +1). Similarly, (S - e^x +1) = 3e^x / (S + e^x -1)But when I substitute into F:F = [ (S - e^x -1) ] / [ (S - e^x +1) ] = [ -e^x / (S + e^x +1) ] / [ 3e^x / (S + e^x -1) ) ] = [ -e^x / (S + e^x +1) ] * [ (S + e^x -1) / 3e^x ) ] = [ -1 / (S + e^x +1) ] * [ (S + e^x -1)/3 ]So yes, F = - (S + e^x -1)/(3(S + e^x +1))But then, this seems a bit more complicated. However, maybe this can be simplified further. Let me see:Let me expand numerator and denominator:Numerator: S + e^x -1Denominator: 3(S + e^x +1)So, F = - [ (S + e^x -1) / (S + e^x +1) ] /3But that is equal to - (1 - 2/(S + e^x +1))/3But perhaps not helpful. Alternatively, maybe notice that S + e^x +1 is a term we can relate to the original expression.Alternatively, perhaps this form isn't helpful, and I should instead consider that the original expression simplifies to something else. Let me check:Alternatively, perhaps the original expression inside the log simplifies to a hyperbolic function or something similar. Let me see.Alternatively, maybe consider that 1 + e^x + e^{2x} can be rewritten as (e^{2x} + e^x +1). Hmm, which is similar to the denominator in some hyperbolic identities, but not exactly. Alternatively, factor e^{2x}:1 + e^x + e^{2x} = e^{2x} + e^x +1 = e^{2x} + e^x +1. Doesn't factor nicely over the reals. Hmm.Alternatively, maybe let me set t = e^x, then S = sqrt(1 + t + t^2). Then, F becomes [sqrt(t^2 + t +1) - t -1]/[sqrt(t^2 + t +1) - t +1]Let me compute this for specific t to see if it simplifies. For example, let t = 1:Numerator: sqrt(1 +1 +1) -1 -1 = sqrt(3) - 2 ≈ 1.732 - 2 ≈ -0.268Denominator: sqrt(3) -1 +1 = sqrt(3) ≈ 1.732So F ≈ -0.268 / 1.732 ≈ -0.1547But not sure if that helps.Alternatively, maybe express sqrt(t^2 + t +1) in terms of t. Let me see:sqrt(t^2 + t +1) can be written as sqrt( (t + 0.5)^2 + 3/4 ), but not sure if that helps.Alternatively, perhaps substitute t = e^x, and see if we can express F in terms of hyperbolic functions. Let me try:sqrt(t^2 + t +1). Hmm, hyperbolic functions usually involve sqrt(t^2 +1), but here we have t^2 + t +1. Maybe not.Alternatively, perhaps use substitution u = t + 0.5, then t^2 + t +1 = u^2 + 3/4.But again, not sure.Alternatively, perhaps this approach isn't helpful and I should proceed with differentiation as I started before.Let me recap: I need to compute dy/dx = (A'/A) - (B'/B). Since A' = B', this becomes A'*(1/A - 1/B) = A'*(B - A)/(AB). We found that B - A = 2, so dy/dx = 2A'/(AB)So, if I can compute A' and AB, then I can find dy/dx.First, let's compute A'.A = sqrt(1 + e^x + e^{2x}) - e^x -1Let me compute A':A' = derivative of sqrt(1 + e^x + e^{2x}) - derivative of e^x - derivative of 1= [ (e^x + 2e^{2x}) / (2*sqrt(1 + e^x + e^{2x})) ] - e^x - 0So,A' = (e^x + 2e^{2x})/(2*sqrt(1 + e^x + e^{2x})) - e^xLet me factor out e^x in the numerator of the first term:A' = e^x [1 + 2e^x]/(2*sqrt(1 + e^x + e^{2x})) - e^x= e^x [ (1 + 2e^x)/(2*sqrt(1 + e^x + e^{2x})) - 1 ]Let me combine the terms:= e^x [ (1 + 2e^x - 2*sqrt(1 + e^x + e^{2x})) / (2*sqrt(1 + e^x + e^{2x})) ) ]Hmm, not sure if this helps. Let's keep A' as:A' = (e^x + 2e^{2x})/(2*sqrt(1 + e^x + e^{2x})) - e^xNow, AB is the product of A and B, which are:A = sqrt(1 + e^x + e^{2x}) - e^x -1B = sqrt(1 + e^x + e^{2x}) - e^x +1From earlier, we found that AB = (sqrt(C) - e^x)^2 -1 where C = 1 + e^x + e^{2x}Let me compute that:AB = (sqrt(C) - e^x)^2 -1 = C - 2e^x sqrt(C) + e^{2x} -1But C = 1 + e^x + e^{2x}, so substituting:AB = (1 + e^x + e^{2x}) - 2e^x sqrt(C) + e^{2x} -1Simplify:1 -1 cancels, e^x + e^{2x} + e^{2x} remains:AB = e^x + 2e^{2x} - 2e^x sqrt(C)But C = 1 + e^x + e^{2x}, so sqrt(C) is sqrt(1 + e^x + e^{2x})Therefore, AB = e^x + 2e^{2x} - 2e^x sqrt(1 + e^x + e^{2x})Alternatively, AB can be written as e^x(1 + 2e^x) - 2e^x sqrt(1 + e^x + e^{2x})Hmm, notice that in A', we have a term similar to the numerator:In A', the first term is (e^x + 2e^{2x})/(2*sqrt(C)). Let me note that e^x + 2e^{2x} = e^x(1 + 2e^x)Therefore, AB = e^x(1 + 2e^x) - 2e^x sqrt(C)But then, AB can be written as e^x [ (1 + 2e^x) - 2 sqrt(C) ]Hmm, interesting. Let me see.Now, looking back at dy/dx = 2A' / (AB)We have A' = (e^x + 2e^{2x})/(2*sqrt(C)) - e^xWhich can be written as A' = [ e^x(1 + 2e^x) ] / (2 sqrt(C)) - e^x= e^x [ (1 + 2e^x)/(2 sqrt(C)) - 1 ]Similarly, AB = e^x [ (1 + 2e^x) - 2 sqrt(C) ]So, putting them together:dy/dx = 2 * [ e^x ( (1 + 2e^x)/(2 sqrt(C)) - 1 ) ] / [ e^x ( (1 + 2e^x) - 2 sqrt(C) ) ) ]The e^x terms cancel:dy/dx = 2 * [ ( (1 + 2e^x)/(2 sqrt(C)) - 1 ) ] / [ (1 + 2e^x - 2 sqrt(C)) ]Let me simplify numerator and denominator:First, numerator inside the brackets:(1 + 2e^x)/(2 sqrt(C)) - 1 = [ (1 + 2e^x) - 2 sqrt(C) ] / (2 sqrt(C))Therefore, dy/dx = 2 * [ (1 + 2e^x - 2 sqrt(C)) / (2 sqrt(C)) ] / [ (1 + 2e^x - 2 sqrt(C)) ]Notice that (1 + 2e^x - 2 sqrt(C)) appears in both numerator and denominator, so they cancel out, provided they are not zero.Assuming 1 + 2e^x - 2 sqrt(C) ≠ 0, then dy/dx simplifies to:2 * [ 1 / (2 sqrt(C)) ] = 1 / sqrt(C)Therefore, dy/dx = 1 / sqrt(1 + e^x + e^{2x})Wait, that's a significant simplification! So all those terms canceled out, leading to dy/dx = 1 / sqrt(1 + e^x + e^{2x})But let's check the steps again to make sure I didn't make a mistake.Start with dy/dx = 2A' / (AB)Computed A' as (e^x + 2e^{2x})/(2 sqrt(C)) - e^xThen AB = e^x(1 + 2e^x) - 2e^x sqrt(C)Then, substituted into dy/dx:2*(A')/(AB) = 2*[ (e^x(1 + 2e^x)/(2 sqrt(C)) - e^x ) ] / [ e^x(1 + 2e^x - 2 sqrt(C)) ]Factor e^x in numerator:= 2*e^x [ (1 + 2e^x)/(2 sqrt(C)) - 1 ] / [ e^x(1 + 2e^x - 2 sqrt(C)) ]Cancel e^x:= 2*[ ( (1 + 2e^x)/(2 sqrt(C)) - 1 ) ] / [ (1 + 2e^x - 2 sqrt(C)) ]Convert the numerator:(1 + 2e^x)/(2 sqrt(C)) - 1 = (1 + 2e^x - 2 sqrt(C)) / (2 sqrt(C))Therefore, dy/dx = 2 * [ (1 + 2e^x - 2 sqrt(C)) / (2 sqrt(C)) ] / [ (1 + 2e^x - 2 sqrt(C)) ]Which is:2 * [1 / (2 sqrt(C)) ] = 1 / sqrt(C)Therefore, dy/dx = 1 / sqrt(1 + e^x + e^{2x})Wow, that simplifies really nicely! So despite the complicated-looking original function, its derivative is simply 1 over the square root of (1 + e^x + e^{2x}).Let me verify this result by checking a specific value. Let's choose x = 0.Compute y and dy/dx at x = 0.First, compute C = 1 + e^0 + e^{0} = 1 +1 +1 =3So sqrt(C) = sqrt(3) ≈ 1.732Original function y = ln [ (sqrt(3) -1 -1)/(sqrt(3) -1 +1) ] = ln [ (sqrt(3) -2)/sqrt(3) ]Compute numerator: sqrt(3) -2 ≈1.732 - 2 ≈ -0.268Denominator: sqrt(3) ≈1.732Therefore, y = ln(-0.268 /1.732). Wait, but ln of a negative number is undefined. That suggests a problem. Wait, but at x=0, the original function is undefined? That can't be. Let's check the original function again.Wait, at x=0, let's compute the numerator and denominator:Numerator: sqrt(1 +1 +1) -1 -1 = sqrt(3) -2 ≈1.732 -2 ≈-0.268Denominator: sqrt(3) -1 +1 = sqrt(3) ≈1.732So the argument of the logarithm is (-0.268)/1.732 ≈ -0.1547But ln of a negative number is undefined in real numbers. Therefore, the original function y is undefined at x=0, which suggests that x=0 is not in the domain. Therefore, the derivative at x=0 is also undefined, which aligns with the result since dy/dx =1/sqrt(C), and at x=0, C=3, so dy/dx =1/sqrt(3) ≈0.577. But since the original function is undefined at x=0, this is not a problem. Perhaps the domain of y is such that the argument of ln is positive. Let's find when (sqrt(1 + e^x + e^{2x}) - e^x -1)/(sqrt(1 + e^x + e^{2x}) - e^x +1) >0.Let me analyze the expression F = [sqrt(1 + e^x + e^{2x}) - e^x -1]/[sqrt(1 + e^x + e^{2x}) - e^x +1]Let me denote S = sqrt(1 + e^x + e^{2x}), then F = (S - e^x -1)/(S - e^x +1)We can analyze the sign of F.First, denominator: S - e^x +1. Since S = sqrt(1 + e^x + e^{2x}) > sqrt(e^{2x}) = e^x. Therefore, S - e^x >0, so denominator S - e^x +1 >0 +1 =1 >0. Therefore, denominator is always positive.Therefore, the sign of F is the same as the sign of the numerator: S - e^x -1.So, F >0 iff S - e^x -1 >0.Compute S - e^x -1 = sqrt(1 + e^x + e^{2x}) - e^x -1Let me check when this is positive.Let me square both sides to check, but need to be cautious as squaring can introduce extraneous solutions.Assume S - e^x -1 >0Then sqrt(1 + e^x + e^{2x}) > e^x +1Square both sides:1 + e^x + e^{2x} > (e^x +1)^2 = e^{2x} + 2e^x +1Subtract 1 + e^x + e^{2x} from both sides:0 > e^{2x} + 2e^x +1 - (1 + e^x + e^{2x}) ) = e^{2x} + 2e^x +1 -1 -e^x -e^{2x} = e^xSo 0 > e^x, which is never true since e^x >0 for all real x.Therefore, the inequality S - e^x -1 >0 is never true. Therefore, the numerator is always negative, denominator is always positive, so F is always negative. But ln of a negative number is undefined in real numbers. Wait, this is a problem. Therefore, the original function y is undefined for all real x? That can't be. Let me check with x approaching infinity.As x approaches infinity, e^{2x} dominates, so S = sqrt(1 + e^x + e^{2x}) ≈ e^x. Therefore, numerator: e^x - e^x -1 = -1. Denominator: e^x - e^x +1 =1. So F≈ -1/1 = -1. Therefore, ln(-1) is undefined. As x approaches negative infinity, e^{2x} approaches 0, so S ≈ sqrt(1 +0 +0) =1. Then numerator:1 -0 -1 =0. Denominator:1 -0 +1=2. So F=0/2=0. But ln(0) is undefined as well. So in fact, the original function y is undefined for all real x? That can't be right. Wait, there must be a mistake here. Let me check my earlier steps.Wait, maybe I made a mistake in simplifying F. Let's reconsider.Original F = [sqrt(1 + e^x + e^{2x}) - e^x -1]/[sqrt(1 + e^x + e^{2x}) - e^x +1]Let me check at x=1:Compute S = sqrt(1 + e + e^{2})Numerator: S - e -1Denominator: S - e +1Compute numerically:e ≈2.718, e^2≈7.389So S = sqrt(1 +2.718 +7.389) = sqrt(11.107)≈3.333Numerator≈3.333 -2.718 -1≈-0.385Denominator≈3.333 -2.718 +1≈1.615Therefore, F≈-0.385/1.615≈-0.238, so ln(-0.238) is undefined. So at x=1, y is undefined.Wait, but the problem statement gives y as ln of that fraction. If the problem is given, then maybe there's a typo, or perhaps we are to consider the derivative in the complex plane, but the question likely assumes real-valued functions. Therefore, perhaps there's a mistake in the problem statement, or perhaps I made an error in earlier steps.Alternatively, maybe the original expression is supposed to have absolute value inside the logarithm? Because otherwise, the argument is negative for all real x, making y undefined.Assuming that the problem is correct as stated, perhaps the answer is as derived, but the derivative only exists in complex plane. But since the problem asks for the derivative, and given the solution dy/dx =1 / sqrt(1 + e^x + e^{2x}), which is real and positive for all real x, but y itself is undefined for real x, this is a contradiction. Therefore, I must have made a mistake in the simplification steps.Wait, let's check again the step where I canceled (1 + 2e^x - 2 sqrt(C)) in numerator and denominator. But if (1 + 2e^x - 2 sqrt(C)) is zero, then the simplification is invalid. Let me check when that term is zero.Set 1 + 2e^x - 2 sqrt(C) =0=> 1 + 2e^x = 2 sqrt(1 + e^x + e^{2x})Square both sides:(1 + 2e^x)^2 = 4(1 + e^x + e^{2x})Left side:1 +4e^x +4e^{2x}Right side:4 +4e^x +4e^{2x}Subtract right side from left side:1 +4e^x +4e^{2x} -4 -4e^x -4e^{2x} = -3 ≠0Therefore, 1 + 2e^x - 2 sqrt(C) = -3/ something?Wait, no. Wait, when I set 1 + 2e^x - 2 sqrt(C) =0 and squared, we arrived at -3=0, which is a contradiction. Therefore, there is no real x for which 1 + 2e^x - 2 sqrt(C)=0. Hence, the cancellation is valid for all real x, and dy/dx =1 / sqrt(1 + e^x + e^{2x})But then, even though the original function y is undefined for all real x, the derivative expression is real and defined. This seems contradictory. Therefore, there must be a mistake in the problem statement, or perhaps my calculations.Alternatively, maybe I miscalculated when simplifying F. Let me check the step where I multiplied numerator and denominator by conjugates.Earlier, I found that:(S - e^x -1)(S + e^x +1) = -e^x(S - e^x +1)(S + e^x -1) = 3e^xThen, F = [S - e^x -1]/[S - e^x +1] = [ -e^x/(S + e^x +1) ] / [ 3e^x/(S + e^x -1) ] = [ -1/(S + e^x +1) ] * [ (S + e^x -1)/3 ]But that gives F = - (S + e^x -1)/(3(S + e^x +1))But S + e^x +1 = sqrt(1 + e^x + e^{2x}) + e^x +1This term is definitely positive for all real x. Similarly, S + e^x -1 is also positive? Let's check:At x=0, S = sqrt(3)≈1.732, so S + e^0 -1≈1.732 +1 -1=1.732>0At x approaches infinity, S≈e^x, so S + e^x -1≈2e^x -1>0At x approaches negative infinity, S≈1, so S + e^x -1≈1 +0 -1=0. Wait, as x approaches negative infinity, e^x approaches 0, so S approaches sqrt(1+0+0)=1, so S + e^x -1 approaches 1 +0 -1=0. So S + e^x -1 approaches 0 from the positive side as x approaches negative infinity.Therefore, S + e^x -1 is positive for all real x. Therefore, F = - (positive)/(positive) = negative, so F is negative for all real x, hence ln(F) is undefined in real numbers.Therefore, the problem as stated is invalid, since the function y is undefined for all real x.But since the problem asks to find the derivative, perhaps the intended answer is 1 / sqrt(1 + e^x + e^{2x}), assuming that the expression inside the log is taken as absolute value, or perhaps there's a typo in the original problem. Alternatively, maybe the original problem is a trick question, where despite y being undefined, the formal derivative simplifies to that expression.Alternatively, perhaps I made a mistake in the sign during simplification. Let me check again:From the original problem:y = ln [ (sqrt(1 + e^x + e^{2x}) - e^x -1)/(sqrt(1 + e^x + e^{2x}) - e^x +1) ]But when I analyzed F = numerator/denominator, I found that F is negative for all real x. Therefore, the argument of ln is negative, which is undefined. Therefore, the original function y is undefined for all real x, which contradicts the problem statement.This suggests that there's a mistake in the problem statement. Perhaps the numerator and denominator were meant to have different signs? For example, if the numerator was sqrt(1 + e^x + e^{2x}) + e^x +1 or something else.Alternatively, maybe I misread the problem. Let me check the original problem statement again.The problem states:y = ln [ (sqrt(1 + e^x + e^{2x}) - e^x -1)/(sqrt(1 + e^x + e^{2x}) - e^x +1) ]Yes, that's what was given. So according to this, y is undefined for real x. Therefore, the problem might be ill-posed, or perhaps the derivative is intended in the complex plane, which is beyond the scope here.Given that the problem asks for the derivative, and according to my calculations, despite y being undefined, the derivative simplifies to 1 / sqrt(1 + e^x + e^{2x}), which is a valid expression. So perhaps the expected answer is this expression.Alternatively, maybe the problem has a typo, and the correct function is the inverse hyperbolic function or similar, whose derivative matches this.Alternatively, let's check if 1 / sqrt(1 + e^x + e^{2x}) can be simplified further. Let me see:1 + e^x + e^{2x} = e^{2x} + e^x +1, which is similar to the denominator of the derivative of inverse hyperbolic functions. For example, derivative of inverse sinh is 1 / sqrt(1 + x^2), but here we have 1 / sqrt(1 + e^x + e^{2x})Alternatively, let me note that 1 + e^x + e^{2x} = (e^{x} + 0.5)^2 + 3/4, but this might not help.Alternatively, perhaps factor 1 + e^x + e^{2x} as follows:Multiply numerator and denominator by e^{-x}:1 + e^x + e^{2x} = e^{2x} + e^x +1 = e^{-x} (e^{3x} + e^{2x} + e^x )But this also doesn't seem helpful.Alternatively, notice that 1 + e^x + e^{2x} = (e^{3x} -1)/(e^x -1) when e^x ≠1, but for x ≠0, but not sure if this helps.Alternatively, the expression 1 + e^x + e^{2x} is similar to the expansion of (e^x + ω + ω^2) where ω is a cube root of unity, but this is complex and likely not relevant here.Given that, and given that the derivative simplifies to 1 / sqrt(1 + e^x + e^{2x}), and assuming that the problem is correctly stated, despite the domain issues, I think this is the answer expected.Therefore, after careful consideration and multiple checks, the derivative dy/dx is equal to 1 divided by the square root of (1 plus e^x plus e^{2x}).Final AnswerThe derivative is boxed{dfrac{1}{sqrt{1 + e^{x} + e^{2x}}}}.</think>To find the derivative of the function y = ln left( frac{sqrt{1 + e^x + e^{2x}} - e^x - 1}{sqrt{1 + e^x + e^{2x}} - e^x + 1} right),we start by using the properties of logarithms to rewrite the function as:y = ln(text{numerator}) - ln(text{denominator}),where the numerator is ( A = sqrt{1 + e^x + e^{2x}} - e^x - 1 ) and the denominator is ( B = sqrt{1 + e^x + e^{2x}} - e^x + 1 ).The derivative of ( y ) with respect to ( x ) is given by:frac{dy}{dx} = frac{A'}{A} - frac{B'}{B}.Since ( A ) and ( B ) differ only by a constant term, their derivatives ( A' ) and ( B' ) are the same. We compute the derivative of ( A ):A' = frac{d}{dx} left( sqrt{1 + e^x + e^{2x}} right) - frac{d}{dx} (e^x) - frac{d}{dx} (1).The derivative of ( sqrt{1 + e^x + e^{2x}} ) is:frac{e^x + 2e^{2x}}{2sqrt{1 + e^x + e^{2x}}}.Thus,A' = frac{e^x + 2e^{2x}}{2sqrt{1 + e^x + e^{2x}}} - e^x.Simplifying ( A' ):A' = frac{e^x + 2e^{2x} - 2e^x sqrt{1 + e^x + e^{2x}}}{2sqrt{1 + e^x + e^{2x}}}.Next, we note that the difference ( B - A = 2 ). Using this, we find:frac{dy}{dx} = A' left( frac{B - A}{AB} right) = A' left( frac{2}{AB} right).The product ( AB ) simplifies to:AB = (sqrt{1 + e^x + e^{2x}} - e^x - 1)(sqrt{1 + e^x + e^{2x}} - e^x + 1) = -e^x.However, after careful simplification and cancellation, we find that the terms involving ( A' ) and ( AB ) cancel out, leading to:frac{dy}{dx} = frac{1}{sqrt{1 + e^x + e^{2x}}}.Thus, the derivative of the given function is:boxed{dfrac{1}{sqrt{1 + e^{x} + e^{2x}}}}.```,<think>Okay, let's see. I need to find the derivative of this function y with respect to x. The function is a natural logarithm of a fraction, and both the numerator and the denominator have similar square root expressions involving e^x and e^(2x). Hmm, this looks a bit complicated, but maybe I can simplify it before jumping into differentiation.First, let me write down the function again to make sure I have it right:y = ln [ (sqrt(1 + e^x + e^(2x)) - e^x - 1) / (sqrt(1 + e^x + e^(2x)) - e^x + 1) ]So, it's the natural log of a fraction. The numerator is sqrt(1 + e^x + e^(2x)) minus e^x minus 1, and the denominator is the same sqrt expression minus e^x plus 1. I remember that the derivative of ln(u/v) is (u'/u - v'/v), right? Because ln(u/v) = ln u - ln v, so the derivative would be (u'/u) - (v'/v). So maybe I can split the logarithm into two parts first. Let's try that.Let me denote the numerator as A and the denominator as B. So, y = ln(A) - ln(B). Then, dy/dx = (A'/A) - (B'/B). But before differentiating, perhaps simplifying A and B would help. Let me look at A and B:A = sqrt(1 + e^x + e^(2x)) - e^x - 1B = sqrt(1 + e^x + e^(2x)) - e^x + 1So, the only difference between A and B is the constant term at the end: -1 vs. +1. That might be useful. Maybe there's a way to relate A and B or combine them somehow. Alternatively, maybe the entire fraction inside the log can be simplified. Let's check.First, let's look at the expression inside the square root: 1 + e^x + e^(2x). That looks like a quadratic in e^x. Let me write it as e^(2x) + e^x + 1. Hmm, maybe that factors or has some other structure. Let me check:If I let t = e^x, then the expression becomes t^2 + t + 1. That's a quadratic, which doesn't factor over real numbers since the discriminant is 1 - 4 = -3. So, it's irreducible. Therefore, sqrt(t^2 + t + 1) is just sqrt(e^(2x) + e^x + 1). Maybe completing the square? Let's try:t^2 + t + 1 = t^2 + t + (1/4) + 3/4 = (t + 1/2)^2 + 3/4Hmm, but that doesn't seem particularly helpful here. Maybe not necessary. Alternatively, maybe there's a substitution that can simplify the expression.Alternatively, maybe rationalizing the numerator or denominator of the fraction inside the log. Since the numerator and denominator both have sqrt(1 + e^x + e^(2x)) - something, maybe multiplying numerator and denominator by the conjugate to simplify.Wait, let's consider the entire fraction:[ sqrt(1 + e^x + e^(2x)) - e^x - 1 ] / [ sqrt(1 + e^x + e^(2x)) - e^x + 1 ]If I let C = sqrt(1 + e^x + e^(2x)) - e^x, then the numerator is C - 1 and the denominator is C + 1. So, the fraction becomes (C - 1)/(C + 1). That's a simpler expression. Let me verify:C = sqrt(1 + e^x + e^(2x)) - e^xThen numerator: C - 1 = sqrt(...) - e^x - 1Denominator: C + 1 = sqrt(...) - e^x + 1Yes, exactly. So the fraction simplifies to (C - 1)/(C + 1). Maybe this is helpful. Let's see.So, y = ln[(C - 1)/(C + 1)] = ln(C - 1) - ln(C + 1). Then dy/dx = [ (C' )/(C - 1) ] - [ (C')/(C + 1) ] = C' [ 1/(C - 1) - 1/(C + 1) ]Combine the terms: C' [ (C + 1 - (C - 1)) / ( (C -1)(C +1) ) ] = C' [ (2) / (C^2 - 1) ]So dy/dx = (2C') / (C^2 - 1)Hmm, that seems simpler. So if I can compute C and C', then substitute back, that would work.First, let me compute C:C = sqrt(1 + e^x + e^(2x)) - e^xLet me compute C^2:C^2 = [sqrt(1 + e^x + e^(2x)) - e^x]^2 = (1 + e^x + e^(2x)) - 2 e^x sqrt(1 + e^x + e^(2x)) + e^(2x)Simplify:= 1 + e^x + e^(2x) + e^(2x) - 2 e^x sqrt(...) = 1 + e^x + 2 e^(2x) - 2 e^x sqrt(1 + e^x + e^(2x))But perhaps I can also compute C^2 - 1:C^2 - 1 = [1 + e^x + 2 e^(2x) - 2 e^x sqrt(1 + e^x + e^(2x))] - 1= e^x + 2 e^(2x) - 2 e^x sqrt(1 + e^x + e^(2x))Hmm, not sure if that helps. Maybe let's compute C + e^x = sqrt(1 + e^x + e^(2x))So, sqrt(1 + e^x + e^(2x)) = C + e^xTherefore, 1 + e^x + e^(2x) = (C + e^x)^2 = C^2 + 2 C e^x + e^(2x)Subtract e^(2x) from both sides:1 + e^x = C^2 + 2 C e^xSo, 1 + e^x = C^2 + 2 C e^xLet me rearrange this:C^2 + 2 C e^x - (1 + e^x) = 0Not sure if helpful. Maybe solving for C in terms of e^x? Alternatively, perhaps using this relation to express C^2 in terms of C and e^x:From above, C^2 = 1 + e^x - 2 C e^xWait, that might be useful when substituting back into dy/dx.Given that dy/dx = 2 C' / (C^2 - 1). And C^2 -1 was calculated earlier as e^x + 2 e^(2x) - 2 e^x sqrt(...). But maybe using the relation from above:From 1 + e^x = C^2 + 2 C e^x, so C^2 = 1 + e^x - 2 C e^xTherefore, C^2 - 1 = e^x - 2 C e^xSo, dy/dx = 2 C' / (e^x - 2 C e^x ) = 2 C' / [ e^x (1 - 2 C) ]Alternatively, let's see if we can find C'. Let's compute C' now.C = sqrt(1 + e^x + e^(2x)) - e^xCompute derivative C':First term: derivative of sqrt(1 + e^x + e^(2x)) is (1/(2 sqrt(...))) * (e^x + 2 e^(2x))Second term: derivative of -e^x is -e^xTherefore:C' = [ (e^x + 2 e^(2x)) / (2 sqrt(1 + e^x + e^(2x))) ] - e^xBut sqrt(1 + e^x + e^(2x)) is equal to C + e^x, as established earlier.Therefore, substitute that in:C' = [ (e^x + 2 e^(2x)) / (2 (C + e^x)) ] - e^xLet me simplify this expression:First term: (e^x + 2 e^(2x)) / (2 (C + e^x))Second term: - e^xSo, combine them:= [ (e^x + 2 e^(2x)) - 2 e^x (C + e^x) ] / [ 2 (C + e^x) ]Expand the numerator:= [ e^x + 2 e^(2x) - 2 e^x C - 2 e^(2x) ] / [ 2 (C + e^x) ]Simplify numerator:e^x + 2 e^(2x) - 2 e^x C - 2 e^(2x) = e^x - 2 e^x CFactor e^x:= e^x (1 - 2 C)Therefore, C' = [ e^x (1 - 2 C) ] / [ 2 (C + e^x) ]So, going back to dy/dx = 2 C' / (C^2 - 1). Substitute C' here:dy/dx = 2 * [ e^x (1 - 2 C) / (2 (C + e^x)) ] / (C^2 - 1 )The 2's cancel:= [ e^x (1 - 2 C) / (C + e^x) ] / (C^2 - 1 )But earlier, we found that C^2 - 1 = e^x - 2 C e^xSo, substitute that denominator:= [ e^x (1 - 2 C) / (C + e^x) ] / ( e^x (1 - 2 C ) )Wait, hold on. Let me check:From above, C^2 -1 = e^x - 2 C e^x. So that's equal to e^x (1 - 2 C )Therefore, dy/dx becomes:[ e^x (1 - 2 C ) / (C + e^x) ] / [ e^x (1 - 2 C ) ] = [ numerator / denominator ] / [ same as numerator ]Wait, the numerator of the first fraction is e^x (1 - 2 C ), and the denominator is (C + e^x). Then the entire expression is divided by e^x (1 - 2 C ). So, this simplifies to:[ e^x (1 - 2 C ) / (C + e^x) ] * [ 1 / (e^x (1 - 2 C )) ] = 1 / (C + e^x )Because the e^x (1 - 2 C ) terms cancel out, leaving 1/(C + e^x )But earlier, we had that sqrt(1 + e^x + e^(2x)) = C + e^x. So, 1/(C + e^x ) = 1 / sqrt(1 + e^x + e^(2x))Therefore, dy/dx = 1 / sqrt(1 + e^x + e^(2x))Wait, is that possible? That seems surprisingly simple. Let me verify each step to make sure I didn't make an error.First, we set C = sqrt(1 + e^x + e^(2x)) - e^x. Then, we found that sqrt(...) = C + e^x. So, 1/(C + e^x) = 1 / sqrt(...). Then, through the differentiation steps, we ended up with dy/dx = 1 / sqrt(...). That seems too straightforward. Let me check.Starting from dy/dx = 2C' / (C^2 - 1). Then, substituting C':C' = [ e^x (1 - 2C ) ] / [ 2 (C + e^x ) ]Therefore, 2C' = [ e^x (1 - 2C ) ] / (C + e^x )Thus, dy/dx = [ e^x (1 - 2C ) / (C + e^x ) ] / (C^2 - 1 )But C^2 - 1 = e^x (1 - 2C )Thus, substituting:dy/dx = [ e^x (1 - 2C ) / (C + e^x ) ] / [ e^x (1 - 2C ) ] = 1 / (C + e^x )But C + e^x = sqrt(1 + e^x + e^(2x)), so dy/dx = 1 / sqrt(1 + e^x + e^(2x)).That does check out. So, all those steps, even though there were substitutions and simplifications, led to a simple result. That's pretty neat.Alternatively, maybe there's a smarter substitution from the start. Let me consider the original function again:y = ln [ ( sqrt(1 + e^x + e^(2x}) - e^x - 1 ) / ( sqrt(1 + e^x + e^(2x}) - e^x + 1 ) ]Suppose I let u = sqrt(1 + e^x + e^(2x}) - e^x. Then, the numerator is u - 1 and denominator is u + 1, so the fraction is (u - 1)/(u + 1). Therefore, y = ln [ (u -1)/(u +1) ].Then, dy/dx = derivative of ln[(u -1)/(u +1)] = [ ( (u' )/(u -1) ) - ( (u')/(u +1) ) ] = u' [ 1/(u -1) - 1/(u +1) ]Which simplifies to u' [ (u +1 - u +1 ) / ( (u -1)(u +1) ) ] = u' [ 2 / (u^2 -1) ]So, dy/dx = 2 u' / (u^2 -1 )But if I compute u^2:u = sqrt(1 + e^x + e^(2x}) - e^xSo, u + e^x = sqrt(1 + e^x + e^(2x})Then, squaring both sides: (u + e^x)^2 = 1 + e^x + e^(2x)Expanding left side: u^2 + 2 u e^x + e^(2x) = 1 + e^x + e^(2x)Subtract e^(2x) from both sides: u^2 + 2 u e^x = 1 + e^xThus, u^2 -1 = -2 u e^x + e^xSo, u^2 -1 = e^x (1 - 2u )Therefore, dy/dx = 2 u' / [ e^x (1 - 2u ) ]But we can also compute u':u = sqrt(1 + e^x + e^(2x}) - e^xSo, u' = derivative of sqrt(...) - derivative of e^xDerivative of sqrt(1 + e^x + e^(2x)) is [ (e^x + 2 e^(2x)) / (2 sqrt(...)) ] = [ e^x (1 + 2 e^x ) ] / (2 sqrt(...)) But sqrt(...) is u + e^x, so:u' = [ e^x (1 + 2 e^x ) ] / (2 (u + e^x )) - e^xCombine terms:= [ e^x (1 + 2 e^x ) - 2 e^x (u + e^x ) ] / [ 2 (u + e^x ) ]Expand numerator:= e^x + 2 e^(2x) - 2 e^x u - 2 e^(2x )Simplify:= e^x - 2 e^x uFactor out e^x:= e^x (1 - 2 u )Therefore, u' = e^x (1 - 2 u ) / [ 2 (u + e^x ) ]So, substituting back into dy/dx:dy/dx = 2 * [ e^x (1 - 2 u ) / ( 2 (u + e^x ) ) ] / [ e^x (1 - 2 u ) ]Simplify numerator:2 * [ numerator ] becomes [ e^x (1 - 2 u ) / (u + e^x ) ]Divide by denominator e^x (1 - 2 u ):= [ e^x (1 - 2 u ) / (u + e^x ) ] / [ e^x (1 - 2 u ) ]Cancel out e^x (1 - 2 u ) terms:= 1 / (u + e^x )But u + e^x = sqrt(1 + e^x + e^(2x)), so dy/dx = 1 / sqrt(1 + e^x + e^(2x))Same result as before. So, regardless of substituting C or u, I end up with the same answer. Therefore, the derivative is 1 over the square root of (1 + e^x + e^{2x}).But let me verify this result with a quick check. Let's pick a value of x and compute both the original function and the derivative numerically to see if they match.Suppose x = 0. Let's compute y and dy/dx at x=0.First, compute sqrt(1 + e^0 + e^(0)) = sqrt(1 + 1 + 1) = sqrt(3)Numerator: sqrt(3) - e^0 -1 = sqrt(3) -1 -1 = sqrt(3) -2 ≈ 1.732 - 2 ≈ -0.268Denominator: sqrt(3) - e^0 +1 = sqrt(3) -1 +1 = sqrt(3) ≈ 1.732Thus, y = ln( (-0.268)/1.732 ) ≈ ln(-0.1547). Wait, ln of a negative number is undefined. Hmm, that can't be right. Wait, what's happening here.Wait, at x=0, the numerator is sqrt(3) -1 -1 ≈ 1.732 - 2 ≈ -0.268, denominator is sqrt(3) -1 +1 = sqrt(3) ≈1.732. Therefore, the fraction is (-0.268)/1.732 ≈ -0.1547. But ln of a negative number is undefined. So, the function y is not real at x=0. So, perhaps x=0 is not in the domain. Let's check.Let me check the original expression. The argument of the logarithm must be positive. So, let's see when is [ sqrt(1 + e^x + e^(2x) ) - e^x -1 ] / [ sqrt(1 + e^x + e^(2x) ) - e^x +1 ] positive.Let me denote the numerator as N and denominator as D.So, need N/D >0. Therefore, N and D must have the same sign.Let me compute N and D for x=1.At x=1:sqrt(1 + e + e^2) ≈ sqrt(1 + 2.718 + 7.389) ≈ sqrt(11.107) ≈ 3.333N = 3.333 - e^1 -1 ≈ 3.333 - 2.718 -1 ≈ -0.385D = 3.333 - e^1 +1 ≈ 3.333 -2.718 +1 ≈ 1.615So, N/D ≈ -0.385 /1.615 ≈ -0.238, which is negative. So again, log of negative number. Hmm, maybe this function is defined only for certain x where N and D are both positive or both negative. Let's check.Alternatively, perhaps the expression inside the log simplifies to something positive. Let me check.Let me analyze N and D.N = sqrt(1 + e^x + e^{2x}) - e^x -1D = sqrt(1 + e^x + e^{2x}) - e^x +1Let me denote S = sqrt(1 + e^x + e^{2x})Therefore, N = S - e^x -1, D = S - e^x +1Compute N/D:(S - e^x -1)/(S - e^x +1)Let me compute this expression. Let me set t = S - e^x, so N = t -1, D = t +1. So N/D = (t -1)/(t +1). Let's see when this is positive.(t -1)/(t +1) >0 when (t -1 and t +1 have the same sign). So either:Case 1: t -1 >0 and t +1 >0 ⇒ t >1 and t > -1. Since t >1 already implies t > -1, so t >1.Case 2: t -1 <0 and t +1 <0 ⇒ t <1 and t < -1. But t = sqrt(1 + e^x + e^{2x}) - e^x. Let's see what t is.Compute t = sqrt(1 + e^x + e^{2x}) - e^x. Let's rationalize:Multiply numerator and denominator by sqrt(1 + e^x + e^{2x}) + e^x:t = [ (sqrt(1 + e^x + e^{2x}) - e^x ) * (sqrt(1 + e^x + e^{2x}) + e^x ) ] / [ sqrt(1 + e^x + e^{2x}) + e^x ]Numerator: (1 + e^x + e^{2x}) - e^{2x} =1 + e^xDenominator: sqrt(...) + e^xTherefore, t = (1 + e^x ) / [ sqrt(1 + e^x + e^{2x}) + e^x ]Since 1 + e^x and denominator are both positive for all real x, t is positive. Therefore, t >0 for all x. Therefore, t +1 >1 >0. So for N/D = (t -1)/(t +1) to be positive, we need t -1 >0, i.e., t >1. So when is t >1?t = (1 + e^x ) / [ sqrt(1 + e^x + e^{2x}) + e^x ] >1Multiply both sides by denominator (positive):1 + e^x > sqrt(1 + e^x + e^{2x}) + e^xSubtract e^x:1 > sqrt(1 + e^x + e^{2x})But sqrt(1 + e^x + e^{2x}) > sqrt(e^{2x}) = e^x. So, when x >0, e^x >1, so sqrt(...) > e^x >1, so 1 > sqrt(...) is false. When x =0, sqrt(1 +1 +1)=sqrt(3)≈1.732 >1. For x <0, e^x <1, so sqrt(1 + e^x + e^{2x}) ?Let's compute when x approaches -infty. e^x approaches 0, so sqrt(1 +0 +0 )=1. So sqrt(...) approaches 1. Therefore, t approaches (1 +0)/ (1 +0 )=1. So as x approaches -infty, t approaches1. Therefore, for x approaching -infty, t approaches1 from above or below?Wait, let's take x approaching -infty. Then e^x ≈0, so t ≈ (1 +0)/ (1 +0 )=1. Let me check x approaching negative values. Let x = -10, so e^-10 ≈0. So t≈1/1=1. Let's take x = -1:e^-1 ≈0.3679sqrt(1 + 0.3679 + (0.3679)^2 ) ≈ sqrt(1 +0.3679 +0.1353) ≈ sqrt(1.5032)≈1.226Therefore, t = (1 +0.3679 ) / (1.226 +0.3679 ) ≈1.3679 /1.5939≈0.858So t ≈0.858 <1. Therefore, for x negative, t <1, so N/D is (t -1)/(t +1), which is negative. But t +1 >0, so (negative)/(positive)=negative. Therefore, N/D is negative for x negative. Which would make the argument of ln negative. But ln is undefined for negative numbers. Therefore, the function y is only defined when N/D >0, which requires t >1. But from above, t >1 only if x is such that t >1. However, when x is positive, t is (1 + e^x ) / [sqrt(1 + e^x + e^{2x}) + e^x ]Let's check x approaching +infty. Then e^x dominates. So sqrt(1 + e^x + e^{2x}) ≈ e^x sqrt(1 + e^{-x} + e^{-2x}) ≈ e^x (1 + (1/2)e^{-x} + ... )Therefore, sqrt(...) ≈ e^x + (1/2) + ...Then, t = (1 + e^x ) / [ e^x + (1/2) + ... + e^x ] ≈ (e^x ) / (2 e^x ) ≈1/2. So t approaches1/2 as x approaches +infty. Therefore, t approaches1/2 <1. Therefore, for x positive, t <1. When x approaches -infty, t approaches1 from below. Wait, when x approaches negative infinity, e^x approaches0, so t approaches (1 +0 )/(1 +0 )=1. But from x=-1 example, t was≈0.858 <1. So, actually, as x approaches -infty, t approaches1 from below. So t is always less than1, except maybe at some finite x?Wait, let's check at x= -infty, t approaches1, but when x is very large negative, say x approaching -infty, e^x approaches0, so sqrt(1 +0 +0 )=1, so t=(1 +0 )/(1 +0 )=1. So, as x approaches -infty, t approaches1 from below. But when x is finite, t is less than1. Therefore, t never exceeds1. Therefore, N/D = (t -1)/(t +1 ) is always negative. Therefore, the argument of ln is negative, which is undefined in reals. Wait, that's a problem.But the original problem statement didn't specify the domain, so perhaps assuming that we're working in a complex plane? Or maybe there's a mistake in the problem. Alternatively, maybe there's an error in my analysis.Wait, let's compute for x= ln(1), so x=0, which we saw gives N/D negative. For x=ln(2), x≈0.693:Compute sqrt(1 + e^0.693 + e^(1.386))≈sqrt(1 +2 +4)=sqrt(7)≈2.6458N=2.6458 -2 -1≈-0.3542D=2.6458 -2 +1≈1.6458Thus, N/D≈-0.3542/1.6458≈-0.215, still negative. So even at x=ln(2), it's negative. Hmm. What about x= -10, as before:sqrt(1 + e^-10 + e^-20 )≈1. So N≈1 -0 -1=0, D≈1 -0 +1=2. So N/D≈0/2=0. So ln(0) is undefined. Hmm. So maybe there is no real x where N/D is positive? But that contradicts the problem statement. Unless I made a mistake.Wait, maybe there was a typo in the problem statement. Let me check again.Original problem:y = ln [ (sqrt(1 + e^x + e^{2x}) - e^x -1 ) / (sqrt(1 + e^x + e^{2x}) - e^x +1 ) ]Alternatively, maybe there's a sign error. Suppose the numerator was sqrt(...) - e^x +1 and denominator sqrt(...) - e^x -1. Then, at x=0, numerator≈sqrt(3) -1 +1≈sqrt(3)≈1.732, denominator≈sqrt(3) -1 -1≈-0.268. Then N/D would be negative. Still undefined. Hmm.Alternatively, maybe the problem is correctly stated, but the domain is complex. But in that case, the derivative would still be as computed. But the problem says "Find the derivative," so maybe expecting a real derivative, but it's possible that the expression inside the log is negative, but treated as complex logarithm. But usually in calculus problems, unless stated otherwise, we assume real functions. Therefore, this seems contradictory.Wait, perhaps the expression inside the log can be rewritten in terms of hyperbolic functions or something else, leading to a real expression.Alternatively, perhaps there's an error in my earlier analysis. Let me check when N/D is positive.Wait, N = sqrt(1 + e^x + e^{2x}) - e^x -1Let me write sqrt(1 + e^x + e^{2x}) as sqrt( (e^{2x} + e^x +1 ) )But 1 + e^x + e^{2x} is similar to e^{2x} + e^x +1, which is a quadratic in e^x. Let me set t = e^x, then the expression becomes t^2 + t +1. The sqrt of that is sqrt(t^2 + t +1). So, N = sqrt(t^2 + t +1) - t -1Similarly, D = sqrt(t^2 + t +1) - t +1So, N/D = [sqrt(t^2 + t +1) - t -1]/[sqrt(t^2 + t +1) - t +1]Let me analyze this expression for t >0.Let me denote s = sqrt(t^2 + t +1). Then N = s - t -1, D = s - t +1So N/D = (s - t -1)/(s - t +1). Let me compute this:Multiply numerator and denominator by (s - t +1):Wait, or maybe rationalize:Multiply numerator and denominator by (s - t -1):Wait, no. Alternatively, express s in terms of t.Alternatively, let's consider:s = sqrt(t^2 + t +1)Let me compute s - t:s - t = sqrt(t^2 + t +1) - tMultiply numerator and denominator by sqrt(t^2 + t +1) + t:s - t = [ (sqrt(t^2 + t +1) - t ) (sqrt(t^2 + t +1) + t ) ] / (sqrt(t^2 + t +1) + t )= [ (t^2 + t +1 ) - t^2 ] / (sqrt(t^2 + t +1 ) + t )= (t +1 ) / (sqrt(t^2 + t +1 ) + t )Therefore, s - t = (t +1 ) / (s + t )Therefore, s - t -1 = [ (t +1 ) / (s + t ) ] -1 = [ t +1 - (s + t ) ] / (s + t ) = (1 - s ) / (s + t )Similarly, s - t +1 = [ (t +1 ) / (s + t ) ] +1 = [ t +1 + s + t ] / (s + t ) = (s + 2 t +1 ) / (s + t )Therefore, N/D = [ (1 - s ) / (s + t ) ] / [ (s + 2 t +1 ) / (s + t ) ] = (1 - s ) / (s + 2 t +1 )But s = sqrt(t^2 + t +1 ), so 1 - s =1 - sqrt(t^2 + t +1 )But sqrt(t^2 + t +1 ) > sqrt(t^2 ) = |t |. Since t >0 (since t = e^x >0), sqrt(t^2 + t +1 ) > t. Therefore, 1 - sqrt(t^2 + t +1 ) <1 - t. Since t>0, if t >=1, 1 - t <=0. If t <1, 1 - t >0, but sqrt(t^2 + t +1 ) >1, so 1 - sqrt(...) <0. Therefore, 1 - s is always negative, and denominator s + 2 t +1 is always positive (since s >0, t>0). Therefore, N/D is negative for all t>0. Therefore, the argument of ln is negative for all real x. Therefore, the function y is undefined for real numbers. Hence, the derivative doesn't exist in real numbers.But the problem asks to "find the derivative," so perhaps there's a miscalculation here. Alternatively, maybe the original problem has a typo, or perhaps the expression inside the log is actually positive if we consider different simplifications.Wait, maybe I miscalculated N/D. Let me check:Wait, s - t -1 = sqrt(t^2 + t +1 ) - t -1Let me compute this for t=1 (x=0):sqrt(1 +1 +1 ) -1 -1 = sqrt(3) -2 ≈1.732 -2≈-0.268Similarly, for t approaching0:sqrt(0 +0 +1 ) -0 -1 =1 -0 -1=0. So N approaches0.D= sqrt(1 +0 +1 ) -0 +1= sqrt(2) +1≈1.414 +1=2.414Therefore, N/D approaches0 as t approaches0. So at x approaching -infty, N/D approaches0 from the negative side (since N approaches0 from below? Wait, when t approaches0, s approaches1, so N=1 -0 -1=0. Wait, s=1 when t=0, so N=1 -0 -1=0. D=1 -0 +1=2. Therefore, N/D=0/2=0. So as x approaches -infty, t approaches0, and N/D approaches0. But since N approaches0 from below? Wait, when t approaches0 from positive side, s approaches1 from above? Wait, s= sqrt(t^2 + t +1 ). For t approaching0+, s= sqrt(0 +0 +1 )=1. So N=1 -0 -1=0. So N approaches0. D approaches2. So N/D approaches0. But the sign of N when approaching t=0 from the right?Let me take t=0.1:s=sqrt(0.01 +0.1 +1 )=sqrt(1.11)≈1.053N=1.053 -0.1 -1≈-0.047D=1.053 -0.1 +1≈1.953So N/D≈-0.047 /1.953≈-0.024. So approaches0 from below.Similarly, for t approaching0 from the left? But t=e^x >0 for all real x, so t is always positive. Therefore, N/D approaches0 from below (negative side). Therefore, the expression inside the log approaches0 from below, which would be ln(0^-), which is undefined (complex infinity). Therefore, the function y is undefined for all real x. But that contradicts the problem statement. So either there's a mistake in the problem or I have a miscalculation.Alternatively, perhaps the expression is supposed to have absolute value inside the log? If so, then the derivative would still be the same, except for the sign. But the problem doesn't mention absolute value.Alternatively, maybe the problem is correctly stated, and despite the argument being negative, we proceed formally and find the derivative, which we did, resulting in dy/dx =1 / sqrt(1 + e^x + e^{2x} ). Perhaps even though the original function is undefined for real numbers, the derivative expression is valid in complex analysis, but the problem is given in a real analysis context. This is confusing.Alternatively, maybe I made a mistake in the derivative calculation. Let me rederive it quickly.Given y = ln [ (sqrt(1 + e^x + e^{2x}) - e^x -1 ) / (sqrt(1 + e^x + e^{2x}) - e^x +1 ) ]Let me use the property that ln(a/b) = ln a - ln b, so:y = ln(A) - ln(B), where A = sqrt(...) -e^x -1, B= sqrt(...) -e^x +1Then, dy/dx = (A'/A) - (B'/B)Compute A and B:A = sqrt(1 + e^x + e^{2x}) - e^x -1B = sqrt(1 + e^x + e^{2x}) - e^x +1So, A and B differ by the constant term: -1 vs +1.Compute A' and B':First, derivative of sqrt(1 + e^x + e^{2x}):Let me denote f(x) = sqrt(1 + e^x + e^{2x})Then f'(x) = [ (e^x + 2 e^{2x}) ] / (2 f(x) )Therefore, A' = f'(x) - e^x -0 = [ (e^x + 2 e^{2x}) / (2 f(x) ) ] - e^xSimilarly, B' = f'(x) - e^x +0 = same as A'Therefore, A' = B'So, dy/dx = A' (1/A - 1/B )Let me compute A':A' = [ (e^x + 2 e^{2x}) / (2 f(x) ) ] - e^x= [ e^x (1 + 2 e^x ) / (2 f(x) ) ] - e^xFactor e^x:= e^x [ (1 + 2 e^x ) / (2 f(x) ) -1 ]Combine terms:= e^x [ (1 + 2 e^x - 2 f(x) ) / (2 f(x) ) ]Now, compute 1/A -1/B:1/A -1/B = [ B - A ] / (A B )But B - A = [ sqrt(...) - e^x +1 ] - [ sqrt(...) - e^x -1 ] = 2Therefore, 1/A -1/B = 2 / (A B )Thus, dy/dx = A' * 2 / (A B )But A' = e^x [ (1 + 2 e^x - 2 f(x) ) / (2 f(x) ) ]Therefore:dy/dx = e^x [ (1 + 2 e^x - 2 f(x) ) / (2 f(x) ) ] * 2 / (A B )Simplify:= e^x (1 + 2 e^x - 2 f(x) ) / ( f(x) A B )But A = f(x) - e^x -1B = f(x) - e^x +1Therefore, A B = (f(x) - e^x )^2 -1Let me compute (f(x) - e^x )^2 = [ sqrt(1 + e^x + e^{2x}) - e^x ]^2 =1 + e^x + e^{2x} -2 e^x sqrt(...) + e^{2x}Wait, earlier computed:C^2 = [sqrt(...) -e^x]^2 =1 + e^x + e^{2x} -2 e^x sqrt(...) + e^{2x} =1 + e^x + 2 e^{2x} - 2 e^x sqrt(...)Wait, not sure. Let me compute (f(x) - e^x )^2:= f(x)^2 - 2 e^x f(x) + e^{2x}But f(x)^2 =1 + e^x + e^{2x}Therefore:=1 + e^x + e^{2x} - 2 e^x f(x) + e^{2x} =1 + e^x + 2 e^{2x} - 2 e^x f(x)Therefore, A B = (f(x) - e^x )^2 -1 = [1 + e^x + 2 e^{2x} - 2 e^x f(x) ] -1 = e^x + 2 e^{2x} - 2 e^x f(x )= e^x (1 + 2 e^x ) - 2 e^x f(x )= e^x [1 + 2 e^x - 2 f(x ) ]Therefore, A B = e^x (1 + 2 e^x - 2 f(x ) )Notice that in the expression for dy/dx, we have:dy/dx = e^x (1 + 2 e^x - 2 f(x ) ) / ( f(x ) A B )But since A B = e^x (1 + 2 e^x - 2 f(x ) ), substituting:dy/dx = e^x (1 + 2 e^x - 2 f(x ) ) / ( f(x ) * e^x (1 + 2 e^x - 2 f(x ) ) )Cancel e^x and (1 + 2 e^x - 2 f(x ) ) terms:dy/dx =1 / f(x )Since f(x )= sqrt(1 + e^x + e^{2x})Therefore, dy/dx =1 / sqrt(1 + e^x + e^{2x})So, regardless of the domain issues, the derivative simplifies to this expression. So despite the original function being undefined for real x, the derivative expression is valid where the original function is defined (which might be in complex plane). But since the problem just asks to find the derivative, presumably in the real domain where it's defined, even though such x might not exist, we can still give the answer as 1 over sqrt(1 + e^x + e^{2x} )Alternatively, maybe there is a domain where the expression inside log is positive. Let me check again for some x. Suppose x is such that e^x is very small, approaching zero. Then sqrt(1 +0 +0 )=1, so numerator≈1 -0 -1=0, denominator≈1 -0 +1=2. So N/D approaches0. So near x approaching -infty, the expression inside log approaches0 from the negative side. For x where e^x is large, sqrt(1 + e^x + e^{2x}) ≈e^x, so numerator≈e^x - e^x -1= -1, denominator≈e^x - e^x +1=1. So N/D approaches-1/1= -1. So always negative.Therefore, in real numbers, the function is undefined. However, if we consider complex logarithm, the derivative still holds, but the problem likely expects the answer in real analysis. However, since the problem doesn't specify, and given that the derivative expression is valid where the function is differentiable (in complex plane), and the problem just says "find the derivative," the answer is 1/sqrt(1 + e^x + e^{2x} )Therefore, I think the answer is 1 over sqrt(1 + e^x + e^{2x} ), which can be written as:boxed{ dfrac{1}{sqrt{1 + e^{x} + e^{2x}}} }Final Answerboxed{ dfrac{1}{sqrt{1 + e^{x} + e^{2x}}} }</think>To find the derivative of the given function:y = ln frac{sqrt{1 + e^x + e^{2x}} - e^x - 1}{sqrt{1 + e^x + e^{2x}} - e^x + 1}we start by simplifying the expression inside the logarithm. Let ( C = sqrt{1 + e^x + e^{2x}} - e^x ). Then, the function can be rewritten as:y = ln left( frac{C - 1}{C + 1} right)Using the property of logarithms, this becomes:y = ln(C - 1) - ln(C + 1)The derivative of ( y ) with respect to ( x ) is:frac{dy}{dx} = frac{d}{dx} ln(C - 1) - frac{d}{dx} ln(C + 1)Applying the chain rule, we get:frac{dy}{dx} = frac{C'}{C - 1} - frac{C'}{C + 1}Factoring out ( C' ):frac{dy}{dx} = C' left( frac{1}{C - 1} - frac{1}{C + 1} right)Combining the fractions:frac{dy}{dx} = C' left( frac{(C + 1) - (C - 1)}{(C - 1)(C + 1)} right) = C' left( frac{2}{C^2 - 1} right)Next, we compute ( C' ). Given ( C = sqrt{1 + e^x + e^{2x}} - e^x ), we find the derivative:C' = frac{d}{dx} sqrt{1 + e^x + e^{2x}} - frac{d}{dx} e^xThe derivative of the square root term is:frac{e^x + 2e^{2x}}{2 sqrt{1 + e^x + e^{2x}}}Thus,C' = frac{e^x + 2e^{2x}}{2 sqrt{1 + e^x + e^{2x}}} - e^xSimplifying, we get:C' = frac{e^x (1 + 2e^x)}{2 sqrt{1 + e^x + e^{2x}}} - e^x = frac{e^x (1 + 2e^x - 2 sqrt{1 + e^x + e^{2x}})}{2 sqrt{1 + e^x + e^{2x}}}We also need ( C^2 - 1 ). Using the relation ( C + e^x = sqrt{1 + e^x + e^{2x}} ), we find:C^2 - 1 = e^x (1 - 2C)Substituting ( C' ) and ( C^2 - 1 ) back into the derivative expression:frac{dy}{dx} = frac{2 left( frac{e^x (1 + 2e^x - 2 sqrt{1 + e^x + e^{2x}})}{2 sqrt{1 + e^x + e^{2x}}} right)}{e^x (1 - 2C)}Simplifying further, we find that the terms cancel out, leading to:frac{dy}{dx} = frac{1}{sqrt{1 + e^x + e^{2x}}}Thus, the final answer is:boxed{dfrac{1}{sqrt{1 + e^{x} + e^{2x}}}}]

🔑:## Solutionbegin{aligned}& y^{prime}=left(ln frac{sqrt{1+e^{x}+e^{2 x}}-e^{x}-1}{sqrt{1+e^{x}+e^{2 x}}-e^{x}+1}right)= & =frac{sqrt{1+e^{x}+e^{2 x}}-e^{x}+1}{sqrt{1+e^{x}+e^{2 x}}-e^{x}-1} cdotleft(frac{sqrt{1+e^{x}+e^{2 x}}-e^{x}-1}{sqrt{1+e^{x}+e^{2 x}}-e^{x}+1}right)^{prime}= & =frac{sqrt{1+e^{x}+e^{2 x}}-e^{x}+1}{sqrt{1+e^{x}+e^{2 x}}-e^{x}-1} cdotleft(1-frac{2}{sqrt{1+e^{x}+e^{2 x}}-e^{x}+1}right)^{prime}= & =frac{sqrt{1+e^{x}+e^{2 x}}-e^{x}+1}{sqrt{1+e^{x}+e^{2 x}}-e^{x}-1} cdotleft(frac{1}{left(sqrt{1+e^{x}+e^{2 x}}-e^{x}+1right)^{2}} cdotleft(frac{1}{2 sqrt{1+e^{x}+e^{2 x}}} cdotleft(e^{x}+2 e^{2 x}right)-e^{x}right)right)= & =frac{1}{left(sqrt{1+e^{x}+e^{2 x}}-e^{x}-1right) cdotleft(sqrt{1+e^{x}+e^{2 x}}-e^{x}+1right)} cdotleft(frac{e^{x}+2 e^{2 x}}{sqrt{1+e^{x}+e^{2 x}}}-2 e^{x}right)= & =frac{1}{left(sqrt{1+e^{x}+e^{2 x}}-e^{x}right)^{2}-1} cdot frac{e^{x}+2 e^{2 x}-2 e^{x} sqrt{1+e^{x}+e^{2 x}}}{sqrt{1+e^{x}+e^{2 x}}}= & =frac{1}{1+e^{x}+e^{2 x}-2 e^{x} cdot sqrt{1+e^{x}+e^{2 x}}+e^{2 x}-1} cdot frac{e^{x}+2 e^{2 x}-2 e^{x} sqrt{1+e^{x}+e^{2 x}}}{sqrt{1+e^{x}+e^{2 x}}}= & =frac{1}{e^{x}+2 e^{2 x}-2 e^{x} cdot sqrt{1+e^{x}+e^{2 x}}} cdot frac{e^{x}+2 e^{2 x}-2 e^{x} sqrt{1+e^{x}+e^{2 x}}}{sqrt{1+e^{x}+e^{2 x}}}=frac{1}{sqrt{1+e^{x}+e^{2 x}}}end{aligned}

🧠:[<think>Okay, so I need to find the total number of households selected in this stratified sampling survey. Let me read the problem again carefully.There are 480 households total. They are divided into three categories: middle-income, low-income, and high-income. Specifically, there are 200 middle-income families, 160 low-income families, and the rest are high-income. So first, I should check if the numbers add up. Let me compute the number of high-income families.Total households = 480Middle-income = 200Low-income = 160So high-income = 480 - 200 - 160 = 480 - 360 = 120.Alright, so high-income families are 120. Then, the problem says that in a stratified sampling survey, 6 high-income families were selected. I need to find the total number of households selected in the survey.Wait, stratified sampling means that they sample each stratum (each income group) separately. So they probably took samples from each of the three income groups. But the problem only mentions that 6 high-income families were selected. Hmm. Is there information missing?Wait, maybe the problem assumes that the same sampling fraction was used for each stratum? Because in stratified sampling, you can have proportional allocation where the number sampled from each stratum is proportional to the size of the stratum. Alternatively, it could be equal allocation or some other method, but the problem doesn't specify. Wait, but they only tell us that 6 high-income families were selected. Maybe we need to infer the sampling fraction from the high-income stratum and apply it to the other strata?Hmm, if that's the case, let's see. The high-income families are 120, and 6 were sampled. So the sampling fraction is 6/120 = 0.05, which is 5%. If they used the same sampling fraction for all strata, then we can compute the number sampled from middle and low income by applying 5% to their populations.Middle-income: 200 * 0.05 = 10Low-income: 160 * 0.05 = 8High-income: 6Total sampled would be 10 + 8 + 6 = 24.But wait, does the problem specify that the sampling was proportional? It just says stratified sampling. Stratified sampling can be done with different allocation methods. If they didn't specify proportional allocation, maybe we can't assume that. But given that only the number of high-income samples is given, maybe we have to assume proportional allocation?Alternatively, maybe the problem is expecting that the number of samples from each stratum is the same, so 6 from each? But that would be 18 total, but that doesn't seem right because the strata sizes are different.Wait, the problem might be missing some information. Let me check again."In a stratified sampling survey conducted for building a happy community, 6 high-income families were selected. The total number of households selected in this survey is ______."So, the problem doesn't mention how many were selected from the other strata. That's odd. Maybe there was a fixed number selected from each stratum, but only the high-income number is given? But that can't be. Wait, maybe in Chinese stratified sampling problems, sometimes they give the number for one stratum and expect you to figure out the rest based on equal proportions? Maybe there's a standard way they do it in exams.Alternatively, maybe the total number is just 6? But that seems unlikely because 6 is only high-income. The problem must have intended that the sampling fraction is the same for all strata. So 6 out of 120 high-income is 5%, so 5% of each stratum is sampled. Then middle-income would be 10, low-income 8, total 24. That seems plausible. But I need to verify.Alternatively, maybe the problem is using a different allocation method, like Neyman allocation, which considers the variability in each stratum. But since we don't have information on variability, that's impossible. So the default assumption is proportional allocation. So given that, the answer is 24.Alternatively, maybe the problem is using a different sample size, but since only high-income is given, perhaps the question expects you to realize that if high-income is 120 and 6 were sampled, then the sampling ratio is 1:20, so total sample size is 480 / 20 = 24. But that is equivalent to proportional allocation, since 480 total, 6 from 120 (which is 1/20), so 480*(1/20)=24. So that's another way to think about it.Therefore, the total number is 24. Let me confirm again. If the sampling ratio is 6 out of 120, that's 1/20. Then the total sample size would be 480*(1/20)=24. Alternatively, for each stratum, middle: 200*(1/20)=10, low:160*(1/20)=8, high:6. Total 10+8+6=24. Yep, that's consistent. So the answer should be 24.But let me check if there's another interpretation. If the problem said "6 high-income families were selected, and the same number from each stratum," then total would be 6*3=18, but the problem doesn't specify that. So that's not the case here.Alternatively, maybe the problem is referring to a different method. Wait, maybe they selected 6 high-income families, and in total, how many were selected? But without knowing how many from the other strata, we can't compute. Therefore, the problem must be expecting that we use proportional allocation, which is the standard approach when no other information is given. So 24 is the answer.I think that makes sense.Final Answerboxed{24}</think>The total number of households is 480, divided into 200 middle-income, 160 low-income, and the remaining 120 high-income families. In a stratified sampling survey, 6 high-income families were selected. To determine the total number of households selected, we first calculate the sampling fraction for the high-income families:[text{Sampling fraction} = frac{6}{120} = 0.05 text{ (or 5%)}]Using this same sampling fraction for all strata, we calculate the number of households selected from each income group:- Middle-income: ( 200 times 0.05 = 10 )- Low-income: ( 160 times 0.05 = 8 )- High-income: 6 (given)Adding these together gives the total number of households selected:[10 + 8 + 6 = 24]Thus, the total number of households selected in the survey is (boxed{24}).,<think>Okay, let's see here. The problem is about stratified sampling, right? So, there are 480 households total, divided into three income categories: middle, low, and high. The numbers given are 200 middle-income families, 160 low-income families, and the rest are high-income. They mention that in the survey, 6 high-income families were selected, and we need to find the total number of households selected in the survey.First, I need to recall what stratified sampling is. If I remember correctly, stratified sampling is when the population is divided into subgroups (or strata) and a random sample is taken from each stratum. The key point is that each stratum is sampled separately. This is different from a simple random sample where you just randomly select from the entire population without dividing into groups.So, in this case, the strata are the three income levels: middle, low, and high. The problem states that 6 high-income families were selected. But to find the total number of households selected, we need to know how many were selected from each stratum (middle, low, high) and then sum them up. However, the problem only mentions that 6 high-income families were selected. It doesn't directly say how many were selected from the other two groups. Hmm, so maybe there's something missing here?Wait, maybe there's a standard approach in stratified sampling where the number selected from each stratum is proportional to the size of the stratum. That is, if high-income families make up a certain percentage of the total population, then the sample from that stratum would be the same percentage of the total sample. But the problem doesn't mention the sampling method explicitly, like whether it's proportional stratified sampling or something else. But maybe we can assume proportional sampling unless stated otherwise?Alternatively, maybe the problem is giving us all the information needed. Let me check again.Total households: 480Middle-income: 200Low-income: 160High-income: 480 - 200 - 160 = 120Wait, yes, 200 + 160 is 360, so 480 - 360 = 120 high-income families.So, high-income: 120 households.In the stratified sampling, 6 high-income families were selected. So, the sample size for high-income is 6. To find the total sample size, we need to know how many were selected from middle and low-income families.But how? The problem doesn't state the sample sizes for middle and low. Unless there's some missing information here. Wait, maybe the problem assumes that the same sampling fraction is used for each stratum. That is, the proportion of the sample taken from each stratum is the same as the proportion of the population.If that's the case, then we can compute the sampling fraction from the high-income stratum and apply it to the other strata.So, the high-income population is 120, and the sample selected is 6. So the sampling fraction would be 6/120 = 0.05, or 5%. Then, if the same fraction is applied to middle and low-income families, we can compute their sample sizes.Middle-income sample: 200 * 0.05 = 10Low-income sample: 160 * 0.05 = 8Then total sample size would be 10 + 8 + 6 = 24.But wait, the problem doesn't explicitly state that the sampling was proportional. So is this a valid assumption? Hmm.Alternatively, maybe the problem is expecting us to realize that since it's stratified sampling and they told us how many were selected from one stratum, but without more information about the other strata, we can't compute the total? That seems unlikely because the problem is asking for a numerical answer.Therefore, probably the intended approach is to assume proportional sampling. So, the answer would be 24.Alternatively, maybe the problem is using equal sample sizes from each stratum? But that would be another method. For example, if they took 6 from each stratum, then total would be 6*3=18. But the problem says "6 high-income families were selected," so that might not be the case.Alternatively, maybe they used a different sampling fraction for each stratum, but unless specified, how would we know? So perhaps the answer is 24, assuming proportional stratified sampling.Alternatively, maybe the problem is in Chinese, and there's a translation issue. The original problem says "the total number of households selected in this survey is ______." Maybe in the original problem, there was more information, but in translation, some parts got lost. But given the information we have, we need to make an assumption.Alternatively, maybe the problem is expecting that the sample size for each stratum is the same, but the problem specifically says "stratified sampling," which generally can be proportional or disproportional. Without specific mention, maybe the standard approach is proportional.But let's check.If we do proportional stratified sampling, the sample size for each stratum is (number in stratum / total population) * total sample size. Wait, but here we only know the sample size for one stratum. So we can compute the total sample size if we know the proportion.Wait, but actually, if the sample is proportional, then the sampling fraction is the same for all strata. So if 6 high-income families were sampled, which is 6 out of 120, that's 5%. Then, middle-income would be 200 * 5% = 10, low-income 160 * 5% = 8, so total 24. So that seems logical.Alternatively, if they used a different sampling fraction, but given that they only told us about the high-income sample, how could we know? So, the problem must be assuming proportional sampling.Therefore, the total number is 24. So, I think that's the answer.Final AnswerThe total number of households selected in this survey is boxed{24}.</think>The problem involves stratified sampling of households in a community. The total number of households is 480, divided into 200 middle-income, 160 low-income, and 120 high-income families. In the survey, 6 high-income families were selected. To find the total number of households selected, we assume proportional stratified sampling, where the sampling fraction is the same for each stratum. 1. Calculate the sampling fraction for high-income families: [ text{Sampling fraction} = frac{6}{120} = 0.05 ]2. Apply this fraction to the middle-income and low-income families: - Middle-income sample: (200 times 0.05 = 10) - Low-income sample: (160 times 0.05 = 8)3. Sum the samples from all strata: [ 10 + 8 + 6 = 24 ]Thus, the total number of households selected in the survey is (boxed{24}).]

🔑:AnalysisThis question tests stratified sampling.Given the information, we can find that there are 480 - 200 - 160 = 120 high-income families. We solve the problem using the definition of stratified sampling.SolutionThere are 480 households, among which there are 200 middle-income families, 160 low-income families, and thus 480 - 200 - 160 = 120 high-income families.6 high-income families were selected. Let the total number of households selected in this survey be n,Then frac{6}{n} = frac{120}{480}, solving this gives n = 24.Therefore, the answer is boxed{24}.

🧠:[<think>Alright, let's see. I need to figure out by how many rubles the price of the carousel ride was reduced in March. Okay, let's start by understanding the problem step by step.In February, the price was 300 rubles per person. Let's denote the number of visitors in February as, say, V. Then in March, the price was reduced by some amount—let's call that reduction x rubles. So the new price in March would be (300 - x) rubles. The problem says that in March, the number of visitors increased by 50%. So if February had V visitors, March would have V plus 50% of V, which is 1.5V visitors. The revenue in February would be the price multiplied by the number of visitors, which is 300 * V. In March, the revenue increased by 25%. So the new revenue is 300 * V plus 25% of that, which is 1.25 * (300 * V). Alternatively, the revenue in March can also be calculated as the new price (300 - x) multiplied by the new number of visitors (1.5V). So, setting up the equation: (300 - x) * 1.5V = 1.25 * (300 * V)Hmm, okay. Let's write that out:1.5V(300 - x) = 1.25 * 300VFirst, I can simplify both sides. Let's divide both sides by V, since V is not zero (assuming there were some visitors in February). That cancels out V:1.5(300 - x) = 1.25 * 300Now compute the right-hand side. 1.25 times 300. Let me calculate that:1.25 * 300 = 375So the equation becomes:1.5(300 - x) = 375Now, let's solve for x. First, divide both sides by 1.5:300 - x = 375 / 1.5Compute 375 divided by 1.5. Let's see:375 divided by 1.5. Hmm, 1.5 goes into 375 how many times? Well, 1.5 times 200 is 300, and 1.5 times 50 is 75, so total 250. So 375 / 1.5 = 250. So:300 - x = 250Subtract 250 from 300:300 - 250 = xSo:50 = xTherefore, x is 50. So the price was reduced by 50 rubles. Wait, let me double-check my calculations to be sure. Let's verify:Original price: 300 rubles. Reduced by 50 rubles, so new price is 250 rubles.Number of visitors increased by 50%, so if originally V people came, now it's 1.5V.Revenue in February: 300 * VRevenue in March: 250 * 1.5V = 375VOriginal revenue was 300V, new revenue is 375V. The increase is 75V, which is (75V / 300V)*100% = 25%. That's correct. So 25% increase in revenue. Yes, that checks out. So the price was reduced by 50 rubles. Therefore, the answer is 50. I think that's solid. Let me see if there's another way to approach it, just to confirm.Alternatively, let's assign variables:Let original price P = 300 rubles.Original number of visitors = N.Revenue in February: R = P * N = 300N.In March, price is P' = 300 - x.Visitors in March: N' = N + 0.5N = 1.5N.Revenue in March: R' = (300 - x) * 1.5N.According to the problem, R' = R + 0.25R = 1.25R = 1.25 * 300N = 375N.Thus, equation:(300 - x) * 1.5N = 375NDivide both sides by N (assuming N ≠ 0):1.5(300 - x) = 375Same equation as before. Solving gives x = 50. Same result. Definitely, the price was reduced by 50 rubles.Final AnswerThe price was reduced by boxed{50} rubles.</think>Let the number of visitors in February be ( V ). The price in February was 300 rubles per person, and the revenue was ( 300V ) rubles.In March, the price was reduced by ( x ) rubles, making the new price ( 300 - x ) rubles. The number of visitors increased by 50%, resulting in ( 1.5V ) visitors. The revenue in March increased by 25%, so the new revenue is ( 1.25 times 300V ).Setting up the equation for the revenue in March:[(300 - x) times 1.5V = 1.25 times 300V]Dividing both sides by ( V ):[1.5(300 - x) = 1.25 times 300]Calculating the right-hand side:[1.25 times 300 = 375]Thus, the equation becomes:[1.5(300 - x) = 375]Dividing both sides by 1.5:[300 - x = frac{375}{1.5} = 250]Solving for ( x ):[300 - 250 = x implies x = 50]Therefore, the price was reduced by (boxed{50}) rubles.,<think>Alright, let's tackle this problem step by step. So, the carousel ride was 300 rubles per person in February. Then in March, the price was reduced, leading to a 50% increase in visitors and a 25% increase in revenue. We need to find out by how many rubles the price was reduced.Hmm, okay. Let me break this down. First, let's recall that revenue is calculated by multiplying the price per unit by the number of units sold. In this case, revenue would be the price per ride multiplied by the number of visitors.In February, the price was 300 rubles per person. Let's denote the number of visitors in February as V. Then, the revenue for February would be 300 * V.In March, the price was reduced by some amount. Let's call that reduction x rubles. So, the new price in March would be (300 - x) rubles. The number of visitors increased by 50%, so that would be V plus 50% of V, which is 1.5V. The revenue in March would then be the new price multiplied by the new number of visitors: (300 - x) * 1.5V.According to the problem, the revenue increased by 25% in March compared to February. So, March's revenue is February's revenue plus 25% of it. Let's write that as an equation:March revenue = February revenue + 25% of February revenueWhich translates to:(300 - x) * 1.5V = 300V + 0.25 * 300VLet me compute the right side first. 0.25 * 300V is 75V, so the total revenue in March should be 300V + 75V = 375V.So, the equation becomes:(300 - x) * 1.5V = 375VHmm, since V is present on both sides of the equation and assuming V is not zero, we can divide both sides by V to simplify. That gives:(300 - x) * 1.5 = 375Now, let's solve for x. First, divide both sides by 1.5:300 - x = 375 / 1.5Calculating 375 divided by 1.5. Let me think, 375 divided by 1.5 is the same as 375 multiplied by 2/3. Let's see:375 * 2 = 750750 / 3 = 250So, 375 / 1.5 = 250Therefore, the equation simplifies to:300 - x = 250Subtracting 250 from 300 gives:300 - 250 = x => 50 = xWait, that seems straightforward. So x is 50. Therefore, the price was reduced by 50 rubles.But let me double-check to make sure I didn't make any mistakes. Let's verify with actual numbers.Suppose in February, the price was 300 rubles, and let's say the number of visitors was, for example, 100 people. Then February's revenue would be 300 * 100 = 30,000 rubles.In March, the price is reduced by 50 rubles to 250 rubles. The number of visitors increases by 50%, so 100 * 1.5 = 150 visitors. March's revenue would be 250 * 150 = 37,500 rubles.Now, let's check the percentage increase in revenue. The original revenue was 30,000, and the new revenue is 37,500. The increase is 7,500 rubles. 7,500 / 30,000 = 0.25, which is a 25% increase. That matches the problem's statement.So, yes, the calculations seem correct. The price was reduced by 50 rubles.Wait a second, but the question says "By how many rubles was the price reduced?" So the answer is 50 rubles. Let me just check once again.Alternative approach: Let's denote February's visitors as V. Revenue in Feb: 300V.In March, price is (300 - x), visitors are 1.5V. Revenue: (300 - x)*1.5V.This is supposed to be 25% higher than Feb's revenue, so 1.25 * 300V = 375V.Therefore, (300 - x)*1.5V = 375V. Divide both sides by V: (300 - x)*1.5 = 375.300 - x = 375 / 1.5 = 250. So x = 300 - 250 = 50. Yep, same result.Alternatively, using algebra without plugging numbers.So, all the checks lead to the same conclusion. The price was reduced by 50 rubles. Therefore, the answer should be 50.Final AnswerThe price was reduced by boxed{50} rubles.</think>The price for a ride on the "Voskhod" carousel in February 2020 was 300 rubles per person. In March, the price was reduced, leading to a 50% increase in visitors and a 25% increase in revenue. We need to find the amount by which the price was reduced.Let ( V ) be the number of visitors in February. The revenue in February is ( 300V ) rubles.In March, the price was reduced by ( x ) rubles, making the new price ( 300 - x ) rubles. The number of visitors increased by 50%, becoming ( 1.5V ). The revenue in March is ( (300 - x) cdot 1.5V ) rubles.The revenue in March is 25% higher than in February, so we have:[(300 - x) cdot 1.5V = 1.25 cdot 300V]Simplifying the equation:[(300 - x) cdot 1.5 = 1.25 cdot 300]Calculating the right-hand side:[1.25 cdot 300 = 375]Dividing both sides by 1.5:[300 - x = frac{375}{1.5} = 250]Solving for ( x ):[x = 300 - 250 = 50]Thus, the price was reduced by (boxed{50}) rubles.]

🔑:Answer: 50.Solution. The entry fee for two people in February 2020 was 600 rubles. In March, instead of every two people, the stadium is visited by three people, as the number of visitors increased by 50 %. Since the total collection increased by 25 %, they will pay 600+0.25 cdot 600=750 (rub.), and thus one ticket will cost 750: 3=250 (rub.), which is 50 rubles less than in February.Comment. Answer without justification - 1 point.

🧠:[<think>Okay, so I need to solve this problem where they give me that the tangent of alpha is 2, and then I have to find the value of this expression: (2 sin alpha + cos alpha) divided by (sin alpha minus cos alpha). The answer choices are A through D, which are numbers 5, 4, 3, and 2. Let me think about how to approach this.First, I remember that tangent of an angle is sine over cosine. So if tan alpha is 2, that means sin alpha divided by cos alpha is 2. Maybe I can express sine and cosine in terms of each other? Let me write that down: tan α = sin α / cos α = 2. So that would mean sin α = 2 cos α. Hmm, that could be helpful. Maybe I can substitute that into the original expression.Let me try substituting sin α with 2 cos α in both the numerator and the denominator. So the numerator becomes 2*(2 cos α) + cos α. Let me calculate that: 2*2 is 4, so 4 cos α + cos α. That adds up to 5 cos alpha. Okay, so the numerator simplifies to 5 cos alpha.Now the denominator is sin alpha minus cos alpha. Again, substituting sin alpha with 2 cos alpha, that becomes 2 cos alpha minus cos alpha. Which is just cos alpha. So the denominator simplifies to cos alpha.So now the entire expression becomes 5 cos alpha divided by cos alpha. The cos alpha terms should cancel out, right? So 5 divided by 1 is 5. That would make the answer 5, which is option A. Wait, but let me check to make sure I didn't make a mistake here. Maybe there's a different approach to confirm.Alternatively, another method could be to use the fact that tan alpha = 2, so we can think of a right triangle where the opposite side is 2 and the adjacent side is 1, so that the tangent is 2/1. Then, the hypotenuse would be sqrt(2^2 + 1^2) = sqrt(5). So then, sin alpha would be opposite over hypotenuse, which is 2/sqrt(5), and cos alpha would be adjacent over hypotenuse, which is 1/sqrt(5). Let's plug those values into the original expression.First, compute the numerator: 2 sin alpha + cos alpha. That's 2*(2/sqrt(5)) + (1/sqrt(5)). Calculating each term: 2*(2/sqrt(5)) is 4/sqrt(5), and adding 1/sqrt(5) gives a total of 5/sqrt(5). Then the denominator is sin alpha minus cos alpha, which is (2/sqrt(5)) - (1/sqrt(5)) = (2 - 1)/sqrt(5) = 1/sqrt(5).So now the expression is (5/sqrt(5)) divided by (1/sqrt(5)). When you divide fractions, you multiply by the reciprocal, so 5/sqrt(5) * sqrt(5)/1. The sqrt(5) terms cancel out, leaving 5*1 = 5. So that also gives 5. Hmm, that's the same result as before, so that seems to confirm that the answer is A:5.Wait, but let me check again because sometimes when I'm dealing with trigonometric substitutions, signs can be tricky depending on the quadrant. The problem didn't specify the quadrant where alpha is, but since tan alpha is positive, alpha must be in either the first or third quadrant. However, since they are asking for the value of the expression, and we're not given any additional information, maybe the answer is the same regardless of the quadrant?Wait, let's test that. If alpha is in the third quadrant, both sine and cosine would be negative. So let's see. If tan alpha is 2, but in the third quadrant, sin alpha = -2/sqrt(5) and cos alpha = -1/sqrt(5). Let's plug those into the expression.Numerator: 2*(-2/sqrt(5)) + (-1/sqrt(5)) = -4/sqrt(5) -1/sqrt(5) = -5/sqrt(5). Denominator: (-2/sqrt(5)) - (-1/sqrt(5)) = (-2 +1)/sqrt(5) = -1/sqrt(5). So then the expression is (-5/sqrt(5)) divided by (-1/sqrt(5)) which is (-5/sqrt(5)) * (sqrt(5)/-1) = (-5)*sqrt(5)/sqrt(5)*(-1) = (-5)*(-1) = 5. So even in the third quadrant, the result is still 5. Therefore, regardless of the quadrant, the answer is 5.Therefore, the correct answer should be A:5. But wait, the options given are A:5, B:4, C:3, D:2. So according to both methods, substituting sin and cos in terms of each other, and using the right triangle approach, and even checking in different quadrants, the answer is 5, which is option A. So that must be the correct answer.But wait, hold on. Let me check once again because sometimes in exams, the answer might not be as straightforward. Let me re-derive the first substitution method. So tan alpha = 2, so sin alpha = 2 cos alpha. Then substituting into the expression:(2 sin alpha + cos alpha)/(sin alpha - cos alpha) = (2*(2 cos alpha) + cos alpha)/(2 cos alpha - cos alpha) = (4 cos alpha + cos alpha)/(cos alpha) = 5 cos alpha / cos alpha = 5. So yes, 5.Alternatively, another way to approach this could be to divide numerator and denominator by cos alpha. Let's try that. So:Original expression: [2 sin alpha + cos alpha] / [sin alpha - cos alpha]Divide numerator and denominator by cos alpha:[2 (sin alpha / cos alpha) + (cos alpha / cos alpha)] / [ (sin alpha / cos alpha) - (cos alpha / cos alpha) ]Which simplifies to [2 tan alpha + 1] / [tan alpha - 1]Since tan alpha is given as 2, substitute that in:[2*(2) + 1] / [2 - 1] = (4 +1)/1 = 5/1 = 5.Oh, that's a quicker method. Instead of substituting sin alpha in terms of cos alpha, we can just express everything in terms of tan alpha by dividing numerator and denominator by cos alpha. That's smart. So this also gives 5. So that's three different methods all leading to 5. So definitely, the answer is 5, which is option A.But wait, the problem is presented as a multiple-choice question with options A to D. Let me check once more to ensure I didn't miscalculate. Let me verify the second method again. Dividing numerator and denominator by cos alpha:Original expression: (2 sin alpha + cos alpha)/(sin alpha - cos alpha)Divide numerator and denominator by cos alpha:(2 (sin alpha / cos alpha) + 1)/( (sin alpha / cos alpha) - 1 )Which becomes (2 tan alpha + 1)/(tan alpha - 1). Since tan alpha is 2:(2*2 + 1)/(2 -1) = (4 +1)/1 = 5/1 =5. Yes, that's correct.Therefore, all methods lead to answer A:5. So the correct choice is A. Therefore, I think that's the answer.Final Answerboxed{A}</think>Given (tan alpha = 2), we need to find the value of (frac{2sin alpha + cos alpha}{sin alpha - cos alpha}).First, we know that (tan alpha = frac{sin alpha}{cos alpha} = 2), which implies (sin alpha = 2 cos alpha).Substituting (sin alpha = 2 cos alpha) into the expression:[frac{2sin alpha + cos alpha}{sin alpha - cos alpha} = frac{2(2 cos alpha) + cos alpha}{2 cos alpha - cos alpha}]Simplifying the numerator and the denominator:[frac{4 cos alpha + cos alpha}{cos alpha} = frac{5 cos alpha}{cos alpha} = 5]Another method is to divide the numerator and the denominator by (cos alpha):[frac{2 sin alpha + cos alpha}{sin alpha - cos alpha} = frac{2 frac{sin alpha}{cos alpha} + 1}{frac{sin alpha}{cos alpha} - 1} = frac{2 tan alpha + 1}{tan alpha - 1}]Substituting (tan alpha = 2):[frac{2 cdot 2 + 1}{2 - 1} = frac{4 + 1}{1} = 5]Both methods confirm that the value is (5).Thus, the correct answer is (boxed{A}).,<think>Okay, so I need to find the value of this expression: (2 sin α + cos α) divided by (sin α - cos α), given that tan α is 2. Hmm, let me think. Alright, since tan α is given, maybe I can use that to express sin α and cos α in terms of tan. I remember that tan α is sin α over cos α. So if tan α = 2, that means sin α / cos α = 2. Therefore, sin α = 2 cos α. That seems useful. Maybe I can substitute sin α with 2 cos α in the original expression. Let me try that.So, replacing sin α with 2 cos α in the numerator: 2 sin α becomes 2*(2 cos α) = 4 cos α. Then, adding cos α gives 4 cos α + cos α = 5 cos α. In the denominator, sin α is 2 cos α, and then subtracting cos α gives 2 cos α - cos α = cos α. So the entire expression becomes 5 cos α divided by cos α. Then the cos α cancels out, leaving 5. Wait, that would be 5. But the options are A:5, B:4, C:3, D:2. So is it A? But let me check again to make sure I didn't make a mistake.Alternatively, maybe I can approach this by dividing numerator and denominator by cos α. Let me see. If I divide each term by cos α, the expression becomes [2 sin α / cos α + cos α / cos α] divided by [sin α / cos α - cos α / cos α]. Simplifying that, sin α / cos α is tan α, which is 2. So the numerator becomes 2*2 + 1 = 4 + 1 = 5. The denominator becomes 2 - 1 = 1. So 5 divided by 1 is 5. That's the same result. So again, 5, which is option A.Wait, but the answer is 5? Let me verify once more. Let's suppose that tan α = 2. Then, we can consider a right triangle where the opposite side is 2 and the adjacent side is 1, so the hypotenuse would be sqrt(1^2 + 2^2) = sqrt(5). Therefore, sin α is opposite over hypotenuse, which is 2/sqrt(5), and cos α is adjacent over hypotenuse, which is 1/sqrt(5). Let's plug these into the original expression.Numerator: 2 sin α + cos α = 2*(2/sqrt(5)) + (1/sqrt(5)) = 4/sqrt(5) + 1/sqrt(5) = 5/sqrt(5) = sqrt(5). Denominator: sin α - cos α = (2/sqrt(5)) - (1/sqrt(5)) = 1/sqrt(5). So sqrt(5) divided by (1/sqrt(5)) equals sqrt(5) * sqrt(5) = 5. Yep, that checks out. So the answer is 5, which is option A. Hmm, that seems right. But why did the problem give options? Let me check once again to make sure I didn't miscalculate.Another way: if tan α = 2, then α is some angle whose tangent is 2. Let me just compute sin α and cos α using tan α = 2. As I did before, in a right triangle with opposite side 2, adjacent 1, hypotenuse sqrt(5). So sin α = 2/sqrt(5), cos α = 1/sqrt(5). So substituting into the expression: (2*(2/sqrt(5)) + 1/sqrt(5)) / (2/sqrt(5) - 1/sqrt(5)) = (4/sqrt(5) + 1/sqrt(5)) / (1/sqrt(5)) = (5/sqrt(5)) / (1/sqrt(5)) = 5/sqrt(5) * sqrt(5)/1 = 5. Yep, same answer. So I think that's correct. So the answer is A:5.But wait, let me check if there's any alternative method where maybe the answer is different. For example, sometimes people might use the identity involving dividing numerator and denominator by cos α, but I did that already. Alternatively, using the original expression: Let’s consider the original expression (2 sin α + cos α)/(sin α - cos α). Let me factor out cos α in numerator and denominator: [cos α (2 tan α + 1)] / [cos α (tan α - 1)]. Then the cos α cancels out, leaving (2 tan α +1)/(tan α -1). Since tan α = 2, substitute that: (2*2 +1)/(2 -1) = (5)/(1) =5. Yep, same answer. So regardless of the method, it's 5. Therefore, the answer is A.Final Answerboxed{A}</think>Given (tan alpha = 2), we need to find the value of (frac{2sin alpha + cos alpha}{sin alpha - cos alpha}).First, we use the identity (tan alpha = frac{sin alpha}{cos alpha}). Since (tan alpha = 2), we have (sin alpha = 2 cos alpha).Substituting (sin alpha = 2 cos alpha) into the expression:[frac{2(2 cos alpha) + cos alpha}{2 cos alpha - cos alpha} = frac{4 cos alpha + cos alpha}{cos alpha} = frac{5 cos alpha}{cos alpha} = 5]Another method involves dividing the numerator and denominator by (cos alpha):[frac{2 sin alpha + cos alpha}{sin alpha - cos alpha} = frac{2 tan alpha + 1}{tan alpha - 1}]Substituting (tan alpha = 2):[frac{2 cdot 2 + 1}{2 - 1} = frac{5}{1} = 5]Both methods confirm that the value is (5).[boxed{A}]]

🔑:AnalysisThis question mainly tests the application of basic trigonometric identities, and it is a basic question.SolutionGiven the information, we have:frac{2sin alpha + cos alpha}{sin alpha - cos alpha} = frac{2tan alpha + 1}{tan alpha - 1} = 5Therefore, the correct answer is boxed{A}.