🧠:[<think>Okay, so I need to figure out the value of m such that the probability that a randomly selected number x from the interval [-1, 5] satisfies |x| ≤ m is 1/2. Let me break this down step by step.First, let's understand the problem. We have a continuous uniform distribution over the interval [-1, 5]. That means every number in this interval has an equal chance of being selected. The total length of the interval is 5 - (-1) = 6 units. So, the probability density function is constant over this interval, specifically 1/6 per unit length, since the total probability must be 1.Now, we need to find a real number m such that the probability that |x| ≤ m is 1/2, which is 0.5. In other words, the length of the interval where |x| ≤ m divided by the total length of the interval [-1, 5] should be 0.5. So, mathematically, this can be written as:Length of {x ∈ [-1, 5] : |x| ≤ m} / 6 = 1/2Therefore, Length of {x ∈ [-1, 5] : |x| ≤ m} = 3So now the task reduces to finding m such that the length of the set where |x| ≤ m within [-1, 5] is 3. Let's visualize the interval [-1, 5]. The absolute value inequality |x| ≤ m translates to x being between -m and m. But since our interval starts at -1, we need to consider where -m and m fall within [-1, 5].Wait, but m has to be a real number. Let's think about possible cases based on the value of m.Case 1: m ≤ 0. If m is negative, then |x| ≤ m would imply that x is between -m and m. But if m is negative, -m is positive, so the interval would be [m, -m], but m is negative, so this interval would be from a negative number to a smaller negative number. However, since our original interval starts at -1, which is already a negative number, but if m is negative, then m might be less than -1, which is outside the original interval. So maybe in this case, the set {x ∈ [-1, 5] : |x| ≤ m} would be empty. But m can't be less than -1, because the interval starts at -1. Wait, but m is a real number here, and the problem states that x is selected from [-1,5]. So m can be any real number, but we need to find m such that the probability is 1/2. Let me check the options given: 0,1,2,3. So m is likely between 0 and 3. But maybe I need to verify.Alternatively, maybe m can be from 0 up to 5. Let me think again. The inequality |x| ≤ m can be split into two parts: -m ≤ x ≤ m. But x is in [-1,5]. So depending on m, the overlap between [-m, m] and [-1,5] will vary.Let me analyze the possible ranges for m.If m is between 0 and 1: Then the interval [-m, m] is within [-1,1], but since m is positive, [-m, m] is symmetric around 0. For example, if m is 0.5, the interval would be [-0.5, 0.5]. But our original interval starts at -1, so the overlap would be from -0.5 to 0.5, which is within [-1,5]. So the length here would be m - (-m) = 2m. Wait, but actually, the overlap between [-m, m] and [-1,5] is [-m, m] if m ≤1. Wait, no. If m is less than 1, then the left end of the interval [-m, m] is -m, which is greater than -1 (since m is positive and less than 1). So the overlap would be from -m to m. So length is 2m.But if m is greater than 1, then the interval [-m, m] would extend beyond -1 on the left (since -m would be less than -1 when m >1), but our original interval starts at -1. So in that case, the overlap would be from -1 to m. Wait, but m can be up to 5. Let me clarify.So, the interval where |x| ≤ m is [-m, m]. However, x must be in [-1,5]. So the intersection of [-m, m] and [-1,5] is [max(-m, -1), min(m,5)]. Therefore, the length of this interval is min(m,5) - max(-m, -1).But since m is a real number, let's consider different cases:Case 1: m ≤ -1. But m is a real number, and since we have x in [-1,5], the probability that |x| ≤ m when m is negative would be zero, because |x| is non-negative. So m must be non-negative. Therefore, m is in [0, ∞). But since the options are 0,1,2,3, we can restrict m to [0,5].Case 2: 0 ≤ m ≤1. Then, since m ≤1, the overlap between [-m, m] and [-1,5] is [-m, m]. Because -m ≥ -1 (since m ≤1) and m ≤5. So the length is 2m.Case 3: 1 < m ≤5. Here, the overlap between [-m, m] and [-1,5] is [-1, m]. Because -m < -1, so the left bound is -1, and the right bound is m (since m ≤5). So the length is m - (-1) = m +1.Case 4: m >5. Then, the overlap would be the entire interval [-1,5], which has length 6. But since we need the probability to be 1/2, which is 3, m can't be greater than 5. So m must be ≤5.Given that, the length is:- For 0 ≤ m ≤1: length = 2m- For 1 < m ≤5: length = m +1We need the length to be 3. Let's check which case 3 falls into.If m ≤1: 2m =3 ⇒ m=1.5, but 1.5 >1, which contradicts m ≤1. Therefore, it must be in the second case.For 1 <m ≤5: length = m +1 =3 ⇒ m=2.So m=2. Let's verify this.If m=2, then the interval where |x| ≤2 is [-2,2]. But our original interval is [-1,5]. The intersection is [-1,2]. The length is 2 - (-1) =3, which is 3. The total interval is 6, so the probability is 3/6=1/2. Correct.Therefore, the answer is C:2.Wait, but let me check if I considered all possibilities correctly. Let me think again.If m=2, then |x| ≤2 translates to x between -2 and 2. But the original x is in [-1,5], so the overlapping region is from -1 to 2. The length is 2 - (-1)=3, which is half of the total length (6). So yes, probability 1/2. Therefore, m=2.Alternatively, if m=1, then the interval is [-1,1]. The length is 1 - (-1)=2. Then the probability is 2/6=1/3, which is less than 1/2. So m needs to be larger than 1.If m=3, the interval would be [-3,3]. Intersection with [-1,5] is [-1,3]. Length is 3 - (-1)=4. Probability is 4/6=2/3, which is more than 1/2. So m=3 is too big.Hence, the correct answer is 2, which is option C.Final Answerboxed{C}</think>To determine the value of ( m ) such that the probability that a randomly selected number ( x ) from the interval ([-1, 5]) satisfies ( |x| leqslant m ) is (frac{1}{2}), we need to consider the length of the interval where ( |x| leqslant m ) within ([-1, 5]).1. The total length of the interval ([-1, 5]) is ( 5 - (-1) = 6 ).2. The probability that ( |x| leqslant m ) is given by the length of the interval ([-m, m]) intersecting with ([-1, 5]) divided by 6. This probability must be (frac{1}{2}), so the length of the intersection must be 3.We analyze different cases for ( m ):- Case 1: ( 0 leqslant m leqslant 1 ) The interval ([-m, m]) is within ([-1, 5]). The length is ( 2m ). Setting ( 2m = 3 ) gives ( m = 1.5 ), which is outside this range. - Case 2: ( 1 < m leqslant 5 ) The intersection of ([-m, m]) with ([-1, 5]) is ([-1, m]). The length is ( m - (-1) = m + 1 ). Setting ( m + 1 = 3 ) gives ( m = 2 ).Verification:- For ( m = 2 ), the interval ([-2, 2]) intersects ([-1, 5]) to form ([-1, 2]). The length is ( 2 - (-1) = 3 ), which is (frac{3}{6} = frac{1}{2}) of the total interval length.Thus, the correct answer is (boxed{C}).,<think>Okay, let's see. I need to figure out the value of m such that the probability that a randomly selected number x from the interval [-1, 5] satisfies |x| ≤ m is 1/2. The options are A through D: 0, 1, 2, 3. Hmm. First, let's understand the problem. We have the interval from -1 to 5. So the total length of this interval is 5 - (-1) = 6 units. So the entire interval is 6 units long. The probability that a randomly selected number x is in a certain subinterval is equal to the length of that subinterval divided by the total length, which is 6. Since the probability here is 1/2, that means the length of the subinterval where |x| ≤ m must be (1/2)*6 = 3 units. So, the subinterval where |x| ≤ m must have a length of 3. Now, |x| ≤ m translates to the interval [-m, m]. But wait, our original interval is from -1 to 5. So we have to consider where [-m, m] overlaps with [-1, 5]. Let's visualize this. If m is a positive number, the interval [-m, m] will extend from negative m to positive m. However, since our original interval starts at -1, the leftmost point of the overlapping region would be the maximum of -m and -1, and the rightmost point would be the minimum of m and 5. But in this case, since we need the length of the overlap between [-m, m] and [-1, 5] to be 3, let's analyze how this overlap works. First, if m is less than or equal to -1, which can't be since m is a real number and the options given are 0,1,2,3. So m is non-negative. So m is between 0 and 5, but let's check the options given: 0,1,2,3.Wait, but m has to be such that the interval [-m, m] overlaps with [-1,5]. However, if m is, say, 0, then the interval is just [0,0], which is a single point. The length is 0, so probability 0. So that's not 1/2. So A is 0, which is not the answer. If m is 1, then the interval [-1,1]. But wait, our original interval starts at -1, so the overlap is from -1 to 1. The length is 1 - (-1) = 2. Then 2/6 = 1/3 probability, which is less than 1/2. So B is 1, which gives probability 1/3. Not 1/2. If m is 2, then the interval [-2,2]. But since our original interval starts at -1, the overlap is from -1 to 2. The length is 2 - (-1) = 3. Then 3/6 = 1/2. That's exactly the probability we need. So that's answer C: 2. Wait, but let me check if m is 3. Then the interval [-3,3]. Overlapping with [-1,5] is from -1 to 3. The length is 3 - (-1) = 4. Then 4/6 = 2/3, which is more than 1/2. So D: 3 gives probability 2/3. Therefore, the answer should be C: 2. Wait, let me confirm again. If m is 2, the interval where |x| ≤ 2 is [-2,2]. However, the original interval is [-1,5]. The intersection of these two intervals is [-1,2], which has a length of 3. The total interval length is 6, so 3/6 = 1/2. That's correct. Alternatively, is there another way this could be approached? Let's consider the distribution. Since x is selected uniformly from [-1,5], the probability density function is constant over this interval. The probability that x is in some subinterval is proportional to the length of that subinterval. We need P(|x| ≤ m) = 1/2. So the set of x where |x| ≤ m is the interval [-m, m]. But since x can't be less than -1, the lower bound of the interval is actually max(-m, -1). The upper bound is min(m, 5). But since m is a positive number, and our options are 0,1,2,3. Let's check each:For m = 0: interval is [0,0], length 0. Probability 0.m =1: interval is [-1,1], since max(-1, -1) is -1, min(1,5) is 1. Length 2. 2/6=1/3.m=2: interval is [-1,2], since max(-2, -1) is -1, min(2,5) is 2. Length 3. 3/6=1/2. Perfect.m=3: interval is [-1,3]. Length 4. 4/6=2/3.So indeed, m=2 gives the required probability. Therefore, the answer is C.But let me check once again to make sure I didn't make a mistake. Maybe I should write out the equation formally.We need the length of {x ∈ [-1,5] | |x| ≤ m} divided by 6 = 1/2.So, the length is 3.The set {x | |x| ≤ m} is [-m, m]. However, overlapping with [-1,5], so:Left endpoint: max(-m, -1)Right endpoint: min(m,5)Thus, length = min(m,5) - max(-m, -1)We need this length to be 3.So:Case 1: m ≤ -1. But m is non-negative, so no.Case 2: -1 < m ≤ 5.So here, since m is positive, max(-m, -1) = -1 (since -m ≥ -1 only if m ≤1. Wait, actually:Wait, if m is between 0 and 1, then -m is between -1 and 0. So max(-m, -1) would be -m if m <1? Wait, no. Wait, for m in [0,1), -m is in (-1,0]. Since our interval is from -1, which is less than -m (if m <1). Wait, for example, m=0.5. Then -m=-0.5. The original interval starts at -1, so the left endpoint is -1. The right endpoint is min(m,5)=0.5. So the length is 0.5 - (-1) =1.5. Wait, but if m is 0.5, then the interval where |x| ≤0.5 is [-0.5,0.5]. But since original interval is from -1 to5, the overlap is from max(-0.5, -1)= -0.5 to min(0.5,5)=0.5. Wait, so length is 0.5 - (-0.5)=1. So 1/6≈0.1667. Which is not 1/2.Wait, maybe my initial analysis was wrong. Wait, perhaps I need to clarify.Wait, the set {x ∈ [-1,5] | |x| ≤ m} is the intersection of [-m, m] and [-1,5]. So depending on the value of m, the intersection changes.If m <1: Then [-m, m] is within [-1,1]. But since original interval is [-1,5], the intersection is [-m, m] if m ≤1? Wait, for m=0.5, intersection would be [-0.5, 0.5]. But original interval starts at -1, so we can have from -0.5 to0.5. Wait, but -0.5 is greater than -1, so the intersection is indeed [-0.5,0.5]. So length is 1. So for m <1, the length is 2m. But since m is positive, if m is less than1, then the intersection is [-m, m], which has length 2m. But when m >=1, the left side is fixed at -1? Wait, no. Wait, if m >=1, the interval [-m, m] would start at -m, but original interval starts at -1. So if m >=1, then the intersection is [-1, m], since -m <= -1. So in that case, the left endpoint is -1, and the right endpoint is m (if m <=5). So length is m - (-1) = m +1.Therefore, to summarize:If m <=1: length is 2m.If 1 < m <=5: length is m +1.So the total length is:If m <=1: 2mIf 1 < m <=5: m +1We need this length to be 3 (since 3/6=1/2).So:Case 1: If m <=1, then 2m =3 ⇒ m=1.5. But 1.5>1, which contradicts m<=1. So invalid.Case 2: If 1 < m <=5, then m +1=3 ⇒ m=2. Which is within the interval (1,5]. So valid.Therefore, m=2. Which is option C.So that's the correct approach. So earlier reasoning was correct.Therefore, the answer is C: 2.Final Answerboxed{C}</think>To determine the value of ( m ) such that the probability that a randomly selected number ( x ) from the interval ([-1, 5]) satisfies ( |x| leqslant m ) is (frac{1}{2}), we need to find the interval where the length of the subinterval ([-m, m]) overlapping with ([-1, 5]) is 3 (since the total length of the interval ([-1, 5]) is 6, and half of 6 is 3).1. Understanding the interval overlap: - The interval ([-m, m]) overlaps with ([-1, 5]). - The left endpoint of the overlapping interval is (max(-m, -1)). - The right endpoint of the overlapping interval is (min(m, 5)).2. Calculating the length for different values of ( m ): - If ( m leq 1 ), the overlapping interval is ([-m, m]) with length (2m). - If ( 1 < m leq 5 ), the overlapping interval is ([-1, m]) with length (m + 1).3. Setting up the equation for the required probability: - We need the length of the overlapping interval to be 3. - For ( 1 < m leq 5 ), the equation is ( m + 1 = 3 ), which gives ( m = 2 ).Thus, the value of ( m ) that satisfies the condition is ( 2 ).[boxed{C}]]

🔑:To solve this, we use the geometric probability model, where the measure is the length of a segment.Since x in [-1,5] and |x| leqslant m, we get -m leqslant x leqslant m.Therefore, the probability that |x| leqslant m is:P(|x| leqslant m) = dfrac{l}{5-(-1)} = dfrac{1}{2},Solving this, we find l=3,which means m-(-1)=3, thus m=2.Therefore, the correct choice is boxed{C}.In this geometric probability model, the measure is the length of a segment. By using P(|x| leqslant m) = dfrac{1}{2}, we can find the value of m.This question mainly examines the calculation of probabilities in geometric models, where the probability of an event occurring is proportional to the length of the region constituting the event. It is a basic question.

🧠:[<think>Alright, let's tackle this problem step by step. So, we have triangle ABC with angle A equal to 30 degrees, and the equation 2 times the dot product of vectors AB and AC equals 3 times the square of vector BC. We need to find the cosine of the largest angle of this triangle. Hmm, okay. First, I need to recall some vector operations and how they relate to triangles.Let me start by writing down what is given. Angle A is 30°, and 2AB · AC = 3BC². The vectors here are AB and AC, so their dot product is involved. Also, BC squared is the square of the vector BC's magnitude, which would be the same as the square of the side BC in the triangle. So, BC² is just the square of the length of side BC. In a triangle, the largest angle is opposite the longest side. So, first, maybe I need to figure out which side is the longest. To do that, I might need to find the lengths of the sides in terms of each other using the given equation. Let's denote the sides opposite angles A, B, and C as a, b, and c respectively. Wait, actually, in standard notation, side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. So, in this case, angle A is 30°, so side a is BC, side b is AC, and side c is AB. Wait, is that right? Let me confirm: in triangle ABC, vertex A is opposite side a, which is BC. Yes, so side BC is a, AC is b, AB is c.So, given that, angle A is 30°, so we have sides a, b, c opposite their respective angles. Now, the equation given is 2AB · AC = 3BC². Translating this into the sides, AB is vector c, AC is vector b, and BC is vector a. Wait, but vectors AB and AC are vectors from point A to B and from point A to C. So their dot product can be expressed in terms of the sides and the angle between them. The angle between vectors AB and AC is angle A, which is 30°. So, the dot product AB · AC is equal to |AB||AC|cos(30°), which is c*b*(√3/2). So, 2AB · AC would be 2*(c*b*(√3/2)) = c*b*√3.On the other side of the equation, we have 3BC², which is 3 times the square of side a. Therefore, the equation becomes c*b*√3 = 3a². So, simplifying, a² = (c*b*√3)/3 = (c*b)/√3. Hmm, interesting. So, a squared is equal to (c times b) divided by the square root of 3. Hmm, okay.Now, perhaps I can use the Law of Cosines here. Since angle A is known, 30°, maybe we can relate the sides using the Law of Cosines. The Law of Cosines states that a² = b² + c² - 2bc cos A. Since angle A is 30°, this becomes a² = b² + c² - 2bc*(√3/2) = b² + c² - bc√3. But from the given equation, we also have a² = (c*b)/√3. Therefore, setting these two expressions for a² equal to each other:(c*b)/√3 = b² + c² - bc√3Let me write that equation down again:b² + c² - bc√3 = (bc)/√3Let me rearrange terms:b² + c² - bc√3 - (bc)/√3 = 0Combine the terms with bc:b² + c² - bc(√3 + 1/√3) = 0Hmm, √3 + 1/√3 can be simplified. Let's compute that:√3 + 1/√3 = (√3 * √3)/√3 + 1/√3 = (3 + 1)/√3 = 4/√3. Wait, no, that's not correct. Wait, √3 + 1/√3 is equal to (√3 * √3 + 1)/√3 = (3 + 1)/√3 = 4/√3. Wait, actually, that's not correct. Wait, let's compute it properly. Let's rationalize 1/√3 first: 1/√3 = √3/3. Therefore, √3 + 1/√3 = √3 + √3/3 = (3√3 + √3)/3 = (4√3)/3. Yes, that's right. So, √3 + 1/√3 = 4√3 / 3. Therefore, the equation becomes:b² + c² - bc*(4√3 / 3) = 0So, b² + c² = (4√3 / 3) bcHmm, that's a relation between sides b and c. Let me see if I can find a ratio between sides b and c here. Maybe we can divide both sides by bc to get:(b/c) + (c/b) = 4√3 / 3Let me set k = b/c. Then, the equation becomes:k + 1/k = 4√3 / 3Multiply both sides by k:k² + 1 = (4√3 / 3) kWhich is a quadratic equation in k:k² - (4√3 / 3) k + 1 = 0Let's solve this quadratic equation for k. Let's compute the discriminant:D = [ (4√3 / 3) ]² - 4*1*1 = (16*3)/9 - 4 = 48/9 - 4 = 16/3 - 12/3 = 4/3Therefore, the solutions are:k = [ (4√3 / 3 ) ± √(4/3) ] / 2Compute √(4/3) = 2/√3 = 2√3 / 3So,k = [ (4√3 / 3 ) ± (2√3 / 3 ) ] / 2First, the positive sign:k = [ (4√3 + 2√3 ) / 3 ] / 2 = (6√3 / 3 ) / 2 = (2√3)/2 = √3Second, the negative sign:k = [ (4√3 - 2√3 ) / 3 ] / 2 = (2√3 / 3 ) / 2 = √3 / 3Therefore, the ratio k = b/c is either √3 or √3 / 3. Therefore, either b = √3 c or b = (√3 / 3) c. So, two possibilities. So, case 1: b = √3 cCase 2: b = (√3 / 3) cLet me analyze both cases.Case 1: If b = √3 c, then side AC (which is b) is √3 times side AB (which is c). So, sides b and c are related by √3. Let's note that. Then, using the Law of Sines, perhaps we can find the other angles. Wait, or maybe use the Law of Cosines again. Alternatively, since we have angle A and relations between sides, perhaps we can compute angles B and C.Alternatively, since we need to find the largest angle, we can first determine which side is the largest. If angle A is 30°, then the other two angles must sum to 150°, so one of them could be greater than 90°, making it the largest angle.But first, let's see with the two cases.Case 1: b = √3 c. So, side AC is longer than side AB. So, sides: a, b = √3 c, c. We need to find which is the largest side. Let's see. Also, we have from the earlier equation, a² = (c*b)/√3 = (c * √3 c)/√3 = c². So, a² = c², so a = c. Wait, so in this case, side a (BC) is equal to side c (AB). So, sides are a = c, b = √3 c. Therefore, the sides are c, √3 c, c. Therefore, the largest side is b = √3 c, so the largest angle would be angle B, which is opposite side AC (which is b). Wait, no, wait. Wait, angle opposite side b is angle B? Wait, no. Wait, side a is BC, opposite angle A; side b is AC, opposite angle B; side c is AB, opposite angle C. So, in case 1, sides: BC = a = c, AC = b = √3 c, AB = c. So, the largest side is AC = √3 c, so the largest angle is angle B, which is opposite side AC. So, angle B would be the largest angle.Case 2: b = (√3 / 3) c. So, side AC is (√3 / 3) times side AB. So, sides: a, b = (√3 / 3) c, c. Then, a² = (c*b)/√3 = (c * (√3 / 3)c)/√3 = ( (√3 / 3)c² ) / √3 = (c² / 3 ) / √3 = c² / (3√3 ). Hmm, so a² = c² / (3√3 ), so a = c / ( (3√3 )^(1/2) ). That's a bit complicated, but in any case, let's see the relative lengths. If b = (√3 / 3)c, then since √3 ≈ 1.732, √3 /3 ≈ 0.577. So, side AC is about 0.577 times AB. So, sides AB = c, AC ≈0.577c, and BC = a ≈ c / sqrt(3*3^(1/2)), which is a bit messy. Wait, perhaps we can compute a in terms of c. Let's do that:a² = c² / (3√3 ) => a = c / (3√3 )^(1/2) = c / (3^(1/2) * 3^(1/4)) )? Wait, maybe better to rationalize:a² = c² / (3√3 ) = c² / (3^(3/2)) = c² * 3^(-3/2). Therefore, a = c * 3^(-3/4). But 3^(-3/4) is approximately 1 / 3^(0.75) ≈ 1 / 2.279 ≈0.438. So, a ≈0.438c, so sides are a ≈0.438c, b≈0.577c, and c. Therefore, the largest side is c (AB), so the largest angle is angle C, which is opposite side AB.Therefore, in case 1, the largest angle is angle B, and in case 2, it's angle C. Therefore, we need to check both cases.But before that, perhaps we can check which case is possible. Because in a triangle, the sum of two sides must be greater than the third. Let's check for both cases.Case 1: sides are a = c, b = √3 c, c. So, sides c, c, √3 c. Check triangle inequalities:c + c > √3 c => 2c > √3 c => 2 > √3 ≈1.732, which is true.c + √3 c > c => √3 c >0, which is true.Similarly, the other inequalities hold. So, case 1 is possible.Case 2: sides a ≈0.438c, b≈0.577c, c. Check triangle inequalities:0.438c +0.577c > c? 1.015c > c, which is true, just barely.0.438c +c >0.577c: 1.438c >0.577c, true.0.577c +c >0.438c: 1.577c >0.438c, true.So, both cases are possible. Therefore, both cases need to be considered. So, we need to compute the cosine of the largest angle in both cases and see which one is valid. Wait, but the problem doesn't mention that the triangle is unique. So, perhaps both solutions are valid, but we need to check which one actually satisfies the original equation. Wait, but maybe not. Wait, the problem states "the cosine of the largest angle", implying that there is a unique answer. Therefore, maybe one of the cases leads to a contradiction, or perhaps both cases result in the same cosine value. Let's check.First, let's handle Case 1: sides a = c, b = √3 c, c. So, sides BC = a = c, AC = b = √3 c, AB = c. Then, the largest angle is angle B, opposite side AC = √3 c. Let's compute the cosine of angle B using the Law of Cosines:cos B = (a² + c² - b²)/(2ac)Plug in the values:a = c, b = √3 c, so:cos B = (c² + c² - ( (√3 c)^2 )) / (2 * c * c ) = (2c² - 3c²)/ (2c² ) = (-c²)/(2c² ) = -1/2So, cos B = -1/2. Therefore, angle B is 120°, since arccos(-1/2) is 120°.Case 2: sides a = c / √(3√3 ), but wait, in case 2, we have a² = (c*b)/√3, and b = (√3 / 3)c. So, a² = (c * (√3 /3)c ) / √3 = ( (√3 / 3 )c² ) / √3 = (c² / 3 ) / √3 = c² / (3√3 ). Therefore, a = c / (3√3 )^(1/2) = c / ( 3^(1/2) * 3^(1/4) ) = c / 3^(3/4 ). Let's rationalize that, but perhaps it's easier to use actual lengths. Let's let c = 1 for simplicity. Then, in case 2, sides are:AB = c =1, AC = b = √3 /3 ≈0.577, BC = a = sqrt(1 / (3√3 )) ≈ sqrt(1 / 5.196 ) ≈0.438.So, sides: AB=1, AC≈0.577, BC≈0.438. Therefore, the largest side is AB=1, so the largest angle is angle C, which is opposite AB. Let's compute the cosine of angle C using Law of Cosines:cos C = (a² + b² - c² ) / (2ab )Plug in the values:a ≈0.438, b≈0.577, c=1.So,cos C = (0.438² + 0.577² -1² ) / (2 *0.438 *0.577 )Compute numerator:0.438² ≈0.191, 0.577²≈0.333, so total ≈0.191 +0.333 -1 ≈0.524 -1 ≈-0.476Denominator:2 *0.438 *0.577 ≈2 *0.253 ≈0.506Therefore, cos C ≈-0.476 /0.506 ≈-0.940.So, angle C ≈ arccos(-0.940 )≈160 degrees. Wait, but in a triangle, angles can't be more than 180°, and the sum of angles is 180°. If angle A is 30°, angle C is 160°, then angle B would be negative, which is impossible. Wait, that can't be. Therefore, this case leads to an impossible triangle. Therefore, case 2 is invalid. Therefore, only case 1 is valid. Therefore, the answer is cos 120° = -1/2.Wait, but how did that happen? If we set c=1, in case 2, angle C ≈160°, angle A=30°, then angle B≈ -10°, which is impossible, so case 2 is invalid. Therefore, only case 1 is valid, so the cosine of the largest angle is -1/2.But wait, why did the quadratic equation give us two solutions, but one of them leads to an impossible triangle? That must mean that even though mathematically the equations can have two solutions, geometrically only one is possible. So, when we solved the quadratic equation, we found two possible ratios, but one of them leads to a triangle where the angles don't add up. Therefore, only case 1 is valid. Therefore, the answer is -1/2.Therefore, the cosine of the largest angle is -1/2.But let me confirm this again. Let's recap:Given angle A =30°, and 2AB · AC =3BC². We translated the dot product into bc√3 =3a². Then, using the Law of Cosines, we arrived at a quadratic equation in the ratio of sides b/c, leading to two possible solutions. However, one of them (case 2) leads to an impossible triangle where angles would exceed 180°, so only case 1 is valid. Therefore, the largest angle is 120°, and its cosine is -1/2. Therefore, the answer is boxed{-dfrac{1}{2}}.Final Answerboxed{-dfrac{1}{2}}</think>Given triangle ( triangle ABC ) with ( angle A = 30^circ ) and the equation ( 2 overrightarrow{AB} cdot overrightarrow{AC} = 3 overrightarrow{BC}^2 ), we need to find the cosine of the largest angle.First, we translate the vector equation into scalar terms. The dot product ( overrightarrow{AB} cdot overrightarrow{AC} ) is ( |AB||AC|cos(30^circ) ). Given ( 2 overrightarrow{AB} cdot overrightarrow{AC} = 3 overrightarrow{BC}^2 ), we have:[2 cdot |AB||AC| cos(30^circ) = 3 |BC|^2]Substituting ( cos(30^circ) = frac{sqrt{3}}{2} ), we get:[2 cdot |AB||AC| cdot frac{sqrt{3}}{2} = 3 |BC|^2 implies |AB||AC|sqrt{3} = 3 |BC|^2]Let ( a = |BC| ), ( b = |AC| ), and ( c = |AB| ). Then:[bcsqrt{3} = 3a^2 implies a^2 = frac{bc}{sqrt{3}}]Using the Law of Cosines for ( angle A ):[a^2 = b^2 + c^2 - 2bc cos(30^circ) = b^2 + c^2 - bcsqrt{3}]Equating the two expressions for ( a^2 ):[frac{bc}{sqrt{3}} = b^2 + c^2 - bcsqrt{3}]Rearranging terms:[b^2 + c^2 - bcsqrt{3} - frac{bc}{sqrt{3}} = 0 implies b^2 + c^2 - bc left( sqrt{3} + frac{1}{sqrt{3}} right) = 0]Simplifying ( sqrt{3} + frac{1}{sqrt{3}} = frac{4sqrt{3}}{3} ):[b^2 + c^2 = frac{4sqrt{3}}{3} bc]Dividing by ( bc ):[frac{b}{c} + frac{c}{b} = frac{4sqrt{3}}{3}]Let ( k = frac{b}{c} ), then:[k + frac{1}{k} = frac{4sqrt{3}}{3}]Solving the quadratic equation ( k^2 - frac{4sqrt{3}}{3}k + 1 = 0 ), we find:[k = sqrt{3} text{ or } k = frac{sqrt{3}}{3}]For ( k = sqrt{3} ), ( b = sqrt{3}c ). Substituting back, we find ( a = c ). The sides are ( a = c ), ( b = sqrt{3}c ), and ( c ). The largest side is ( b ), opposite angle ( B ). Using the Law of Cosines:[cos B = frac{a^2 + c^2 - b^2}{2ac} = frac{c^2 + c^2 - 3c^2}{2c^2} = frac{-c^2}{2c^2} = -frac{1}{2}]For ( k = frac{sqrt{3}}{3} ), the resulting triangle would have angles summing to more than ( 180^circ ), which is impossible. Thus, the valid solution is ( cos B = -frac{1}{2} ).The cosine of the largest angle is (boxed{-dfrac{1}{2}}).,<think>Okay, so I need to find the cosine of the largest angle in triangle ABC where angle A is 30 degrees, and there's this vector equation: 2 times the dot product of vectors AB and AC equals 3 times the square of vector BC. Hmm, vectors can be tricky, but maybe I can translate this into something more familiar with trigonometry or the law of cosines.First, let me recall that the dot product of two vectors is equal to the product of their magnitudes and the cosine of the angle between them. So, if I have vectors AB and AC, their dot product would be |AB| * |AC| * cos(angle between them). But in triangle ABC, the angle between vectors AB and AC is actually angle A, right? Because AB is from A to B, and AC is from A to C, so the angle between them is angle BAC, which is 30 degrees. So, the dot product of AB and AC is |AB| |AC| cos(30°).But wait, in the problem statement, it's 2 times that dot product equals 3 times the square of vector BC. Vector BC squared would just be the magnitude squared of vector BC, which is the same as BC squared. So, in terms of sides of the triangle, BC is side opposite angle A, which is angle 30 degrees. So, BC is side a, opposite angle A. Wait, in standard notation, usually a is BC, b is AC, and c is AB. So, sides are a, b, c opposite angles A, B, C respectively. But here, angle A is 30°, so side a is BC.So, rewriting the given equation:2 * (AB · AC) = 3 * (BC)^2Translating the dot product:2 * |AB| |AC| cos(30°) = 3 * |BC|^2So, in terms of sides, if AB is side c, AC is side b, and BC is side a, then:2 * b * c * cos(30°) = 3 * a²So, 2bc * (√3 / 2) = 3a²Simplifying the left side: 2bc*(√3/2) = bc√3So, bc√3 = 3a²Thus, bc√3 = 3a² => a² = (bc√3)/3 => a² = (bc)/√3Hmm, that's an equation relating sides a, b, and c. Since we have a triangle with angle A = 30°, maybe we can use the Law of Cosines here. The Law of Cosines says that a² = b² + c² - 2bc cos A. Since angle A is 30°, we can write:a² = b² + c² - 2bc cos(30°)But from earlier, we have a² = (bc)/√3. So:(bc)/√3 = b² + c² - 2bc*(√3/2)Simplify the right-hand side:= b² + c² - bc√3So, we have:(bc)/√3 = b² + c² - bc√3Let me bring all terms to one side:b² + c² - bc√3 - (bc)/√3 = 0Factor bc terms:b² + c² - bc(√3 + 1/√3) = 0Simplify √3 + 1/√3:√3 + 1/√3 = (3 + 1)/√3 = 4/√3Wait, no. Wait, √3 + 1/√3 is equal to ( (√3)^2 + 1 ) / √3 ) = (3 + 1)/√3 = 4/√3. Wait, actually, that's correct. Because √3 + 1/√3 = (3 + 1)/√3 = 4/√3.So, then:b² + c² - bc*(4/√3) = 0Hmm. So, we have the equation:b² + c² - (4/√3) bc = 0This seems a bit complicated. Maybe we can use substitution variables. Let me let x = b/c, so that ratio.Let me divide both sides by c² to make it in terms of x:(b/c)² + 1 - (4/√3)(b/c) = 0So, x² + 1 - (4/√3)x = 0So, x² - (4/√3)x + 1 = 0This is a quadratic equation in x. Let's solve for x.Using the quadratic formula:x = [ (4/√3) ± sqrt( (4/√3)^2 - 4*1*1 ) ] / 2Compute discriminant:(4/√3)^2 - 4 = (16 / 3) - 4 = (16/3 - 12/3) = 4/3Thus, sqrt(4/3) = 2/√3Therefore:x = [4/√3 ± 2/√3] / 2So:x = ( (4 ± 2)/√3 ) / 2 = ( (6/√3) or (2/√3) ) / 2Thus:First solution: (6/√3)/2 = (6)/(2√3) = 3/√3 = √3Second solution: (2/√3)/2 = 1/√3So, x = √3 or x = 1/√3But x is b/c. So, either b/c = √3 => b = c√3, or b/c = 1/√3 => b = c/√3So, there are two possibilities for the ratio of sides b and c. Let's analyze both cases.Case 1: b = c√3Case 2: b = c/√3We need to see which case leads to the triangle with angle A = 30°, and then determine the largest angle.But in a triangle, the largest angle is opposite the longest side. So, we need to find which side is the longest.First, let's recall that in a triangle, the sides are related to the angles opposite them. The largest side is opposite the largest angle.Given angle A is 30°, which is the smallest angle, so the largest angle will be either angle B or C, depending on the lengths of sides b and c.But in Case 1: b = c√3, so b is longer than c. So side AC is longer than side AB. Therefore, angle B (opposite side AC) is larger than angle C (opposite side AB). So, angle B is larger. Since angle A is 30°, the largest angle is angle B.In Case 2: b = c/√3, so b is shorter than c. So side AC is shorter than side AB. Thus, angle C (opposite side AB) is larger than angle B (opposite side AC). Therefore, angle C is the largest angle.But in both cases, we need to compute the cosine of the largest angle. Let's handle both cases.First, let's compute sides in terms of c.Case 1: b = c√3Then, using Law of Cosines for angle A:a² = b² + c² - 2bc cos(30°)But from earlier, we also have a² = (bc)/√3. Let's verify.Wait, in the problem statement, we derived a² = (bc)/√3. But in this case, if b = c√3, then a² = (c√3 * c)/√3 = c². So, a = c.Therefore, BC = a = c, and AB = c, AC = b = c√3.So, sides are AB = c, BC = c, AC = c√3. So, triangle with sides c, c, c√3. Wait, but angle A is 30°, so this must hold. Let's check.If AB = c, AC = c√3, BC = c. Then angle A is between AB and AC, which are sides of length c and c√3, and opposite side BC = c.Using the Law of Cosines:BC² = AB² + AC² - 2 AB AC cos(angle A)So, c² = c² + (c√3)² - 2 * c * c√3 * cos(30°)Compute:c² = c² + 3c² - 2c²√3 * (√3/2)Simplify:c² = 4c² - 2c²√3*(√3/2)Compute the last term: 2c²√3*(√3/2) = 2c²*(3)/2 = 3c²Thus:c² = 4c² - 3c² = c²Which holds true. So, this case is valid.In this case, sides are AB = c, BC = c, and AC = c√3. Therefore, the largest side is AC = c√3, so the largest angle is angle B, which is opposite AC. So, angle B is the largest angle.Now, let's compute cos(angle B). Using Law of Cosines:cos B = (AB² + BC² - AC²)/(2 AB BC)Plugging in:AB = c, BC = c, AC = c√3So,cos B = (c² + c² - (c√3)^2 ) / (2 * c * c ) = (2c² - 3c²)/2c² = (-c²)/2c² = -1/2Wait, cos B = -1/2. But in a triangle, angles are between 0 and 180°, so cosine can be negative, which would mean the angle is obtuse. But angle B can't be more than 180°, but in this case, angle B is 120°, because cos 120° = -1/2.But in this triangle, angle A is 30°, angle B is 120°, angle C would be 30°, since 30 + 120 + 30 = 180. Wait, but angle C is opposite side AB, which is length c. But in this case, sides AB and BC are both length c, so triangle is isoceles with AB = BC, so angles opposite them should be equal. Wait, but AB is opposite angle C, and BC is opposite angle A. Wait, no. AB is side c, opposite angle C; BC is side a, opposite angle A. Wait, in standard notation, side a is BC, opposite angle A; side b is AC, opposite angle B; side c is AB, opposite angle C. So, if AB = BC = c, then sides AB and BC are both length c, so sides c and a are equal. Wait, but in standard notation, side a is BC, side c is AB. So, if BC = AB, then sides a and c are equal. Therefore, angles opposite them are equal, i.e., angle A and angle C. But angle A is 30°, so angle C is also 30°, and angle B is 180 - 30 - 30 = 120°. So, yes, that's correct. So, in this case, the largest angle is 120°, whose cosine is -1/2.Case 2: b = c / √3So, in this case, AC = b = c / √3, AB = c, BC = a.Again, from the given equation, a² = (b c)/√3 = (c / √3 * c)/√3 = c² / (3) / √3? Wait, wait. Let me compute a² again.Earlier, we had a² = (bc)/√3. Since in this case, b = c / √3, then a² = (c / √3 * c)/√3 = (c² / √3)/√3 = c² / 3So, a = c / √3Therefore, sides are AB = c, AC = c / √3, BC = a = c / √3Wait, so BC and AC are both equal to c / √3. Therefore, sides AC and BC are equal. Therefore, triangle is isoceles with AC = BC, so angles opposite them are equal. Which angles? Side AC is opposite angle B, and side BC is opposite angle A. But angle A is 30°, so angle B is equal to angle A? Wait, no. Wait, if AC = BC, then angles opposite them are equal. But AC is opposite angle B, and BC is opposite angle A. So, angle B = angle A = 30°, which would make angle C = 120°. But angle C is opposite side AB, which is length c. If angle C is 120°, then using Law of Cosines:AB² = AC² + BC² - 2 AC BC cos(angle C)But AB = c, AC = BC = c / √3So:c² = (c² / 3) + (c² / 3) - 2*(c / √3)*(c / √3)*cos(120°)Simplify:c² = (2c² / 3) - 2*(c² / 3)*cos(120°)Compute cos(120°) = -1/2Thus:c² = (2c² / 3) - 2*(c² / 3)*(-1/2) = (2c² / 3) + (c² / 3) = c²Which holds true. So, this case is valid.In this case, sides AB = c, AC = BC = c / √3. So, the largest side is AB = c, opposite angle C. Therefore, angle C is the largest angle, which is 120°, so its cosine is -1/2.Wait, so in both cases, the largest angle is 120°, so cosine is -1/2. Therefore, regardless of the case, the cosine of the largest angle is -1/2.Therefore, the answer is -1/2, which can be written as boxed{-dfrac{1}{2}}.But wait, let me just confirm once again. Both cases lead to the largest angle being 120°, whose cosine is indeed -1/2. Therefore, regardless of whether b is larger or smaller than c, the largest angle ends up being 120°, so the answer is consistent.Alternatively, maybe there's another approach. Let me think.Alternatively, using vectors, perhaps coordinate geometry.Let me place point A at the origin, vector AB along the x-axis. So, coordinates:Point A: (0, 0)Point B: (c, 0), since AB is a vector of length c along the x-axis.Point C: coordinates (b cos 30°, b sin 30°), because angle at A is 30°, and AC has length b.Wait, but if angle at A is 30°, then coordinates of C can be written as (b cos 30°, b sin 30°). Then vector AB is (c, 0), vector AC is (b cos 30°, b sin 30°)Then, the dot product AB · AC is (c, 0) · (b cos 30°, b sin 30°) = c * b cos 30° + 0 * b sin 30° = bc cos 30°, which matches the previous result.Then, vector BC is C - B = (b cos 30° - c, b sin 30° - 0) = (b cos 30° - c, b sin 30°)The square of vector BC is the magnitude squared:|BC|² = (b cos 30° - c)^2 + (b sin 30°)^2Which is:= b² cos² 30° - 2bc cos 30° + c² + b² sin² 30°= b² (cos² 30° + sin² 30°) - 2bc cos 30° + c²= b² (1) - 2bc cos 30° + c²Which is just the Law of Cosines expression for a² = b² + c² - 2bc cos A, which is consistent.Given that 2 AB · AC = 3 |BC|²So, 2 bc cos 30° = 3 a²But a² = b² + c² - 2bc cos 30°, so substituting:2 bc cos 30° = 3(b² + c² - 2 bc cos 30°)Bring all terms to one side:2 bc cos 30° - 3b² - 3c² + 6 bc cos 30° = 0Combine like terms:(2 bc cos 30° + 6 bc cos 30°) - 3b² - 3c² = 08 bc cos 30° - 3b² - 3c² = 0But cos 30° is √3/2, so:8 bc (√3/2) - 3b² - 3c² = 0Simplify:4 bc √3 - 3b² - 3c² = 0Divide both sides by 3:(4 bc √3)/3 - b² - c² = 0Rearranged:b² + c² - (4√3/3) bc = 0Which is the same equation as before. So, leading to the same quadratic. Therefore, same solutions, so same conclusion. Therefore, in both cases, the largest angle is 120°, cos is -1/2.Therefore, the answer is boxed{-dfrac{1}{2}}.Final Answerboxed{-dfrac{1}{2}}</think>Given triangle ABC with angle A = 30^circ and the vector equation 2 overrightarrow{AB} cdot overrightarrow{AC} = 3 overrightarrow{BC}^2, we need to find the cosine of the largest angle.1. Translate the vector equation into scalar terms: - The dot product overrightarrow{AB} cdot overrightarrow{AC} is |AB||AC|cos 30^circ. - The magnitude squared overrightarrow{BC}^2 is |BC|^2.2. Substitute the known values: - 2 |AB||AC|cos 30^circ = 3 |BC|^2 - Using cos 30^circ = frac{sqrt{3}}{2}, we get: [ 2 |AB||AC| frac{sqrt{3}}{2} = 3 |BC|^2 implies |AB||AC|sqrt{3} = 3 |BC|^2 implies |BC|^2 = frac{|AB||AC|sqrt{3}}{3} ]3. Apply the Law of Cosines: - For triangle ABC, |BC|^2 = |AB|^2 + |AC|^2 - 2|AB||AC|cos 30^circ: [ frac{|AB||AC|sqrt{3}}{3} = |AB|^2 + |AC|^2 - |AB||AC|sqrt{3} ]4. Solve the resulting equation: - Let b = |AC| and c = |AB|. Then: [ frac{bcsqrt{3}}{3} = b^2 + c^2 - bcsqrt{3} ] - Multiply through by 3 and rearrange: [ bcsqrt{3} = 3b^2 + 3c^2 - 3bcsqrt{3} implies 4bcsqrt{3} = 3b^2 + 3c^2 ] - Divide by bc and let x = frac{b}{c}: [ 4sqrt{3} = 3x + frac{3}{x} implies 3x^2 - 4sqrt{3}x + 3 = 0 ] - Solving this quadratic equation gives x = sqrt{3} or x = frac{1}{sqrt{3}}.5. Determine the largest angle: - For x = sqrt{3} or x = frac{1}{sqrt{3}}, the sides b and c are in ratios leading to the largest angle being 120^circ (opposite the longest side). - The cosine of 120^circ is -frac{1}{2}.Thus, the cosine of the largest angle in triangle ABC is boxed{-dfrac{1}{2}}.]

🔑:3. -frac{1}{2}.Let A B=c, A C=b, B C=a.From 2 overrightarrow{A B} cdot overrightarrow{A C}=3 overrightarrow{B C}^{2} Rightarrow 2 c b cos A=3 a^{2}.Since angle A=30^{circ}, we have b c=sqrt{3} a^{2}.Combining with the sine rule, we getbegin{array}{l}sin B cdot sin C=sqrt{3} sin ^{2} A=frac{sqrt{3}}{4} Rightarrow cos (C-B)-cos (C+B)=frac{sqrt{3}}{2} text {. } text { Also } angle C+angle B=pi-angle A=150^{circ} text {, then } cos (C-B)=0 Rightarrow|angle C-angle B|=90^{circ} Rightarrow angle C=120^{circ}, angle B=30^{circ}end{array}or angle C=30^{circ}, angle B=120^{circ}.Therefore, the cosine value of the largest angle in triangle A B C is -frac{1}{2}.

🧠:[<think>Okay, let's see. There are three people: A, B, and C. Only one of them can play the piano. Their statements are:- A says, "I can."- B says, "I can't."- C says, "A can't."And the condition is that only one of these three statements is true. We need to figure out who is the pianist.Hmm, so first, since only one person is telling the truth, and only one person can play the piano. Let me think step by step.Let's consider each case where each person is the pianist and check how many statements are true in each scenario.Case 1: Suppose A is the pianist. Then:- A's statement "I can" is true.- B's statement "I can't" is also true because if A is the pianist, then B can't play, so B is telling the truth.- C's statement "A can't" is false because A can play.So in this case, two statements are true (A and B). But the problem says only one statement is true. Therefore, A cannot be the pianist. So we can eliminate A.Case 2: Suppose B is the pianist. Then:- A's statement "I can" is false because B is the pianist.- B's statement "I can't" is false because B actually can play.- C's statement "A can't" is true because A is not the pianist.So here, only C's statement is true. That fits the condition: only one statement is true. So this seems possible. Let me check again.If B can play, then A's claim is false, B's denial is false (since B can play), and C's statement that A can't is true. So yes, only C is telling the truth. That works. So B could be the answer. Let me check the third case to be thorough.Case 3: Suppose C is the pianist. Then:- A's statement "I can" is false.- B's statement "I can't" is true because if C is the pianist, B can't play.- C's statement "A can't" is true because A is not the pianist.So here, both B and C's statements are true, which makes two truths. But the problem states only one is true. Therefore, C cannot be the pianist.So summarizing:- If A is pianist: 2 truths (A and B) → Not possible.- If B is pianist: 1 truth (C) → Possible.- If C is pianist: 2 truths (B and C) → Not possible.Therefore, the only valid case is when B is the pianist. So the answer should be B.Wait a second, let me just verify once more because sometimes these logic problems can trick you.If B is the pianist, then:- A is lying about being able to play.- B is lying when saying "I can't" because B can play.- C is telling the truth when saying "A can't."Yes, exactly one true statement (C's), which fits the condition. So yes, B is the correct answer. I think that's solid. No contradictions here.Final AnswerThe person who can play the piano is boxed{B}.</think>Let's analyze each case to determine who can play the piano given that only one statement is true:1. Case 1: Suppose A is the pianist. - A's statement "I can" is true. - B's statement "I can't" is true (since B can't play if A is the pianist). - C's statement "A can't" is false. - Result: Two true statements (A and B). This does not satisfy the condition of only one true statement.2. Case 2: Suppose B is the pianist. - A's statement "I can" is false. - B's statement "I can't" is false (since B is the pianist). - C's statement "A can't" is true (since A is not the pianist). - Result: One true statement (C). This satisfies the condition of only one true statement.3. Case 3: Suppose C is the pianist. - A's statement "I can" is false. - B's statement "I can't" is true (since B can't play if C is the pianist). - C's statement "A can't" is true (since A is not the pianist). - Result: Two true statements (B and C). This does not satisfy the condition of only one true statement.The only valid case is when B is the pianist, resulting in exactly one true statement (C's statement). [boxed{B}],<think>Okay, let's see. So there's this problem where there are three people: A, B, and C. Only one of them knows how to play the piano. Then each of them makes a statement. A says, "I can," meaning A claims to be able to play. B says, "I can't," denying their own ability. C says, "A can't," which contradicts A's statement. The key detail here is that only one of these three statements is true. Our job is to figure out who the actual piano player is.Hmm. Let's break this down step by step. Since only one person can play the piano, and only one statement is true, we need to check each possibility and see which scenario fits the condition.First, let's consider each person being the piano player and see how many statements would be true in each case. The correct case should result in exactly one true statement.Case 1: Suppose A is the piano player. Then let's check each statement's truthfulness.- A's statement: "I can." If A is the piano player, this statement is true.- B's statement: "I can't." If B is not the piano player, then B's statement is actually true because B indeed can't play. Wait, but if A is the piano player, then B's statement "I can't" is also true. But the problem states only one statement is true. So in this case, both A and B's statements would be true. That's two true statements, which violates the condition. Therefore, A cannot be the piano player.Case 2: Suppose B is the piano player. Let's check the statements again.- A's statement: "I can." If A is not the piano player, this is false.- B's statement: "I can't." But if B is the piano player, this statement is a lie. So it's false.- C's statement: "A can't." Since A isn't the piano player, this is actually true. So here, only C's statement is true. That fits the condition of only one true statement. Wait, so if B is the piano player, then only C's statement is true. But B is the one who can play, yet B says "I can't," which would be a lie. So that seems to work because only C is telling the truth. So this could be a possible scenario. Let me check again. If B can play, then:- A says "I can" – false.- B says "I can't" – false.- C says "A can't" – true. Yes, exactly one true statement. So this works. So B is a possible candidate.Case 3: Suppose C is the piano player. Let's check the statements.- A's statement: "I can" – false, since C is the piano player.- B's statement: "I can't" – true, because B isn't the piano player.- C's statement: "A can't" – true, because A isn't the piano player. So here, both B and C's statements are true. That's two true statements, which again violates the condition. Therefore, C can't be the piano player.Wait, so when I checked case 2, B being the piano player resulted in only one true statement (C's). Whereas in case 1 and 3, we get two true statements each. Therefore, the answer must be B. But wait, let me double-check my reasoning because sometimes these puzzles can be tricky.Wait, hold on. If B is the piano player, then B's own statement "I can't" is false, which is correct because B can play. Then C's statement "A can't" is true because A isn't the piano player. So only C's statement is true, so that's one. Correct. So that works.But let me think again. Maybe I missed something. Let's go through each case again.Case 1: A is the pianist.- A's statement: True- B's statement: True (because B can't)- C's statement: False (because A can)- So two truths, which is invalid.Case 2: B is the pianist.- A's statement: False- B's statement: False (B can play, so "I can't" is a lie)- C's statement: True (A can't)- So only one true statement. Valid.Case 3: C is the pianist.- A's statement: False- B's statement: True (B can't)- C's statement: True (A can't)- Two truths, invalid.Therefore, the only valid case is when B is the pianist. So the answer should be B. Wait, but hold on. Let me check once more. Maybe I made a mistake in the logic.Wait, another way to approach this is to assume each statement is the true one and see if that leads to a unique pianist.Suppose only A's statement is true. Then:- A can play (since A's statement is true)- B's statement is false, so B can play. But that would mean both A and B can play, which contradicts the problem's condition that only one can play. So this is impossible.Suppose only B's statement is true. Then:- B's statement "I can't" is true, so B cannot play.- A's statement "I can" is false, so A cannot play.- C's statement "A can't" is false, so A can play. But this contradicts A's statement being false. So this is impossible.Suppose only C's statement is true. Then:- C's statement "A can't" is true, so A cannot play.- A's statement "I can" is false, which matches.- B's statement "I can't" is false, so B can play.- Therefore, the pianist is B. Which matches our previous conclusion. So this works.Therefore, when only C's statement is true, B is the pianist. So yes, that's consistent. Therefore, the answer is B. So the person who can play the piano is B. Hmm. But wait, does B's statement being false mean B can play? Yes. Because B says "I can't," so if that's a lie, then B can play. So yes, that's correct.Therefore, the answer is B. Let me check once more. In the scenario where only C's statement is true, that gives:- A cannot play (from C's true statement)- B can play (from B's false statement)- C cannot play (since only one can play, and it's B)So yes, exactly one pianist (B), and only one true statement (C's). Perfect.Therefore, the answer is B.Final AnswerThe person who can play the piano is boxed{B}.</think>Given three people A, B, and C, where only one knows how to play the piano, and their statements:- A says, "I can."- B says, "I can't."- C says, "A can't."We need to determine who the piano player is, given that only one of the three statements is true.Case 1: Suppose A is the piano player.- A's statement is true.- B's statement ("I can't") is true (since B is not the piano player).- C's statement ("A can't") is false.- This results in two true statements, which is invalid.Case 2: Suppose B is the piano player.- A's statement ("I can") is false.- B's statement ("I can't") is false (since B is the piano player).- C's statement ("A can't") is true (since A is not the piano player).- This results in one true statement, which is valid.Case 3: Suppose C is the piano player.- A's statement ("I can") is false.- B's statement ("I can't") is true (since B is not the piano player).- C's statement ("A can't") is true (since A is not the piano player).- This results in two true statements, which is invalid.Thus, the only valid case is when B is the piano player, resulting in exactly one true statement (C's statement).[boxed{B}]]

🔑:1. Suppose A can play the piano. Then, both A and B's statements are true, which contradicts the given condition. Therefore, A cannot play the piano.2. Suppose B can play the piano. Then, C's statement is true, which is consistent with the given condition. Therefore, B can play the piano.3. Suppose C can play the piano. Then, both B and C's statements are true, which contradicts the given condition. Therefore, C cannot play the piano.Comprehensive analysis of 1, 2, and 3: B can play the piano.Therefore, the answer is: boxed{text{B}}.First, understand the problem, then conduct simple logical reasoning, and verify each possibility to find the solution. This question tests the ability to conduct simple logical reasoning and is considered an easy question.

🧠:[<think>Okay, so there's this problem: 48 blacksmiths need to shoe 60 horses. Each blacksmith spends 5 minutes on one horseshoe. We need to find the minimum time they'll spend on the work. Hmm, let me think about this step by step.First, I need to recall that shoeing a horse typically involves attaching horseshoes to all four of its hooves. So each horse would need four horseshoes. Wait, but the problem says "shoe 60 horses." Does that mean each horse needs one shoe or four? The wording is a bit unclear. Let me check the problem again. It says "shoe 60 horses." In real life, shoeing a horse means putting on four shoes, so maybe each horse requires four horseshoes. But the problem might be simplified here. Let me see how it's presented. The blacksmith spends 5 minutes on one horseshoe. So each horseshoe takes 5 minutes. Therefore, if each horse needs four horseshoes, then each horse would take 4*5=20 minutes per blacksmith? Wait, no. Wait, maybe not. Let's parse this carefully.Each blacksmith spends 5 minutes on one horseshoe. So one horseshoe takes 5 minutes of a blacksmith's time. So the time required per horseshoe is 5 minutes, regardless of how many blacksmiths are working. So the total number of horseshoes needed is 60 horses multiplied by 4 shoes per horse, which is 240 horseshoes. Each horseshoe takes 5 minutes of work. So total work time is 240 * 5 = 1200 minutes. Now, if there are 48 blacksmiths working simultaneously, the total time required would be the total work time divided by the number of blacksmiths. So 1200 / 48 = 25 minutes. Therefore, the minimum time is 25 minutes. Hmm, but wait, is that correct? Let me verify.Alternatively, maybe the problem is considering that each horse is shod one shoe at a time, and multiple blacksmiths can work on the same horse? Wait, but in reality, a horse is usually shod by one blacksmith at a time, working on one hoof. But maybe in this problem, they can work on different hooves simultaneously. If that's allowed, then maybe each horse can have up to four blacksmiths working on it at the same time. But the problem doesn't specify any constraints on whether multiple blacksmiths can work on the same horse. Hmm. But in the absence of such constraints, we can assume that they can work on different horseshoes even on the same horse. So, the total number of horseshoes is 60 * 4 = 240. Each takes 5 minutes. So total work is 240 * 5 = 1200 minutes. With 48 blacksmiths working in parallel, dividing the work, the time would be 1200 / 48 = 25 minutes. That seems right.But wait, maybe the problem is simpler. Maybe "shoe a horse" means attaching all four shoes, so each horse requires 4 shoes, each taking 5 minutes. Therefore, per horse, it's 4*5=20 minutes. So 60 horses would be 60*20=1200 minutes. Then with 48 blacksmiths, 1200/48=25 minutes. So same answer. So regardless of whether you think per horseshoe or per horse, the total time comes out the same because if each horseshoe is 5 minutes, the total per horse is 4*5=20, so total time is 60*20=1200. Divided by 48 blacksmiths gives 25 minutes. So that seems to check out.But wait, another thought. If multiple blacksmiths can work on the same horse, but if a single blacksmith can only work on one horseshoe at a time, then each horseshoe is a separate task. So each horseshoe is a 5-minute task. So with 240 tasks (horseshoes) and 48 workers, the time required is ceiling(240 / 48) * 5? Wait, no. Wait, each horseshoe is 5 minutes of work. If you have 48 blacksmiths, each working on separate horseshoes. So in 5 minutes, each can complete one horseshoe. So every 5 minutes, 48 horseshoes are done. Then total number of intervals needed is 240 / 48 = 5. So 5 intervals of 5 minutes each, totaling 25 minutes. So that's the same answer.Alternatively, if they can work in parallel, then yes, 25 minutes. So whether you look at the total time as 240 tasks each taking 5 minutes, divided by 48 workers, gives 240*5 /48 = 25 minutes. That's correct.Therefore, the minimum time is 25 minutes. Hmm. So that seems to be the answer. Let me just check for any possible mistakes. Did I account for all horseshoes? 60 horses * 4 shoes = 240. Yes. Each shoe is 5 minutes. So 240*5 = 1200 minutes total work. Divided by 48 smiths: 1200/48 = 25. Correct. So yeah, 25 minutes. Okay, I think that's solid.</think>The minimum time required is calculated by first determining the total number of horseshoes needed. Each horse requires 4 shoes, so 60 horses need (60 times 4 = 240) horseshoes. Each horseshoe takes 5 minutes of work, resulting in a total work time of (240 times 5 = 1200) minutes. With 48 blacksmiths working simultaneously, the total time is (frac{1200}{48} = 25) minutes.(boxed{25}),<think>Okay, let me try to figure out this problem. So, there are 48 blacksmiths who need to shoe 60 horses. Each blacksmith spends 5 minutes on one horseshoe. The question is asking for the minimum time they will spend on the work. Hmm, right. Let me break this down step by step.First, I need to understand how many horseshoes are required in total. Wait, each horse typically has 4 horseshoes, right? One for each hoof. So, if there are 60 horses, each needing 4 shoes, that would be 60 multiplied by 4. Let me calculate that: 60 × 4 = 240. So, there are 240 horseshoes needed in total.Now, each blacksmith works on one horseshoe for 5 minutes. So, the time required depends on how the work is distributed among the blacksmiths. Since we want the minimum time, we need to maximize parallelism, meaning each blacksmith should be working simultaneously as much as possible.Each horseshoe takes 5 minutes of a blacksmith's time. So, the total amount of work required is 240 horseshoes × 5 minutes per horseshoe. Let's compute that: 240 × 5 = 1200 minutes. That's the total person-minutes required.Now, with 48 blacksmiths working simultaneously, we can divide the total work by the number of blacksmiths to find the time needed. So, 1200 minutes divided by 48 blacksmiths. Let me do that division: 1200 ÷ 48. Hmm, 48 × 25 = 1200. Wait, 48 × 20 = 960, subtract that from 1200, we get 240. 48 × 5 = 240. So 20 + 5 = 25. Therefore, 1200 ÷ 48 = 25. So, the time needed is 25 minutes?But wait, is there a catch here? Let me double-check. Each horse needs four shoes, but can a blacksmith work on multiple shoes for the same horse? Or is there a constraint that a horse can only be shoed by one blacksmith at a time? The problem doesn't specify any constraints on the horses, so I think multiple blacksmiths can work on different shoes of the same horse simultaneously. Therefore, the total time isn't limited by the number of horses but by the total number of horseshoes divided by the number of blacksmiths, adjusted by the time per shoe.So, if each horseshoe is an independent task that any blacksmith can do, then yes, splitting the total workload (240 shoes) among 48 blacksmiths would take 25 minutes. Each blacksmith would handle 240 ÷ 48 = 5 shoes. Since each shoe takes 5 minutes, one blacksmith doing 5 shoes would take 5 × 5 = 25 minutes. So all blacksmiths working in parallel would finish their assigned shoes in 25 minutes. That seems correct.But let me think again. Suppose all blacksmiths start at the same time. Each can work on a different horseshoe. But since there are 240 shoes, and 48 blacksmiths, each would need to do 5 shoes. Each shoe is 5 minutes. If they work sequentially on their assigned shoes, then each blacksmith would take 5 shoes × 5 minutes per shoe = 25 minutes. Since they all finish at the same time, the total time is 25 minutes.Alternatively, is there a way to parallelize even more? For example, if a blacksmith can work on multiple shoes at the same time? But the problem states that each blacksmith spends 5 minutes on one horseshoe. So I think that implies that each horseshoe requires 5 minutes of focused work by a blacksmith. So they can't do multiple at once. Each shoe is 5 minutes, and they can only work on one shoe at a time. Therefore, each blacksmith must handle their shoes one after another. Hence, with 5 shoes each, it's 25 minutes.Therefore, the minimum time required is 25 minutes. Let me check if there's another angle. Maybe each horse needs all four shoes to be done in sequence? But that would depend on the problem statement. The problem doesn't specify that a horse can't be shoed by multiple blacksmiths at the same time. In reality, a horse is shoed one hoof at a time, but unless the problem states that only one blacksmith can work on a horse at a time, we can assume that multiple blacksmiths can work on different hooves of the same horse simultaneously. Therefore, this isn't a constraint here. So our original reasoning holds.Therefore, the minimum time is 25 minutes. That should be the answer.</think>To determine the minimum time required for 48 blacksmiths to shoe 60 horses, we first calculate the total number of horseshoes needed. Each horse requires 4 shoes, so:[60 text{ horses} times 4 text{ shoes/horse} = 240 text{ shoes}.]Each horseshoe takes 5 minutes of work by a blacksmith. The total work required is:[240 text{ shoes} times 5 text{ minutes/shoe} = 1200 text{ person-minutes}.]With 48 blacksmiths working in parallel, the time needed is the total work divided by the number of blacksmiths:[frac{1200 text{ person-minutes}}{48 text{ blacksmiths}} = 25 text{ minutes}.]Each blacksmith handles (frac{240}{48} = 5) shoes, taking (5 times 5 = 25) minutes. Assuming no constraints on concurrent work for the same horse, the minimum time is:[boxed{25}]]

🔑:Notice that four blacksmiths cannot shoe one horse at the same time.## SolutionIt will take no less than 60 cdot 4 cdot 5: 48=25 minutes. For this, it is necessary that none of the blacksmiths are idle. It is sufficient to show that four blacksmiths can shoe five horses in 25 minutes. For this, arrange the five horses in a circle. Four blacksmiths approach the first four horses and shoe (in 5 minutes) one hoof each. Then each moves to the next horse clockwise and so on.Send a comment

🧠:[<think>Okay, let's see. I need to find the area of a sphere that's tangent to lines AB and AD at point A, and also tangent to lines BC and CD at point C. The given information includes AB=1, BD=2, and angles ABC and BAD are both 90 degrees. The four points A, B, C, D are not coplanar. Hmm, this seems a bit complex, but I'll try to break it down step by step.First, let me visualize the setup. Since angles at B and A are right angles, maybe the points form some sort of three-dimensional figure. Let me start by sketching a rough diagram in my mind. Points A, B, C, D are not in the same plane, so it's a tetrahedron-like structure. But given the right angles, maybe it's part of a cube or a rectangular prism?Given that ABC and BAD are 90 degrees, so at point B, angle ABC is 90°, meaning AB is perpendicular to BC. Similarly, angle BAD is 90°, so BA is perpendicular to AD. So AB is perpendicular to both BC and AD. That suggests that AB is like an edge from which two other edges (BC and AD) extend perpendicularly. So maybe AB is orthogonal to both BC and AD.Since AB=1, BD=2. BD is the distance from B to D, which is 2. Let me think about coordinates. Maybe assigning coordinates to the points will help. Let's place point A at the origin (0,0,0). Then, since AB=1 and angle BAD is 90°, we can set point B along the x-axis: B=(1,0,0). Then, since angle BAD is 90°, AD is perpendicular to AB. If AB is along the x-axis, then AD can be along the y-axis or z-axis. Let's choose y-axis for simplicity. So point D would be (0, y, 0). But BD=2. Let's compute BD. The coordinates of B are (1,0,0), D is (0, y, 0). Then BD distance is sqrt[(1-0)^2 + (0 - y)^2 + (0 - 0)^2] = sqrt(1 + y²) = 2. So sqrt(1 + y²)=2 => 1 + y²=4 => y²=3 => y=√3. So point D is (0, √3, 0).Now, point C. Since angle ABC is 90°, which is at point B. So AB is along x-axis from A(0,0,0) to B(1,0,0). Then BC is perpendicular to AB. Since AB is along the x-axis, BC can be in the y-z plane. Let's denote point C as (1, m, n). Since BC is from B(1,0,0) to C(1, m, n), the vector BC is (0, m, n). AB is along (1,0,0), so BC is perpendicular to AB, which is satisfied since their dot product is 1*0 + 0*m + 0*n = 0. So any point C with x-coordinate 1 will satisfy that ABC angle is 90°, which is given. But we also need the sphere to be tangent to lines BC and CD at point C. Hmm, so the sphere is tangent to BC and CD at C, and tangent to AB and AD at A. Also, points A, B, C, D are not coplanar, so point C must not lie in the same plane as A, B, D. Since A, B, D are in the xy-plane (z=0), point C must have a non-zero z-coordinate. So n ≠ 0.Additionally, the sphere is tangent to lines BC and CD at C. Similarly, it's tangent to lines AB and AD at A. So the sphere touches those lines only at points A and C. So the sphere is tangent to four lines: AB, AD, BC, CD, at points A and C. The sphere's surface touches each of these lines at those points, meaning the lines are tangent to the sphere at those points. So the sphere must have tangent lines AB and AD at A, and tangent lines BC and CD at C. To find the area of the sphere, I need to find its radius first. Then area is 4πr².Therefore, the key is to find the radius of the sphere. To find the sphere's radius, maybe we can find its center and then compute the distance from the center to any of the tangent points (A or C). Since the sphere is tangent to lines at those points, the radius at the point of tangency should be perpendicular to the tangent line. So the center of the sphere must lie along the line perpendicular to each tangent line at the point of tangency.So, for point A: the sphere is tangent to lines AB and AD at A. So the center of the sphere, let's call it O, must lie along the line that is perpendicular to both AB and AD at point A. Similarly, at point C, the sphere is tangent to BC and CD, so the center O must also lie along the line perpendicular to both BC and CD at point C.Therefore, the center O lies along both of these perpendicular lines: the one at A and the one at C. Hence, the center is the intersection point of these two lines (the common perpendicular line). Therefore, if we can find the equations of these two lines (the normals at A and C), their intersection will be the center O. Then, the distance from O to A (or O to C) will be the radius.So let's try to formalize this.First, at point A(0,0,0), the sphere is tangent to lines AB and AD. Let's find the direction vectors of AB and AD. AB is from A(0,0,0) to B(1,0,0), so direction vector is (1,0,0). AD is from A(0,0,0) to D(0,√3,0), direction vector is (0,√3,0). So these two lines AB and AD lie in the xy-plane. The sphere is tangent to both these lines at A, so the center O must lie along the line perpendicular to both AB and AD at A. Since AB and AD are in the xy-plane, their cross product will give a vector perpendicular to both. The cross product of AB direction vector (1,0,0) and AD direction vector (0,√3,0) is (0,0,√3). So the line perpendicular to both AB and AD at A is along the z-axis. Therefore, the center O lies somewhere along the z-axis, so coordinates (0,0,k) for some k.Similarly, at point C, the sphere is tangent to lines BC and CD. Let's find the direction vectors of BC and CD. First, we need coordinates of point C. Wait, we haven't determined coordinates of C yet. Let's do that.Earlier, we placed point C at (1, m, n). Since we need to determine m and n. Also, the line CD goes from C(1, m, n) to D(0, √3, 0). The direction vector of CD is (-1, √3 - m, -n). The direction vector of BC is (0, m, n), as BC is from B(1,0,0) to C(1, m, n).Since the sphere is tangent to BC and CD at C, the center O must lie along the line perpendicular to both BC and CD at C. So we need to find the direction of this perpendicular line. The cross product of the direction vectors of BC and CD will give a vector perpendicular to both, which is the direction of the line on which the center lies.First, let's compute direction vectors:Vector BC is (0, m, n).Vector CD is (-1, √3 - m, -n).Compute cross product of BC and CD:Let me denote BC as vector u = (0, m, n)CD as vector v = (-1, √3 - m, -n)Cross product u × v = |i   j   k|                     0   m   n                     -1 √3 - m -nCalculating determinant:i*(m*(-n) - n*(√3 - m)) - j*(0*(-n) - n*(-1)) + k*(0*(√3 - m) - m*(-1))Simplify each component:i*(-mn - n√3 + mn) - j*(0 + n) + k*(0 + m)Simplify:i*(-n√3) - j*(n) + k*(m)Therefore, cross product u × v = (-n√3, -n, m)So the direction vector of the line perpendicular to both BC and CD at C is (-n√3, -n, m). Therefore, the line containing the center O can be parametrized as:C + t*(-n√3, -n, m) = (1, m, n) + t*(-n√3, -n, m)Similarly, the line from A along the z-axis is (0, 0, k). Since the center O must lie on both lines, we can set up equations to find the intersection point.So the center O is on both lines:From A: (0, 0, k)From C: (1 - n√3 * t, m - n t, n + m t)Therefore, setting these equal:1 - n√3 t = 0   (1)m - n t = 0       (2)n + m t = k       (3)From equation (1): 1 = n√3 t => t = 1/(n√3)From equation (2): m = n t => m = n*(1/(n√3)) = 1/√3So m = 1/√3From equation (3): n + m t = k. Substitute m = 1/√3 and t = 1/(n√3):k = n + (1/√3)*(1/(n√3)) = n + 1/(3n)So k = n + 1/(3n)Therefore, the center O has coordinates (0,0,k) = (0,0, n + 1/(3n))But the center O is also at a distance equal to the radius from point A(0,0,0), so OA = radius r = |k| = |n + 1/(3n)|Similarly, the distance from O to point C(1, m, n) must also be equal to the radius. Let's compute OC:Coordinates of O: (0,0,k) = (0,0, n + 1/(3n))Coordinates of C: (1, m, n) = (1, 1/√3, n)Distance OC:sqrt[(1 - 0)^2 + (1/√3 - 0)^2 + (n - (n + 1/(3n)))^2]Simplify:sqrt[1 + (1/3) + (-1/(3n))^2]Which is sqrt[1 + 1/3 + 1/(9n²)] = sqrt[4/3 + 1/(9n²)]But OC must equal OA, which is |n + 1/(3n)|. Since radius is positive, we can drop the absolute value:sqrt[4/3 + 1/(9n²)] = n + 1/(3n)Let me square both sides to eliminate the square root:4/3 + 1/(9n²) = (n + 1/(3n))²Compute the right-hand side:(n + 1/(3n))² = n² + 2*(n)*(1/(3n)) + (1/(3n))² = n² + 2/3 + 1/(9n²)Set equal to left-hand side:4/3 + 1/(9n²) = n² + 2/3 + 1/(9n²)Subtract 1/(9n²) from both sides:4/3 = n² + 2/3Subtract 2/3 from both sides:4/3 - 2/3 = n² => 2/3 = n²Therefore, n² = 2/3 => n = sqrt(2/3) or n = -sqrt(2/3)But since in our coordinate system, point C is at (1, m, n). Since angles and distances are positive, probably n is positive. So n = sqrt(2/3) = √6 / 3Therefore, n = √6 / 3Then, from earlier, m = 1/√3 = √3 / 3So coordinates of point C are (1, √3/3, √6/3)Now, compute k = n + 1/(3n) = √6/3 + 1/(3*(√6/3)) = √6/3 + 1/(√6) = √6/3 + √6/6 = (2√6 + √6)/6 = 3√6/6 = √6/2So k = √6 / 2Therefore, the center O is at (0, 0, √6/2), and the radius r = OA = √6/2Wait, OA is the distance from (0,0,0) to (0,0,√6/2), which is indeed √6/2. Let me check OC distance:Coordinates of O: (0,0,√6/2), coordinates of C: (1, √3/3, √6/3)Distance OC:sqrt[(1 - 0)^2 + (√3/3 - 0)^2 + (√6/3 - √6/2)^2]Compute each term:First term: 1^2 = 1Second term: (√3/3)^2 = (3/9) = 1/3Third term: (√6/3 - √6/2) = √6*(1/3 - 1/2) = √6*(-1/6) => square is (6*(1/36)) = 1/6So adding them up: 1 + 1/3 + 1/6 = (6/6 + 2/6 + 1/6) = 9/6 = 3/2Wait, sqrt(3/2) is equal to √(3)/√2 = √6/2, which matches OA. So yes, OC is sqrt(3/2) = √6/2. So radius is indeed √6/2.Therefore, the radius of the sphere is √6 / 2. Hence, the surface area is 4πr² = 4π*( (√6 / 2 )² ) = 4π*(6/4) = 4π*(3/2) = 6π.Wait, let me check that calculation:(√6 / 2)^2 = (6)/4 = 3/2Then 4π*(3/2) = 6πYes, that's correct. So the area of the sphere is 6π.But before finalizing, let me verify if all steps are correct. Let me retrace:Assigned coordinates correctly with A at origin, B at (1,0,0), D at (0,√3,0). Found coordinates of C as (1, √3/3, √6/3). Found center O at (0,0,√6/2) by finding intersection of two perpendicular lines. Calculated radius, squared it, multiplied by 4π. All steps seem logical. The key was realizing that the center lies on the common perpendiculars at A and C, then solving the equations for their intersection. The algebra worked out, and the distances matched. Seems solid.Therefore, the area of the sphere is 6π.Final Answerboxed{6pi}</think>Given four points (A, B, C, D) not lying in the same plane, we need to find the area of a sphere tangent to the lines (AB) and (AD) at point (A), and to the lines (BC) and (CD) at point (C). The given information includes (AB = 1), (BD = 2), and (angle ABC = angle BAD = 90^circ).1. Coordinate Assignment: - Place (A) at the origin ((0, 0, 0)). - Place (B) at ((1, 0, 0)) since (AB = 1). - Determine (D) using (BD = 2). Since (BAD = 90^circ), (D) is at ((0, sqrt{3}, 0)).2. Coordinates of (C): - (C) must satisfy (angle ABC = 90^circ). Let (C) be ((1, m, n)). - Vector (BC) is ((0, m, n)) and vector (CD) is ((-1, sqrt{3} - m, -n)). - The cross product of vectors (BC) and (CD) gives the direction of the line perpendicular to both at (C).3. Finding (m) and (n): - Solve the system of equations derived from the cross product and the intersection of perpendicular lines at (A) and (C). - Determine (m = frac{1}{sqrt{3}}) and (n = frac{sqrt{6}}{3}).4. Center of the Sphere: - The center (O) lies on the z-axis at ((0, 0, k)) where (k = frac{sqrt{6}}{2}).5. Radius Calculation: - The radius (r) is the distance from (O) to (A) or (C), both equal to (frac{sqrt{6}}{2}).6. Surface Area: - The surface area of the sphere is (4pi r^2 = 4pi left(frac{sqrt{6}}{2}right)^2 = 6pi).Thus, the area of the sphere is (boxed{6pi}).,<think>Alright, let me try to work through this problem step by step. So, we have four points A, B, C, D not lying in the same plane. There's a sphere that's tangent to the lines AB and AD at point A, and also tangent to the lines BC and CD at point C. We need to find the area of this sphere given that AB=1, BD=2, and the angles ABC and BAD are both 90 degrees.First, let's try to visualize the problem. Since the points aren't coplanar, this must be a 3D figure. We have points A, B, C, D with some right angles. Let me try to sketch this mentally.Given that angle BAD is 90 degrees, so BA is perpendicular to AD. Similarly, angle ABC is 90 degrees, so AB is perpendicular to BC. So, points A, B, C form a right angle at B, and points A, B, D form a right angle at A. Hmm, but since the points aren't coplanar, D must be out of the plane formed by A, B, C.Given BD=2. Since BD connects B to D, and BD is 2 units long. Also, AB=1.We need to find the area of the sphere. The area of a sphere is 4πr², so if we can find the radius, we can compute the area.The sphere is tangent to AB and AD at A, and to BC and CD at C. So, the sphere touches each of these lines exactly at points A and C. Since it's tangent at A and C, those points must lie on the sphere. Moreover, the sphere's center must lie along the line perpendicular to both AB and AD at A. Similarly, the center must also lie along the line perpendicular to both BC and CD at C.Wait, let me think. If a sphere is tangent to a line at a point, then the radius to that point is perpendicular to the line. So, the center of the sphere must lie along the line that is perpendicular to the tangent line at the point of tangency. So, for the sphere tangent to AB at A, the center lies along the line perpendicular to AB at A. Similarly, since it's also tangent to AD at A, the center must lie along the line perpendicular to AD at A. But since AB and AD are two different lines meeting at A, the only line that's perpendicular to both is along the cross product of their direction vectors. Wait, but in 3D, two lines intersecting at a point have a common perpendicular line?Alternatively, if the sphere is tangent to two lines at A, then the center of the sphere must lie along the line that is perpendicular to both AB and AD at A. Since AB and AD are perpendicular (because angle BAD is 90 degrees), their direction vectors are perpendicular. Therefore, the line perpendicular to both AB and AD at A would be along the normal vector to the plane formed by AB and AD. But since AB and AD are perpendicular, the normal vector would be the cross product of AB and AD vectors.But also, the sphere is tangent to BC and CD at C, so similarly, the center must lie along the line perpendicular to both BC and CD at C. Therefore, the center of the sphere must lie at the intersection of these two lines: the line perpendicular to AB and AD at A, and the line perpendicular to BC and CD at C.Therefore, if we can find the coordinates of points A, B, C, D, then we can find equations for these two lines (the perpendiculars at A and C), find their intersection point (which is the center of the sphere), compute the distance from the center to A (which is the radius), and then compute the area.So, let's set up a coordinate system. Let's choose point A as the origin. Since AB and AD are perpendicular, let's align AB along the x-axis and AD along the y-axis. Then, since angle BAD is 90 degrees, this makes sense. Then, since angle ABC is 90 degrees, BC is perpendicular to AB. Since AB is along the x-axis, BC would be in the plane perpendicular to AB at B. But since ABC is 90 degrees, BC is along the y-axis? Wait, no. Wait, let's be precise.Let me set coordinates:Let’s place point A at (0,0,0). Since AB is 1 unit, and angle BAD is 90 degrees, let's set point B at (1,0,0), and point D at (0,1,0). Then, AB is along the x-axis from (0,0,0) to (1,0,0), AD is along the y-axis from (0,0,0) to (0,1,0). Now, point C is such that angle ABC is 90 degrees. So, point C is somewhere in 3D space such that BC is perpendicular to AB. Since AB is along the x-axis, BC must be in the plane perpendicular to AB at B. The plane perpendicular to AB at B is the y-z plane passing through B(1,0,0). So, point C must lie in this y-z plane.But BD is given as 2 units. Point D is at (0,1,0), so the distance from B(1,0,0) to D(0,1,0) is sqrt((1-0)^2 + (0-1)^2 + (0-0)^2) = sqrt(2). But the problem states BD=2. Wait, that's a problem. Wait, BD is given as 2, but in my coordinate setup, BD is sqrt(2). Hmm, so this coordinate system might not work. Let me check.Wait, if AB=1, BD=2, angle BAD=90 degrees. So AB and AD are perpendicular. Let me see: If AB=1, angle BAD=90, so if I set A at (0,0,0), B at (1,0,0), then AD is along some direction perpendicular to AB. Let me set D at (0, y, 0) so that AD is along the y-axis. Then BD would be the distance from (1,0,0) to (0, y, 0). So BD = sqrt(1² + y²). The problem states BD=2, so sqrt(1 + y²) = 2 → y² = 3 → y=√3. So point D is at (0, √3, 0). That would make BD=2. Okay, so I need to adjust the coordinates accordingly.So let's redefine coordinates:Let’s place point A at (0,0,0).Point B is at (1,0,0), since AB=1 along the x-axis.Angle BAD is 90 degrees, so AD is perpendicular to AB. Let me set point D at (0, √3, 0) so that BD=2. Then, BD is from (1,0,0) to (0, √3, 0): distance is sqrt((1)^2 + (√3)^2) = sqrt(1 + 3) = 2, which matches BD=2.Now, point C is such that angle ABC is 90 degrees, so BC is perpendicular to AB. Since AB is along the x-axis, BC must lie in the plane x=1 (the plane perpendicular to AB at B). So point C has coordinates (1, y, z). Also, since the sphere is tangent to BC and CD at C, we need to figure out coordinates for C such that CD is a line from C to D(0, √3, 0). But we need more information to determine C.Wait, we might need to use the fact that the sphere is tangent to BC and CD at C. So similar to point A, the center of the sphere must lie along the line perpendicular to BC and CD at C. So, the center is along the line perpendicular to both BC and CD at point C.But we also know that the center lies along the line perpendicular to AB and AD at A. So, we need to find the center that lies on both of these lines. Therefore, we can set up equations for these two lines and find their intersection point.First, let's find the line perpendicular to AB and AD at A. AB is along the x-axis from A(0,0,0) to B(1,0,0). AD is from A(0,0,0) to D(0, √3, 0). Wait, but in this coordinate system, AD is along the y-axis. Wait, but if A is at (0,0,0), and D is at (0, √3, 0), then AD is along the y-axis. Therefore, AB is along the x-axis, AD along the y-axis, both in the x-y plane. Then, the line perpendicular to both AB and AD at A is the z-axis. Because the cross product of AB (x-axis) and AD (y-axis) is the z-axis. So, the line perpendicular to both AB and AD at A is the z-axis. So, the center of the sphere must lie somewhere along the z-axis.Now, the other line: the line perpendicular to BC and CD at C. Let's parametrize point C. Since angle ABC is 90 degrees, BC is perpendicular to AB. Since AB is along the x-axis, BC lies in the plane x=1 (since B is at (1,0,0)). So, point C is (1, y, z). Then, BC is the vector (0, y, z), since from B(1,0,0) to C(1, y, z). Then, CD is from C(1, y, z) to D(0, √3, 0), so vector CD is (-1, √3 - y, -z).The line perpendicular to both BC and CD at C would be along the cross product of vectors BC and CD. So, first, vectors BC and CD are:BC = (0, y, z)CD = (-1, √3 - y, -z)Their cross product is:|i     j         k     ||0     y       z   ||-1  √3 - y -z |Calculating determinant:i * [ y*(-z) - z*(√3 - y) ] - j * [0*(-z) - z*(-1) ] + k * [0*(√3 - y) - y*(-1) ]Simplify each component:i: [ -yz - z√3 + yz ] = -z√3j: - [ 0 + z ] = -zk: [ 0 + y ] = yTherefore, the cross product is ( -z√3, -z, y )So, the direction vector of the line perpendicular to both BC and CD at C is ( -z√3, -z, y ). Therefore, the parametric equation of this line is:x = 1 - z√3 * ty = y_C - z * tz = z_C + y * tWait, point C is (1, y_C, z_C). So, starting at point C(1, y, z), moving along direction vector ( -z√3, -z, y ). So the parametric equations would be:x = 1 - z√3 * ty = y - z * tz = z + y * tSo this is the line perpendicular to BC and CD at C.But the center of the sphere lies on both this line and the z-axis (from the previous condition). Wait, the z-axis is x=0, y=0, z= any. So, the center is at (0,0, k) for some k. Therefore, we need to find t such that:1 - z√3 * t = 0y - z * t = 0z + y * t = kBut since the center is (0,0,k), then the first equation gives 1 - z√3 * t = 0 → z√3 * t = 1Second equation: y - z t = 0 → y = z tThird equation: z + y t = kSo, from the first equation, t = 1/(z√3)From the second equation, y = z * t = z * (1/(z√3)) ) = 1/√3Therefore, y = 1/√3So, the y-coordinate of point C is 1/√3.Now, from the third equation: z + y t = kBut let's express z in terms of t. From the first equation, t = 1/(z√3). So, z = 1/(t√3)But let's substitute y = 1/√3 into the third equation:z + (1/√3) * t = kBut z = 1/(t√3), so substituting:1/(t√3) + (1/√3) * t = kMultiply both sides by √3:1/t + t = k√3So, 1/t + t = k√3But we need to find k. However, k is the z-coordinate of the center, which is also on the z-axis. But we need another equation to relate these variables.Wait, but point C is (1, y, z) where y = 1/√3, so C is (1, 1/√3, z). Also, the sphere passes through point C, so the distance from the center (0,0,k) to point C must be equal to the radius, which is also the distance from the center to point A (0,0,0). Since the sphere is tangent at A, the radius is the distance from center to A, which is sqrt(0² + 0² + k²) = |k|. Similarly, the distance from center (0,0,k) to C(1, 1/√3, z) is sqrt( (1)^2 + (1/√3)^2 + (z - k)^2 ). This must equal |k|.So, set up the equation:sqrt(1 + 1/3 + (z - k)^2 ) = |k|Square both sides:1 + 1/3 + (z - k)^2 = k²Simplify left side:4/3 + (z - k)^2 = k²Then, (z - k)^2 = k² - 4/3Expand left side:z² - 2kz + k² = k² - 4/3Simplify:z² - 2kz = -4/3But from earlier, we had the expression for k in terms of t: 1/t + t = k√3But t is related to z via t = 1/(z√3). Let's substitute t = 1/(z√3) into 1/t + t = k√3:1/(1/(z√3)) + 1/(z√3) = k√3Which simplifies to:z√3 + 1/(z√3) = k√3Multiply both sides by √3:3z + 1/z = 3kSo, 3z + 1/z = 3k → k = z + 1/(3z)Now, substitute k = z + 1/(3z) into the equation z² - 2kz = -4/3:z² - 2z(z + 1/(3z)) = -4/3Simplify inside the parentheses:z² - 2z² - 2*(1/3) = -4/3So,z² - 2z² - 2/3 = -4/3Combine like terms:- z² - 2/3 = -4/3Add 2/3 to both sides:- z² = -4/3 + 2/3 = -2/3Multiply both sides by -1:z² = 2/3Therefore, z = sqrt(2/3) or z = -sqrt(2/3). Since the problem doesn't specify the orientation, but given the right angles, perhaps we can assume z is positive. So, z = sqrt(2/3) = √6 / 3.Now, substitute z = √6 / 3 into k = z + 1/(3z):k = √6 / 3 + 1/(3*(√6 / 3)) = √6 / 3 + 1/√6 = √6 / 3 + √6 / 6 = (2√6 + √6)/6 = 3√6 /6 = √6 / 2Therefore, k = √6 / 2. So, the center of the sphere is at (0, 0, √6 / 2), and the radius is |k| = √6 / 2.Therefore, the area of the sphere is 4πr² = 4π*( (√6 / 2 )² ) = 4π*(6/4) = 4π*(3/2) = 6π.Wait, that seems straightforward, but let me verify the steps again to ensure there are no mistakes.First, coordinates: A(0,0,0), B(1,0,0), D(0,√3,0). Correct, since BD=2.Point C is (1, 1/√3, z), found from y = 1/√3. Then, solving for z and k gives z=√6/3 and k=√6/2.Then, verifying the distance from center (0,0,√6/2) to C(1,1/√3,√6/3):Distance squared is (1)^2 + (1/√3)^2 + (√6/3 - √6/2)^2.Compute each term:1 + 1/3 + ( (√6/3 - √6/2) )²Compute the last term:√6/3 - √6/2 = √6(1/3 - 1/2) = √6(-1/6) = -√6/6So, squared: ( -√6/6 )² = 6/36 = 1/6So total distance squared: 1 + 1/3 + 1/6 = (6/6 + 2/6 + 1/6) = 9/6 = 3/2But the radius squared is (√6/2)^2 = 6/4 = 3/2. So, correct. The distance from center to C is indeed equal to the radius. Similarly, distance from center to A is √(0 + 0 + (√6/2)^2 ) = √(6/4) = √6 / 2, which matches.Therefore, the radius is √6 / 2, so the area is 4π*(√6 / 2)^2 = 4π*(6/4) = 6π. So, the area of the sphere is 6π.I think this makes sense. Let me just recap the steps to ensure everything is consistent.1. Assigned coordinates to A, B, D such that AB=1, AD=√3 (to make BD=2), all in the x-y plane.2. Determined that point C must be in the plane x=1 (since BC is perpendicular to AB), so coordinates (1, y, z).3. The center of the sphere must lie on the z-axis (perpendicular to AB and AD at A) and also on the line perpendicular to BC and CD at C.4. Derived equations based on the cross product direction for the perpendicular at C, leading to expressions for y and z in terms of t, and then related them to the center's coordinates.5. Enforced that the center is equidistant from A and C (since both are on the sphere), leading to solving for z and k.6. Calculated the radius and verified the distances.All steps check out, so I think the area is indeed 6π.Final AnswerThe area of the sphere is boxed{6pi}.</think>Given four points (A, B, C, D) not lying in the same plane, we need to find the area of a sphere tangent to the lines (AB) and (AD) at (A), and to the lines (BC) and (CD) at (C). The given conditions are (AB = 1), (BD = 2), and (angle ABC = angle BAD = 90^circ).1. Coordinate Setup: - Place (A) at the origin ((0, 0, 0)). - Place (B) at ((1, 0, 0)) since (AB = 1). - Place (D) at ((0, sqrt{3}, 0)) to ensure (BD = 2).2. Determining Coordinates of (C): - Since (angle ABC = 90^circ), (C) lies in the plane (x = 1). - Let (C) be ((1, y, z)).3. Finding the Center of the Sphere: - The center of the sphere must be on the z-axis (perpendicular to (AB) and (AD) at (A)). - The center must also lie on the line perpendicular to (BC) and (CD) at (C).4. Using Cross Product for Direction: - Vectors (BC = (0, y, z)) and (CD = (-1, sqrt{3} - y, -z)). - Cross product of (BC) and (CD) gives direction vector ((-zsqrt{3}, -z, y)).5. Parametric Equations and Solving for (y) and (z): - Parametric equations for the line perpendicular to (BC) and (CD) at (C) lead to (y = frac{1}{sqrt{3}}). - Solving for (z) using distance constraints from the center to (A) and (C) gives (z = frac{sqrt{6}}{3}).6. Finding the Radius: - The center of the sphere is at ((0, 0, frac{sqrt{6}}{2})). - Radius (r = frac{sqrt{6}}{2}).7. Calculating the Area: - The area of the sphere is (4pi r^2 = 4pi left(frac{sqrt{6}}{2}right)^2 = 6pi).The area of the sphere is (boxed{6pi}).]

🔑:Pass a plane through points A, C and the center of the sphere.## SolutionLet O be the center of the sphere, r be its radius. Consider the tetrahedron A B C D. Since the sphere touches its edges A B and A D at point A, the line O A is perpendicular to two intersecting lines A B and A D in the plane of face A B D, so the line O A is perpendicular to this plane. Therefore, the sphere touches the plane of face A B D at point A.Similarly, we can prove that the sphere touches the plane of face B C D at point C. Thus, the sphere is inscribed in the dihedral angle formed by the planes of faces A B D and A B C.Let the plane passing through the intersecting lines O A and O C intersect the edge B D of the specified dihedral angle at point K. Since O A is perpendicular to the plane of face A B D, and O C is perpendicular to the plane of face B C D, the line B D is perpendicular to two intersecting lines O A and O C in the plane A K C. Therefore, B D perp A K and B D perp C K, i.e., A K and C K are altitudes of triangles A B D and C B D.In the right triangle A B D, it is known that B D=2 and A B=1, and B D is the hypotenuse, soangle A D B=30^{circ}, A D=sqrt{3}, A K=frac{1}{2} A D=sqrt{3} / 2By the theorem of the equality of segments of tangents drawn from a point to a sphere, B C=B A=1, K C=K A=sqrt{3}/2. Thus, the problem reduces to finding the radius r of the circle that touches the sides K A and K C of the isosceles triangle A K B at points A and B, respectively, andK A=K C=sqrt{A B^{2}+B C^{2}}=sqrt{1+1}=sqrt{2}.Let K M be the altitude of triangle A K C. ThenA M=C M=sqrt{A K^{2}-A M^{2}}=sqrt{(sqrt{3} / 2)^{2}-(sqrt{2} / 2)^{2}}=1 / 2.Denote angle O K C=alpha. From the right triangles A K M and A K O, we find thatr=A O=A K cdot operatorname{tg} alpha=A K cdot A M / K M=(sqrt{3} / 2) cdot(sqrt{2} / 2) /(1 / 2)=sqrt{6} / 2Therefore, the surface area of the sphere is 4 pi r^{2}=6 pi.## Answer6 pi.Submit a comment

🧠:[<think>Okay, let's see. I need to find the value of this expression: (cos α)/√(1 - sin²α) + 2 sin α / √(1 - cos²α) when α is in the third quadrant. Hmm. Alright, first, maybe I can simplify those denominators. I remember that 1 - sin²α is equal to cos²α, right? Because of the Pythagorean identity sin²θ + cos²θ = 1. So 1 - sin²α = cos²α. Similarly, 1 - cos²α should be sin²α. So substituting those in, the expression becomes:cos α / √(cos²α) + 2 sin α / √(sin²α)But then, √(cos²α) is |cos α|, and √(sin²α) is |sin α|. So this simplifies to:cos α / |cos α| + 2 sin α / |sin α|Now, here's where the quadrant comes into play. The problem says α is in the third quadrant. In the third quadrant, both sine and cosine are negative. So sin α < 0 and cos α < 0.Therefore, |cos α| = -cos α (since cos α is negative), and |sin α| = -sin α. So substituting back:cos α / (-cos α) + 2 sin α / (-sin α)Simplify each term:First term: cos α divided by -cos α is -1.Second term: 2 sin α divided by -sin α is -2.So adding them together: -1 + (-2) = -3.Wait, let me check again. Maybe I made a mistake with the signs. Let's go step by step.Starting with the first fraction: cos α over √(1 - sin²α). Since 1 - sin²α is cos²α, the denominator becomes |cos α|. In the third quadrant, cos α is negative, so |cos α| = -cos α. So cos α divided by |cos α| is cos α / (-cos α) = -1. That's correct.Second fraction: 2 sin α over √(1 - cos²α). Similarly, 1 - cos²α is sin²α, so the denominator is |sin α|. In the third quadrant, sin α is negative, so |sin α| = -sin α. Therefore, 2 sin α divided by |sin α| is 2 sin α / (-sin α) = -2. So adding -1 and -2 gives -3. That seems right.Is there any chance I messed up the quadrant signs? Let me recall the quadrants. First quadrant: all positive. Second: sine positive. Third: both sine and cosine negative. Fourth: cosine positive. So yeah, third quadrant both are negative, so absolute values would flip the sign. So substituting gives -1 and -2. Sum is -3. Hmm.Alternatively, maybe there's a trigonometric identity that can be used here, but I think breaking it down into absolute values and considering the quadrant signs is the right approach.Wait, but let me check if there's another way. For example, if I rewrite the original expression:cos α / √(1 - sin²α) + 2 sin α / √(1 - cos²α)Since √(1 - sin²α) is |cos α| and √(1 - cos²α) is |sin α|, so the expression is equivalent to:(cos α)/|cos α| + 2 (sin α)/|sin α|Which is sign(cos α) + 2 sign(sin α). Because cos α over |cos α| is the sign of cos α, same for sin. Since in the third quadrant, both signs are negative. Therefore:sign(cos α) = -1 and sign(sin α) = -1. Therefore, -1 + 2*(-1) = -1 -2 = -3. Same result.So that seems correct. Therefore, the value is -3.Final Answerboxed{-3}</think>Given that (alpha) is in the third quadrant, we need to find the value of the expression (frac{cos alpha}{sqrt{1 - sin^{2} alpha}} + frac{2 sin alpha}{sqrt{1 - cos^{2} alpha}}).First, we use the Pythagorean identities:[1 - sin^2 alpha = cos^2 alpha quad text{and} quad 1 - cos^2 alpha = sin^2 alpha]Substituting these into the expression, we get:[frac{cos alpha}{sqrt{cos^2 alpha}} + frac{2 sin alpha}{sqrt{sin^2 alpha}}]Since (sqrt{cos^2 alpha} = |cos alpha|) and (sqrt{sin^2 alpha} = |sin alpha|), the expression simplifies to:[frac{cos alpha}{|cos alpha|} + frac{2 sin alpha}{|sin alpha|}]In the third quadrant, both (cos alpha) and (sin alpha) are negative. Therefore, (|cos alpha| = -cos alpha) and (|sin alpha| = -sin alpha). Substituting these values in, we get:[frac{cos alpha}{-cos alpha} + frac{2 sin alpha}{-sin alpha}]Simplifying each term:[frac{cos alpha}{-cos alpha} = -1 quad text{and} quad frac{2 sin alpha}{-sin alpha} = -2]Adding these results together:[-1 + (-2) = -3]Thus, the value of the given expression is (boxed{-3}).,<think>Okay, let's see. I need to find the value of this expression: cos α divided by the square root of (1 - sin²α) plus 2 sin α divided by the square root of (1 - cos²α). And α is in the third quadrant. Hmm. Alright, let's start by recalling some trigonometric identities.First, I remember that sin²α + cos²α = 1. So, 1 - sin²α would be cos²α, right? And similarly, 1 - cos²α is sin²α. So, the denominators here can be simplified. Let me write that down.So, the first term is cos α / sqrt(1 - sin²α). Since 1 - sin²α is cos²α, that becomes cos α / sqrt(cos²α). Similarly, the second term is 2 sin α / sqrt(1 - cos²α), which is 2 sin α / sqrt(sin²α).But wait, when we take the square root of a squared trigonometric function, we have to consider the sign. Because sqrt(cos²α) is |cos α|, and sqrt(sin²α) is |sin α|. So, the expressions become cos α / |cos α| and 2 sin α / |sin α|.Now, the problem states that α is in the third quadrant. In the third quadrant, both sine and cosine are negative. So, sin α is negative and cos α is negative. Therefore, |cos α| would be -cos α (since cos α is negative), and |sin α| would be -sin α.Let me substitute those in. So, the first term: cos α / |cos α|. Since cos α is negative, |cos α| = -cos α. Therefore, that term becomes cos α / (-cos α) = -1. Similarly, the second term: 2 sin α / |sin α|. Since sin α is negative, |sin α| = -sin α. So, that term becomes 2 sin α / (-sin α) = 2 * (-1) = -2. Wait, hold on. Let's check that again.Wait, if sin α is negative, then |sin α| = -sin α. So, sin α / |sin α| is sin α / (-sin α) = -1. Therefore, the second term is 2 * (-1) = -2. So, combining both terms: first term is -1 and the second term is -2. So, -1 + (-2) = -3.But let me verify again step by step to make sure I didn't make any mistakes. Starting with the original expression:[cos α / sqrt(1 - sin²α)] + [2 sin α / sqrt(1 - cos²α)]We replace 1 - sin²α with cos²α and 1 - cos²α with sin²α:[cos α / sqrt(cos²α)] + [2 sin α / sqrt(sin²α)]sqrt(cos²α) is |cos α|, sqrt(sin²α) is |sin α|. So:[cos α / |cos α|] + [2 sin α / |sin α|]Since α is in the third quadrant, both cos α and sin α are negative. Therefore:cos α is negative, so |cos α| = -cos α (because the absolute value is positive, and cos α is negative, so you take the negative of cos α to get the positive value). Therefore, cos α / |cos α| = (negative) / (positive) = negative. So, cos α / |cos α| = (cos α) / (-cos α) = -1.Similarly, sin α is negative, so |sin α| = -sin α. Therefore, sin α / |sin α| = (negative) / (positive) = negative. So, sin α / |sin α| = (sin α) / (-sin α) = -1. Therefore, 2 sin α / |sin α| = 2 * (-1) = -2.Adding these together: -1 + (-2) = -3.Therefore, the value of the expression is -3.Wait, but let me check once more to make sure. Is there any possibility that I might have mixed up the signs? For example, sqrt(cos²α) is |cos α|, which is correct. Since in the third quadrant, cosine is negative, so sqrt(cos²α) = -cos α. Therefore, cos α divided by sqrt(cos²α) is cos α divided by (-cos α) which is -1. Same for sine. So, yeah, that seems right.Alternatively, maybe using a specific angle in the third quadrant to test. Let's pick α = 210 degrees, which is in the third quadrant. Let's compute each term.First, cos 210 degrees. 210 degrees is 180 + 30 degrees. The cosine of 210 degrees is -cos(30 degrees) = -√3/2. Similarly, sin 210 degrees is -sin(30 degrees) = -1/2.So, plug into the expression:First term: cos α / sqrt(1 - sin²α) = (-√3/2) / sqrt(1 - (-1/2)²) = (-√3/2) / sqrt(1 - 1/4) = (-√3/2) / sqrt(3/4) = (-√3/2) / (√3/2) = (-√3/2) * (2/√3) = -1.Second term: 2 sin α / sqrt(1 - cos²α) = 2*(-1/2) / sqrt(1 - (-√3/2)^2) = (-1) / sqrt(1 - 3/4) = (-1) / sqrt(1/4) = (-1) / (1/2) = -2.Adding them together: -1 + (-2) = -3. So, with actual numbers, it also results in -3. Therefore, that confirms the answer is indeed -3.I think that's thorough. So the answer should be boxed{-3}.Final Answerboxed{-3}</think>Given that (alpha) is in the third quadrant, we need to find the value of the expression (frac{cos alpha}{sqrt{1 - sin^{2} alpha}} + frac{2 sin alpha}{sqrt{1 - cos^{2} alpha}}).First, we use the Pythagorean identities:- (1 - sin^2 alpha = cos^2 alpha)- (1 - cos^2 alpha = sin^2 alpha)Substituting these into the expression, we get:[frac{cos alpha}{sqrt{cos^2 alpha}} + frac{2 sin alpha}{sqrt{sin^2 alpha}}]Since (alpha) is in the third quadrant, both (cos alpha) and (sin alpha) are negative. Therefore, the square roots simplify to the absolute values:- (sqrt{cos^2 alpha} = |cos alpha| = -cos alpha) (because (cos alpha) is negative)- (sqrt{sin^2 alpha} = |sin alpha| = -sin alpha) (because (sin alpha) is negative)Substituting these into the expression, we have:[frac{cos alpha}{-cos alpha} + frac{2 sin alpha}{-sin alpha}]Simplifying each term:- (frac{cos alpha}{-cos alpha} = -1)- (frac{2 sin alpha}{-sin alpha} = -2)Adding these results together:[-1 + (-2) = -3]Thus, the value of the expression is (boxed{-3}).]

🔑:This problem tests our understanding of trigonometric identities and the properties of sine and cosine functions. Given that alpha is in the third quadrant, we know that both sin alpha < 0 and cos alpha < 0. Using these properties, we can simplify the expression as follows:begin{align*}frac{cos alpha}{sqrt{1 - sin^{2} alpha}} + frac{2 sin alpha}{sqrt{1 - cos^{2} alpha}}&= frac{cos alpha}{sqrt{cos^{2} alpha}} + frac{2 sin alpha}{sqrt{sin^{2} alpha}} &= frac{cos alpha}{-cos alpha} + frac{2 sin alpha}{-sin alpha} &= -1 - 2 &= boxed{-3}end{align*}Hence, the answer is boxed{-3}.