💡Alright, so I have this problem about a weaving factory with 150 workers. They want to add a clothing project to improve their economic efficiency. Let me try to understand what's being asked here.First, each worker can either weave fabric or make clothing. If they weave fabric, they can produce 30 meters per day. If they make clothing, they can make 4 pieces per day. It takes 1.5 meters of fabric to make one piece of clothing. The fabric can be sold for a profit of 2 yuan per meter, and the clothing can be sold for a profit of 25 yuan per piece.We need to define x as the number of workers assigned to make clothing. Then, we have three parts to solve:1. The profit from making clothing in one day, expressed in terms of x.2. The profit from the remaining fabric in one day, expressed in terms of x.3. How many workers need to be assigned to make clothing in one day to earn a profit of 11800 yuan.Okay, let's tackle each part step by step.**Part 1: Profit from Making Clothing**If x workers are making clothing, each of them makes 4 pieces per day. So, the total number of clothing pieces made in a day is 4x.Each piece of clothing gives a profit of 25 yuan. So, the total profit from making clothing would be 25 yuan multiplied by the number of pieces, which is 4x.So, Profit P = 25 * 4x = 100x.Wait, that seems straightforward. So, P = 100x.**Part 2: Profit from Remaining Fabric**Now, the remaining workers are weaving fabric. Since there are 150 workers in total, the number of workers weaving fabric is 150 - x.Each weaver produces 30 meters of fabric per day. So, the total fabric produced by weavers is 30*(150 - x) meters.But, some of this fabric is used to make clothing. Each piece of clothing requires 1.5 meters of fabric, and we're making 4x pieces of clothing. So, the total fabric used for clothing is 1.5*4x = 6x meters.Therefore, the remaining fabric that can be sold is the total fabric produced minus the fabric used for clothing.Remaining fabric = 30*(150 - x) - 6x = 4500 - 30x - 6x = 4500 - 36x meters.Each meter of fabric gives a profit of 2 yuan, so the profit from the remaining fabric is 2*(4500 - 36x) = 9000 - 72x.So, Profit Q = 9000 - 72x.**Part 3: Number of Workers to Achieve 11800 Yuan Profit**Now, we need to find x such that the total profit from both making clothing and selling the remaining fabric is 11800 yuan.Total profit = Profit from clothing + Profit from fabric = P + Q = 100x + (9000 - 72x) = 100x + 9000 - 72x = 28x + 9000.We set this equal to 11800:28x + 9000 = 11800Subtract 9000 from both sides:28x = 2800Divide both sides by 28:x = 100So, 100 workers need to be assigned to make clothing to achieve a profit of 11800 yuan.Wait, let me double-check my calculations to make sure I didn't make any mistakes.For Part 1:- x workers making clothing.- Each makes 4 pieces, so total pieces = 4x.- Profit per piece = 25 yuan.- Total profit P = 25*4x = 100x. That seems correct.For Part 2:- Remaining workers = 150 - x.- Each weaves 30 meters, so total fabric = 30*(150 - x) = 4500 - 30x.- Fabric used for clothing = 1.5 meters per piece * 4x pieces = 6x.- Remaining fabric = 4500 - 30x - 6x = 4500 - 36x.- Profit per meter = 2 yuan.- Total profit Q = 2*(4500 - 36x) = 9000 - 72x. That also seems correct.For Part 3:- Total profit = P + Q = 100x + 9000 - 72x = 28x + 9000.- Set equal to 11800: 28x + 9000 = 11800.- Subtract 9000: 28x = 2800.- Divide by 28: x = 100. That makes sense.So, all parts seem to check out. I think my answers are correct.

💡First, I need to determine how many businessmen drank either coffee or tea. I know that 15 businessmen drank coffee and 12 drank tea, but 6 of them drank both. To avoid double-counting those who drank both, I'll use the inclusion-exclusion principle.I'll add the number of coffee drinkers and tea drinkers: 15 + 12 = 27.Since 6 businessmen are counted in both groups, I'll subtract them once: 27 - 6 = 21.This means 21 businessmen drank either coffee or tea. To find out how many drank neither, I'll subtract this number from the total number of businessmen: 30 - 21 = 9.Therefore, 9 businessmen drank neither coffee nor tea.

💡Alright, so I have this function f(x) = e^x (cos x - sin x), and I need to find all positive numbers x where the derivative f’(x) equals zero. Then, I have to arrange these x values in ascending order to form a sequence {x_n}. After that, I need to prove that the sequence {f(x_n)} is a geometric sequence. Hmm, okay, let me break this down step by step.First, I need to find the derivative of f(x). The function is a product of e^x and (cos x - sin x), so I should use the product rule for differentiation. The product rule states that if you have a function h(x) = u(x)v(x), then h’(x) = u’(x)v(x) + u(x)v’(x). Applying this to f(x):Let u(x) = e^x and v(x) = (cos x - sin x). So, u’(x) = e^x because the derivative of e^x is e^x. Now, v(x) = cos x - sin x. The derivative of cos x is -sin x, and the derivative of -sin x is -cos x. So, v’(x) = -sin x - cos x.Putting it all together, f’(x) = u’(x)v(x) + u(x)v’(x) = e^x (cos x - sin x) + e^x (-sin x - cos x).Let me simplify that:f’(x) = e^x (cos x - sin x) + e^x (-sin x - cos x)Combine like terms:= e^x [ (cos x - sin x) + (-sin x - cos x) ]= e^x [ cos x - sin x - sin x - cos x ]Simplify inside the brackets:cos x - cos x = 0-sin x - sin x = -2 sin xSo, f’(x) = e^x (-2 sin x) = -2 e^x sin x.Okay, so f’(x) = -2 e^x sin x.Now, I need to find all positive x such that f’(x) = 0. So, set f’(x) equal to zero:-2 e^x sin x = 0.Let me solve this equation. Since e^x is always positive for any real x, and -2 is just a constant, the only way this product can be zero is if sin x = 0.So, sin x = 0.When does sin x equal zero? Well, sin x = 0 at x = nπ, where n is an integer. But since we're only considering positive x, n must be a positive integer (1, 2, 3, ...).Therefore, the solutions are x = π, 2π, 3π, 4π, and so on. So, arranging these in ascending order, the sequence {x_n} is x_n = nπ for n = 1, 2, 3, ...Alright, so now I have the sequence {x_n} where each term is x_n = nπ. Next, I need to evaluate f at each x_n and show that {f(x_n)} is a geometric sequence.Let me compute f(x_n):f(x_n) = e^{x_n} (cos x_n - sin x_n) = e^{nπ} (cos(nπ) - sin(nπ)).I know that cos(nπ) is equal to (-1)^n because cosine of integer multiples of π alternates between 1 and -1. For example, cos(π) = -1, cos(2π) = 1, cos(3π) = -1, etc.Similarly, sin(nπ) is always 0 because sine of any integer multiple of π is zero. So, sin(nπ) = 0.Therefore, f(x_n) simplifies to:f(x_n) = e^{nπ} [ (-1)^n - 0 ] = (-1)^n e^{nπ}.So, f(x_n) = (-1)^n e^{nπ}.Now, I need to check if this sequence {f(x_n)} is a geometric sequence. A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a constant called the common ratio.Let me write out the first few terms of {f(x_n)}:For n = 1: f(x_1) = (-1)^1 e^{1π} = -e^π.For n = 2: f(x_2) = (-1)^2 e^{2π} = e^{2π}.For n = 3: f(x_3) = (-1)^3 e^{3π} = -e^{3π}.For n = 4: f(x_4) = (-1)^4 e^{4π} = e^{4π}.And so on.So, the sequence is: -e^π, e^{2π}, -e^{3π}, e^{4π}, ...Let me see if this is a geometric sequence. Let's compute the ratio of consecutive terms.Compute f(x_2)/f(x_1): (e^{2π}) / (-e^π) = -e^{π}.Compute f(x_3)/f(x_2): (-e^{3π}) / (e^{2π}) = -e^{π}.Compute f(x_4)/f(x_3): (e^{4π}) / (-e^{3π}) = -e^{π}.It seems that each term is obtained by multiplying the previous term by -e^{π}. So, the common ratio is -e^{π}.Therefore, the sequence {f(x_n)} is a geometric sequence with the first term f(x_1) = -e^{π} and common ratio q = -e^{π}.Let me just verify this with the general term. The nth term of a geometric sequence is given by a_n = a_1 * q^{n-1}.Here, a_1 = -e^{π}, q = -e^{π}, so:a_n = (-e^{π}) * (-e^{π})^{n-1} = (-e^{π}) * (-1)^{n-1} e^{(n-1)π} = (-1)^n e^{nπ}.Which matches our expression for f(x_n). So, yes, it's confirmed.Therefore, the sequence {f(x_n)} is indeed a geometric sequence with the first term -e^{π} and common ratio -e^{π}.**Final Answer**The sequence {f(x_{n})} is a geometric sequence with the first term -e^{pi} and common ratio -e^{pi}. Therefore, the final answer is boxed{text{The sequence } {f(x_{n})} text{ is a geometric sequence with the first term } -e^{pi} text{ and common ratio } -e^{pi}.}

💡Okay, so I've got this problem here, and I need to figure out the answer step by step. Let me try to understand what each part is saying.First, the universal set U is defined as N*, which I believe stands for the set of all positive natural numbers. So, U = {1, 2, 3, 4, 5, 6, 7, 8, ...} and so on.Next, there's set M, which is defined as M = {x ∈ N* | y = tan(π/2 x)}. Hmm, okay. So, M is the set of positive integers x for which y = tan(π/2 x) is defined. But wait, tan(π/2 x) is the tangent function. I remember that the tangent function has asymptotes where its argument is an odd multiple of π/2. So, tan(θ) is undefined when θ = (2k + 1)π/2 for any integer k.So, if we have tan(π/2 x), it will be undefined when π/2 x = (2k + 1)π/2. Let me solve for x here:π/2 x = (2k + 1)π/2Divide both sides by π/2:x = 2k + 1So, x must be an odd integer for tan(π/2 x) to be undefined. Therefore, M is the set of all positive integers x where tan(π/2 x) is defined, which would be when x is not an odd integer. So, M is the set of even positive integers: M = {2, 4, 6, 8, ...}.Therefore, the complement of M in U, denoted as ∁_U M, would be the set of all positive integers not in M, which are the odd positive integers: ∁_U M = {1, 3, 5, 7, 9, ...}.Okay, got that part. Now, moving on to set N. N is defined as N = {y | y = x + 4/x, 1/2 ≤ x ≤ 4}. So, N is the set of all y values such that y is equal to x plus 4 divided by x, where x ranges from 1/2 to 4.I need to figure out what the range of y is for this function y = x + 4/x over the interval [1/2, 4]. Let me think about how to approach this.First, I know that the function y = x + 4/x is a type of function that has a minimum value somewhere in the interval. To find the minimum, I can take the derivative and set it equal to zero.Let me compute the derivative of y with respect to x:dy/dx = 1 - 4/x²Set this equal to zero to find critical points:1 - 4/x² = 01 = 4/x²Multiply both sides by x²:x² = 4Take square roots:x = ±2But since x is in [1/2, 4], we only consider x = 2.So, x = 2 is a critical point. Now, let's check the value of y at x = 2:y = 2 + 4/2 = 2 + 2 = 4So, the minimum value of y is 4 at x = 2.Now, let's check the endpoints of the interval to find the maximum value.First, at x = 1/2:y = (1/2) + 4/(1/2) = 0.5 + 8 = 8.5 or 17/2.Next, at x = 4:y = 4 + 4/4 = 4 + 1 = 5.So, the function y = x + 4/x on the interval [1/2, 4] ranges from a minimum of 4 to a maximum of 17/2 (which is 8.5). So, N is the set of all real numbers y such that 4 ≤ y ≤ 17/2.But wait, N is defined as {y | y = x + 4/x, 1/2 ≤ x ≤ 4}, so it's a set of real numbers, not necessarily integers. However, the question is about the intersection of ∁_U M and N. Since ∁_U M is a set of positive integers (the odd ones), and N is a set of real numbers between 4 and 17/2, the intersection would be the set of odd positive integers that lie within the interval [4, 17/2].So, let's list the odd positive integers and see which ones fall within [4, 17/2].First, 17/2 is 8.5, so the interval is from 4 to 8.5.Odd positive integers are 1, 3, 5, 7, 9, 11, etc.Now, which of these are between 4 and 8.5?5 and 7 are in that range. 9 is above 8.5, so it's excluded. 1 and 3 are below 4, so they're also excluded.Therefore, the intersection (∁_U M) ∩ N is {5, 7}.Looking back at the options:A: {5,7}B: {4,5,6,7}C: [4,8]D: [4, 17/2]So, the correct answer should be A: {5,7}.Wait, but let me double-check to make sure I didn't make any mistakes.First, confirming that M is the set of even positive integers because tan(π/2 x) is undefined for odd x. That seems correct because tan(π/2 * odd) is undefined.Then, the complement of M would indeed be the odd positive integers.For set N, I found that y ranges from 4 to 17/2, which is 8.5, by analyzing the function y = x + 4/x. The minimum at x=2 is 4, and the maximum at x=1/2 is 17/2. At x=4, y=5, which is less than 17/2, so the maximum is indeed 17/2.Therefore, N is the interval [4, 17/2]. The intersection with ∁_U M, which is the odd positive integers, would be the odd integers within [4, 17/2]. The odd integers in that range are 5 and 7, since 4 is even, 5 is odd, 6 is even, 7 is odd, and 8 is even, but 8.5 is the upper limit, so 9 is beyond that.Yes, that seems correct. So, the answer is A: {5,7}.**Final Answer**boxed{A}

💡First, I need to determine Sydney's age. Since Randolph is 5 years older than Sydney and Randolph is 55 years old, I can calculate Sydney's age by subtracting 5 years from Randolph's age.Sydney's age = Randolph's age - 5 yearsSydney's age = 55 years - 5 yearsSydney's age = 50 yearsNow that I know Sydney is 50 years old and Sherry is 25 years old, I can find the ratio of Sydney's age to Sherry's age by dividing Sydney's age by Sherry's age.Ratio = Sydney's age / Sherry's ageRatio = 50 years / 25 yearsRatio = 2/1Therefore, the ratio of Sydney's age to Sherry's age is 2:1.

💡Okay, so I have this problem about a parallelogram with sides of lengths (a) and (b), and diagonals of lengths (m) and (n). I need to prove that (a^4 + b^4 = m^2 n^2) if and only if the acute angle of the parallelogram is (45^circ). Hmm, okay, let me try to break this down step by step.First, I remember that in a parallelogram, the diagonals bisect each other, and there are some formulas related to the lengths of the diagonals in terms of the sides and the angles. I think the Law of Cosines might come into play here because we're dealing with triangles formed by the diagonals.Let me recall the Law of Cosines. For any triangle with sides (x), (y), and opposite angle (theta), the third side (z) is given by:[z^2 = x^2 + y^2 - 2xy cos theta]But in a parallelogram, the diagonals split the parallelogram into two congruent triangles. So, maybe I can apply the Law of Cosines to these triangles.Let me denote the acute angle of the parallelogram as (alpha). Then, the obtuse angle would be (180^circ - alpha). Now, considering one of the triangles formed by a diagonal, the sides of the triangle would be (a), (b), and the diagonal (m) or (n), depending on which diagonal we're considering.Wait, actually, each diagonal divides the parallelogram into two triangles. So, one diagonal will correspond to the acute angle, and the other to the obtuse angle. Let me clarify that.If I take diagonal (m), it will form a triangle with sides (a), (b), and (m), with the angle between sides (a) and (b) being (alpha). Similarly, the other diagonal (n) will form a triangle with sides (a), (b), and (n), but the angle between (a) and (b) will be (180^circ - alpha), which is the obtuse angle.So, applying the Law of Cosines to both triangles:For diagonal (m):[m^2 = a^2 + b^2 - 2ab cos alpha]Wait, hold on, is that right? Because in a parallelogram, the diagonals are related to the sides and the angles. Actually, I think the formula is slightly different because the diagonals are not just the sides of a triangle but they split the parallelogram into triangles with sides (a), (b), and the diagonal.Wait, no, actually, I think I made a mistake. The formula for the diagonals in a parallelogram is given by:[m^2 = a^2 + b^2 + 2ab cos alpha]and[n^2 = a^2 + b^2 - 2ab cos alpha]Yes, that sounds more familiar. Because when you have the angle (alpha), the diagonal opposite to the angle will have a different sign in the cosine term. So, one diagonal will have a plus and the other a minus.Let me verify that. If I consider the triangle formed by sides (a), (b), and diagonal (m), with angle (alpha) between sides (a) and (b), then by the Law of Cosines:[m^2 = a^2 + b^2 - 2ab cos alpha]But wait, in a parallelogram, the diagonals are not just the sides of a triangle with angle (alpha); actually, the angle between the sides when considering the diagonal is supplementary to (alpha). Hmm, maybe I need to think more carefully.Alternatively, I remember that in a parallelogram, the sum of the squares of the diagonals is equal to twice the sum of the squares of the sides. The formula is:[m^2 + n^2 = 2(a^2 + b^2)]But that's not directly helpful here because the problem is about the product of the squares of the diagonals, not the sum.Wait, the problem is (a^4 + b^4 = m^2 n^2). So, I need to relate (a^4 + b^4) to (m^2 n^2). Maybe I can express (m^2) and (n^2) in terms of (a), (b), and (alpha), then multiply them together and see what condition gives (a^4 + b^4).So, let's write down the expressions for (m^2) and (n^2) again:[m^2 = a^2 + b^2 + 2ab cos alpha][n^2 = a^2 + b^2 - 2ab cos alpha]Okay, so if I multiply (m^2) and (n^2), I get:[m^2 n^2 = (a^2 + b^2 + 2ab cos alpha)(a^2 + b^2 - 2ab cos alpha)]This looks like the product of a sum and difference, so it should simplify to:[m^2 n^2 = (a^2 + b^2)^2 - (2ab cos alpha)^2]Let me compute that:[m^2 n^2 = (a^2 + b^2)^2 - 4a^2 b^2 cos^2 alpha]Expanding ((a^2 + b^2)^2), we get:[a^4 + 2a^2 b^2 + b^4]So, substituting back:[m^2 n^2 = a^4 + 2a^2 b^2 + b^4 - 4a^2 b^2 cos^2 alpha]Simplify the terms:[m^2 n^2 = a^4 + b^4 + 2a^2 b^2 - 4a^2 b^2 cos^2 alpha]Now, the problem states that (a^4 + b^4 = m^2 n^2). So, setting them equal:[a^4 + b^4 = a^4 + b^4 + 2a^2 b^2 - 4a^2 b^2 cos^2 alpha]Subtracting (a^4 + b^4) from both sides:[0 = 2a^2 b^2 - 4a^2 b^2 cos^2 alpha]Factor out (2a^2 b^2):[0 = 2a^2 b^2 (1 - 2 cos^2 alpha)]Since (a) and (b) are lengths of sides, they can't be zero, so (2a^2 b^2) is not zero. Therefore, we have:[1 - 2 cos^2 alpha = 0]Solving for (cos^2 alpha):[2 cos^2 alpha = 1 implies cos^2 alpha = frac{1}{2}]Taking square roots:[cos alpha = pm frac{sqrt{2}}{2}]Since (alpha) is the acute angle, we take the positive value:[cos alpha = frac{sqrt{2}}{2}]Which corresponds to:[alpha = 45^circ]So, that shows that if (a^4 + b^4 = m^2 n^2), then the acute angle (alpha) must be (45^circ).Conversely, if the acute angle is (45^circ), then (cos alpha = frac{sqrt{2}}{2}). Plugging this back into the expressions for (m^2) and (n^2):[m^2 = a^2 + b^2 + 2ab cdot frac{sqrt{2}}{2} = a^2 + b^2 + ab sqrt{2}][n^2 = a^2 + b^2 - 2ab cdot frac{sqrt{2}}{2} = a^2 + b^2 - ab sqrt{2}]Multiplying (m^2) and (n^2):[m^2 n^2 = (a^2 + b^2 + ab sqrt{2})(a^2 + b^2 - ab sqrt{2}) = (a^2 + b^2)^2 - (ab sqrt{2})^2][= a^4 + 2a^2 b^2 + b^4 - 2a^2 b^2 = a^4 + b^4]So, indeed, (a^4 + b^4 = m^2 n^2) when (alpha = 45^circ).Therefore, the condition (a^4 + b^4 = m^2 n^2) holds if and only if the acute angle of the parallelogram is (45^circ).**Final Answer**The acute angle of the parallelogram is boxed{45^circ}.