🧠:<think>Okay, so I have this problem about a jewelry manufacturer who makes synthetic gemstone rings. They have a cost function and a profit function, and I need to figure out how many rings they should produce to maximize their profit. Hmm, sounds like an optimization problem. Let me try to break it down step by step.First, the cost function is given by:[ C(n) = 500 + 20n + 0.02n^2 ]And the profit function is:[ P(n) = 150n - C(n) - 0.05n^3 ]So, I think I need to substitute the cost function into the profit function to get a single equation for profit in terms of n. Let me do that.Substituting C(n) into P(n):[ P(n) = 150n - (500 + 20n + 0.02n^2) - 0.05n^3 ]Let me simplify this step by step. First, distribute the negative sign into the parentheses:[ P(n) = 150n - 500 - 20n - 0.02n^2 - 0.05n^3 ]Now, combine like terms. The 150n and -20n can be combined:150n - 20n = 130nSo, now the equation becomes:[ P(n) = 130n - 500 - 0.02n^2 - 0.05n^3 ]Let me rearrange the terms in descending order of the exponent of n:[ P(n) = -0.05n^3 - 0.02n^2 + 130n - 500 ]Alright, so now I have the profit function in terms of n. The next step is to find the value of n that maximizes this profit. Since it's a continuous function, I can use calculus to find the critical points and then determine which one gives the maximum profit.To find the critical points, I need to take the first derivative of P(n) with respect to n and set it equal to zero.So, let's compute the first derivative P'(n):The derivative of -0.05n^3 is -0.15n^2.The derivative of -0.02n^2 is -0.04n.The derivative of 130n is 130.The derivative of -500 is 0.So putting it all together:[ P'(n) = -0.15n^2 - 0.04n + 130 ]Now, set this equal to zero to find critical points:[ -0.15n^2 - 0.04n + 130 = 0 ]Hmm, this is a quadratic equation in terms of n. Let me write it in standard form:[ -0.15n^2 - 0.04n + 130 = 0 ]I can multiply both sides by -1 to make the coefficients positive:[ 0.15n^2 + 0.04n - 130 = 0 ]Now, this is a quadratic equation of the form:[ ax^2 + bx + c = 0 ]where a = 0.15, b = 0.04, and c = -130.To solve for n, I can use the quadratic formula:[ n = frac{-b pm sqrt{b^2 - 4ac}}{2a} ]Plugging in the values:First, compute the discriminant:[ D = b^2 - 4ac = (0.04)^2 - 4 * 0.15 * (-130) ]Calculating each part:(0.04)^2 = 0.00164 * 0.15 = 0.60.6 * (-130) = -78So, D = 0.0016 - (-78) = 0.0016 + 78 = 78.0016That's a positive discriminant, so we have two real roots.Now, compute the two possible solutions:[ n = frac{-0.04 pm sqrt{78.0016}}{2 * 0.15} ]First, let's compute the square root of 78.0016. Let me approximate that.I know that 8^2 = 64 and 9^2 = 81, so sqrt(78) is between 8 and 9. Let me compute 8.8^2 = 77.44, and 8.83^2 = ?8.83^2 = (8 + 0.83)^2 = 8^2 + 2*8*0.83 + 0.83^2 = 64 + 13.28 + 0.6889 ≈ 64 + 13.28 = 77.28 + 0.6889 ≈ 77.9689Hmm, 8.83^2 ≈ 77.9689, which is very close to 78.0016. So, sqrt(78.0016) ≈ 8.83.So, approximately, sqrt(78.0016) ≈ 8.83.Therefore, plugging back into the formula:First, the positive root:[ n = frac{-0.04 + 8.83}{2 * 0.15} ]Compute numerator:-0.04 + 8.83 = 8.79Denominator:2 * 0.15 = 0.3So, n ≈ 8.79 / 0.3 ≈ 29.3Second, the negative root:[ n = frac{-0.04 - 8.83}{0.3} ]Numerator:-0.04 - 8.83 = -8.87Denominator:0.3So, n ≈ -8.87 / 0.3 ≈ -29.57Since n represents the number of rings produced, it can't be negative. So, we discard the negative solution.Therefore, the critical point is at approximately n = 29.3.But n must be an integer since you can't produce a fraction of a ring. So, we need to check whether n = 29 or n = 30 gives the maximum profit.But before that, let me verify if this critical point is indeed a maximum. For that, I can use the second derivative test.Compute the second derivative P''(n):We already have P'(n) = -0.15n^2 - 0.04n + 130So, the derivative of that is:P''(n) = -0.3n - 0.04Now, evaluate P''(n) at n ≈ 29.3:P''(29.3) = -0.3 * 29.3 - 0.04 ≈ -8.79 - 0.04 ≈ -8.83Since the second derivative is negative, the function is concave down at this point, which means it's a local maximum. So, n ≈ 29.3 is indeed a maximum.But since n must be an integer, we need to check n = 29 and n = 30 to see which one gives a higher profit.Let me compute P(29) and P(30).First, compute P(29):P(n) = -0.05n^3 - 0.02n^2 + 130n - 500So, plug in n = 29:Compute each term:-0.05*(29)^3: First, 29^3 = 29*29*29 = 841*29. Let me compute 800*29 = 23,200 and 41*29 = 1,189. So, 23,200 + 1,189 = 24,389. Then, -0.05*24,389 = -1,219.45-0.02*(29)^2: 29^2 = 841. So, -0.02*841 = -16.82130*29 = 3,770-500So, adding all together:-1,219.45 -16.82 + 3,770 - 500Compute step by step:-1,219.45 -16.82 = -1,236.27-1,236.27 + 3,770 = 2,533.732,533.73 - 500 = 2,033.73So, P(29) ≈ 2,033.73Now, compute P(30):Again, P(n) = -0.05n^3 - 0.02n^2 + 130n - 500n = 30:-0.05*(30)^3 = -0.05*27,000 = -1,350-0.02*(30)^2 = -0.02*900 = -18130*30 = 3,900-500Adding all together:-1,350 -18 + 3,900 - 500Compute step by step:-1,350 -18 = -1,368-1,368 + 3,900 = 2,5322,532 - 500 = 2,032So, P(30) = 2,032Comparing P(29) ≈ 2,033.73 and P(30) = 2,032, so P(29) is slightly higher.Therefore, n = 29 gives a higher profit than n = 30.But wait, the critical point was at n ≈29.3, so between 29 and 30, 29.3 is closer to 29. So, n = 29 is the optimal integer value.But just to be thorough, let me compute P(29.3) to see the maximum profit at that point, even though n must be integer.But maybe it's not necessary, but just to check.Compute P(29.3):First, n =29.3Compute each term:-0.05*(29.3)^3First, compute 29.3^3:29.3 *29.3 = let's compute 29*29 = 841, 29*0.3=8.7, 0.3*29=8.7, 0.3*0.3=0.09So, (29 + 0.3)^2 = 29^2 + 2*29*0.3 + 0.3^2 = 841 + 17.4 + 0.09 = 858.49Then, 29.3^3 = 29.3 * 858.49Compute 29 * 858.49 = let's compute 30*858.49 = 25,754.7, subtract 1*858.49 = 25,754.7 - 858.49 = 24,896.21Then, 0.3 *858.49 = 257.547So, total 24,896.21 + 257.547 ≈ 25,153.757So, 29.3^3 ≈25,153.757Then, -0.05*25,153.757 ≈ -1,257.688Next term: -0.02*(29.3)^2We already computed (29.3)^2 =858.49So, -0.02*858.49 ≈ -17.1698Next term: 130*29.3 = 130*29 + 130*0.3 = 3,770 + 39 = 3,809Last term: -500So, adding all together:-1,257.688 -17.1698 + 3,809 - 500Compute step by step:-1,257.688 -17.1698 ≈ -1,274.8578-1,274.8578 + 3,809 ≈ 2,534.14222,534.1422 - 500 ≈ 2,034.1422So, P(29.3) ≈ 2,034.14Which is slightly higher than P(29) ≈ 2,033.73So, the maximum profit is approximately 2,034.14 at n ≈29.3, but since n must be integer, n=29 gives the closest integer value with a profit of approximately 2,033.73, which is just slightly less than the maximum.Therefore, the manufacturer should produce 29 rings to maximize profit.Wait, but before I conclude, let me double-check my calculations because sometimes when dealing with decimals, rounding errors can occur.Let me recalculate P(29):P(n) = -0.05n^3 -0.02n^2 +130n -500n=29:-0.05*(29)^3 = -0.05*(24,389) = -1,219.45-0.02*(29)^2 = -0.02*(841) = -16.82130*29 = 3,770-500So, total: -1,219.45 -16.82 + 3,770 -500Compute:-1,219.45 -16.82 = -1,236.27-1,236.27 + 3,770 = 2,533.732,533.73 -500 = 2,033.73Yes, that's correct.Similarly, P(30):-0.05*(27,000) = -1,350-0.02*(900) = -18130*30 = 3,900-500Total: -1,350 -18 + 3,900 -500 = (-1,368) + (3,900 -500) = (-1,368) + 3,400 = 2,032So, yes, P(30) is 2,032.Therefore, n=29 gives a slightly higher profit.So, the answer to the first sub-problem is n=29.Now, moving on to the second sub-problem: Calculate the maximum profit and the corresponding cost of gemstone production at this optimal level.We already computed P(29) ≈ 2,033.73, which is the maximum profit.Now, the corresponding cost of gemstone production is C(n) at n=29.Given the cost function:C(n) = 500 + 20n + 0.02n^2So, plug in n=29:C(29) = 500 + 20*29 + 0.02*(29)^2Compute each term:20*29 = 5800.02*(29)^2 = 0.02*841 = 16.82So, C(29) = 500 + 580 + 16.82 = 500 + 580 = 1,080 + 16.82 = 1,096.82Therefore, the cost of production at n=29 is 1,096.82.Let me just verify this:20*29: 20*20=400, 20*9=180, total 580.0.02*(29)^2: 29^2=841, 0.02*841=16.82So, 500 +580=1,080 +16.82=1,096.82. Correct.So, summarizing:Maximum profit is approximately 2,033.73, and the corresponding cost is approximately 1,096.82.But wait, let me check if the profit is indeed P(n) = 150n - C(n) -0.05n^3.Wait, actually, in the original profit function, it's P(n) = 150n - C(n) -0.05n^3.So, let me compute P(n) as 150n - C(n) -0.05n^3 for n=29 to cross-verify.Compute 150*29 = 4,350C(29) = 1,096.820.05*(29)^3 = 0.05*24,389 = 1,219.45So, P(n) = 4,350 - 1,096.82 -1,219.45Compute:4,350 -1,096.82 = 3,253.183,253.18 -1,219.45 = 2,033.73Yes, that's correct. So, P(29) is indeed 2,033.73.Therefore, the calculations are consistent.So, to recap:1. The number of rings that maximizes profit is n=29.2. The maximum profit is approximately 2,033.73, and the corresponding cost is approximately 1,096.82.I think that's all. I don't see any mistakes in my calculations, but let me just go through the steps once more to be sure.First, substituted C(n) into P(n), expanded and simplified correctly.Took the first derivative, set it to zero, solved the quadratic equation, got n≈29.3, which is between 29 and 30.Computed second derivative, found it negative, so it's a maximum.Checked P(29) and P(30), found P(29) slightly higher.Computed C(29) correctly.Yes, everything seems consistent.Final Answer1. The number of rings that maximizes profit is boxed{29}.2. The maximum profit is boxed{2033.73} dollars and the corresponding cost is boxed{1096.82} dollars.</think>

🧠:<think>Alright, so I have this problem where Dr. Elena is studying the impact of cultural exchange on local communities using some mathematical models. There are two parts to the problem. Let me try to tackle them one by one.Starting with the first part: calculating the cultural integration index (CI). The formula given is CI(T, D, S, L) = k * (T^α * D^β * S^γ * L^δ). They've provided all the constants and variables, so it's just a matter of plugging in the numbers, right?Let me list out all the given values:- T = 10,000- D = 0.8- S = 5- L = 0.9- k = 1.2- α = 0.6- β = 0.4- γ = 0.3- δ = 0.2So, I need to compute CI = 1.2 * (10,000^0.6 * 0.8^0.4 * 5^0.3 * 0.9^0.2). Hmm, that looks a bit intimidating, but let's break it down step by step.First, let's compute each part separately.1. Compute T^α: 10,000^0.6 - 10,000 is 10^4, so 10^4^0.6 = 10^(4*0.6) = 10^2.4 - 10^2.4 is the same as 10^2 * 10^0.4 - 10^2 is 100 - 10^0.4 is approximately... Hmm, I remember that 10^0.4 is about 2.51188643150958 - So, 100 * 2.51188643150958 ≈ 251.188643150958 - Let me double-check that with a calculator: 10^0.4 ≈ 2.5118864315, yes, so 100 * that is approximately 251.18862. Compute D^β: 0.8^0.4 - 0.8 is 4/5, so (4/5)^0.4 - Alternatively, I can compute it as e^(0.4 * ln(0.8)) - ln(0.8) is approximately -0.22314 - So, 0.4 * (-0.22314) ≈ -0.089256 - e^(-0.089256) ≈ 0.9148 - Let me verify: 0.8^0.4. Using a calculator, 0.8^0.4 ≈ 0.9148, yes.3. Compute S^γ: 5^0.3 - 5^0.3 is the same as e^(0.3 * ln(5)) - ln(5) ≈ 1.60944 - 0.3 * 1.60944 ≈ 0.482832 - e^0.482832 ≈ 1.620 - Let me check: 5^0.3. 5^(1/3) is about 1.709, but 0.3 is a bit less than 1/3, so maybe around 1.62. Yes, that seems right.4. Compute L^δ: 0.9^0.2 - 0.9^0.2 is e^(0.2 * ln(0.9)) - ln(0.9) ≈ -0.10536 - 0.2 * (-0.10536) ≈ -0.021072 - e^(-0.021072) ≈ 0.9792 - Let me confirm: 0.9^0.2. Since 0.9^0.2 is close to 1, as the exponent is small. Using a calculator, 0.9^0.2 ≈ 0.9792, correct.Now, let me multiply all these computed values together:251.1886 * 0.9148 * 1.620 * 0.9792First, multiply 251.1886 and 0.9148:251.1886 * 0.9148 ≈ Let's compute 251.1886 * 0.9 = 226.07, and 251.1886 * 0.0148 ≈ 3.71. So total ≈ 226.07 + 3.71 ≈ 229.78Wait, that's an approximation. Let me do it more accurately:251.1886 * 0.9148:Multiply 251.1886 * 0.9 = 226.07Multiply 251.1886 * 0.0148:First, 251.1886 * 0.01 = 2.511886251.1886 * 0.0048 ≈ 1.2057So total ≈ 2.511886 + 1.2057 ≈ 3.7176So total ≈ 226.07 + 3.7176 ≈ 229.7876So approximately 229.7876Now, multiply this by 1.620:229.7876 * 1.620 ≈ Let's compute 229.7876 * 1.6 = 367.66016 and 229.7876 * 0.02 = 4.595752So total ≈ 367.66016 + 4.595752 ≈ 372.2559Now, multiply this by 0.9792:372.2559 * 0.9792 ≈ Let's compute 372.2559 * 1 = 372.2559 and subtract 372.2559 * 0.0208 ≈ 7.737So, 372.2559 - 7.737 ≈ 364.5189So, the product of all four terms is approximately 364.5189Now, multiply this by k = 1.2:CI = 1.2 * 364.5189 ≈ 437.4227So, approximately 437.42Wait, let me verify the calculations step by step because sometimes approximations can lead to errors.Alternatively, maybe I should use logarithms or exponents more accurately.But given that all the intermediate steps were approximate, maybe I can use a calculator for each exponentiation.Alternatively, perhaps I can compute each term more precisely.Let me try to compute each term with more precision:1. T^α = 10,000^0.6As 10,000 = 10^4, so 10^(4*0.6) = 10^2.410^2.4 is equal to 10^2 * 10^0.410^2 is 10010^0.4: Let me compute ln(10^0.4) = 0.4 * ln(10) ≈ 0.4 * 2.302585 ≈ 0.921034So, e^0.921034 ≈ 2.511886So, 10^0.4 ≈ 2.511886Thus, 10^2.4 ≈ 100 * 2.511886 ≈ 251.18862. D^β = 0.8^0.4Compute ln(0.8) ≈ -0.22314Multiply by 0.4: -0.089256Exponentiate: e^(-0.089256) ≈ 1 - 0.089256 + (0.089256)^2/2 - (0.089256)^3/6 + ... ≈ 0.9148Alternatively, using calculator: 0.8^0.4 ≈ 0.91483. S^γ = 5^0.3Compute ln(5) ≈ 1.60944Multiply by 0.3: 0.482832Exponentiate: e^0.482832 ≈ 1.620Alternatively, 5^0.3: 5^(3/10) is the 10th root of 5^3. 5^3=125. 10th root of 125 is approximately 1.6204. L^δ = 0.9^0.2Compute ln(0.9) ≈ -0.10536Multiply by 0.2: -0.021072Exponentiate: e^(-0.021072) ≈ 0.9792So, all these exponents are correct.Now, multiplying them:251.1886 * 0.9148 = ?Let me compute 251.1886 * 0.9148:First, 251.1886 * 0.9 = 226.07251.1886 * 0.0148 = ?Compute 251.1886 * 0.01 = 2.511886251.1886 * 0.0048 ≈ 1.2057So, total ≈ 2.511886 + 1.2057 ≈ 3.7176Thus, total ≈ 226.07 + 3.7176 ≈ 229.7876Next, multiply by 1.620:229.7876 * 1.620Compute 229.7876 * 1.6 = 367.66016229.7876 * 0.02 = 4.595752Total ≈ 367.66016 + 4.595752 ≈ 372.2559Then, multiply by 0.9792:372.2559 * 0.9792Compute 372.2559 * 1 = 372.2559372.2559 * 0.0208 ≈ 7.737So, subtract 7.737 from 372.2559: 372.2559 - 7.737 ≈ 364.5189Multiply by k = 1.2:364.5189 * 1.2 ≈ 437.4227So, approximately 437.42Wait, that seems consistent. So, the cultural integration index is approximately 437.42.But let me check if I can compute this more accurately without approximating each multiplication step.Alternatively, perhaps I can compute the product step by step with more precise intermediate results.Alternatively, maybe I can use logarithms to compute the product.But that might be more complicated.Alternatively, perhaps I can use a calculator for each multiplication step.But since I'm doing this manually, let's see:Compute 251.1886 * 0.9148:251.1886 * 0.9148Let me compute 251.1886 * 0.9 = 226.07251.1886 * 0.0148:Compute 251.1886 * 0.01 = 2.511886251.1886 * 0.0048:Compute 251.1886 * 0.004 = 1.0047544251.1886 * 0.0008 = 0.20095088So, 1.0047544 + 0.20095088 ≈ 1.20570528So, total for 0.0148 is 2.511886 + 1.20570528 ≈ 3.71759128Thus, total is 226.07 + 3.71759128 ≈ 229.78759128Now, multiply 229.78759128 * 1.620:Compute 229.78759128 * 1.6 = 367.660146048229.78759128 * 0.02 = 4.5957518256Total ≈ 367.660146048 + 4.5957518256 ≈ 372.255897874Now, multiply 372.255897874 * 0.9792:Compute 372.255897874 * 0.9792Compute 372.255897874 * 0.9 = 335.030308087372.255897874 * 0.07 = 26.057912851372.255897874 * 0.0092 ≈ 3.42633756So, total ≈ 335.030308087 + 26.057912851 + 3.42633756 ≈ 364.514558498Multiply by k = 1.2:364.514558498 * 1.2 ≈ 437.417470198So, approximately 437.4175So, rounding to two decimal places, 437.42.Therefore, the cultural integration index (CI) is approximately 437.42.Now, moving on to the second part: calculating the economic benefit (EB). The formula given is EB(CI, A) = m * CI^θ * A^η.Given values:- CI = 437.42 (from part 1)- A = 1500- m = 0.05- θ = 1.2- η = 0.8So, EB = 0.05 * (437.42)^1.2 * (1500)^0.8Again, let's break this down step by step.First, compute CI^θ: 437.42^1.2Second, compute A^η: 1500^0.8Then, multiply all together with m.Let me compute each part.1. Compute 437.42^1.2This is a bit tricky. Let me see.First, note that 437.42 is approximately 437.42We can write 437.42^1.2 as e^(1.2 * ln(437.42))Compute ln(437.42):We know that ln(400) ≈ 5.99146ln(437.42) is a bit higher.Compute ln(437.42):Let me compute ln(400) = 5.99146Compute ln(437.42) - ln(400) = ln(437.42 / 400) = ln(1.09355)ln(1.09355) ≈ 0.09 (since ln(1.09) ≈ 0.08617, ln(1.1) ≈ 0.09531)Compute 1.09355 - 1 = 0.09355Using Taylor series: ln(1+x) ≈ x - x^2/2 + x^3/3 - x^4/4 + ...x = 0.09355ln(1.09355) ≈ 0.09355 - (0.09355)^2 / 2 + (0.09355)^3 / 3 - (0.09355)^4 / 4Compute each term:0.09355 ≈ 0.09355(0.09355)^2 ≈ 0.008752(0.09355)^3 ≈ 0.000818(0.09355)^4 ≈ 0.0000763So,ln(1.09355) ≈ 0.09355 - 0.008752/2 + 0.000818/3 - 0.0000763/4Compute each term:0.09355- 0.008752 / 2 = -0.004376+ 0.000818 / 3 ≈ +0.0002727- 0.0000763 / 4 ≈ -0.000019075So total ≈ 0.09355 - 0.004376 + 0.0002727 - 0.000019075 ≈0.09355 - 0.004376 = 0.0891740.089174 + 0.0002727 ≈ 0.08944670.0894467 - 0.000019075 ≈ 0.0894276So, ln(1.09355) ≈ 0.0894276Thus, ln(437.42) ≈ ln(400) + ln(1.09355) ≈ 5.99146 + 0.0894276 ≈ 6.0808876So, ln(437.42) ≈ 6.0808876Now, compute 1.2 * ln(437.42) ≈ 1.2 * 6.0808876 ≈ 7.29706512Now, compute e^7.29706512We know that e^7 ≈ 1096.633e^0.29706512 ≈ ?Compute ln(1.344) ≈ 0.297, so e^0.297 ≈ 1.344Thus, e^7.29706512 ≈ e^7 * e^0.29706512 ≈ 1096.633 * 1.344 ≈Compute 1096.633 * 1.344:First, 1000 * 1.344 = 134496.633 * 1.344 ≈ Let's compute 96 * 1.344 = 129.024, and 0.633 * 1.344 ≈ 0.851So total ≈ 129.024 + 0.851 ≈ 129.875Thus, total ≈ 1344 + 129.875 ≈ 1473.875But let me check with a calculator:e^7.29706512We can compute e^7 = 1096.633e^0.29706512 ≈ e^0.297 ≈ 1.344So, 1096.633 * 1.344 ≈ 1096.633 * 1.3 = 1425.6229, 1096.633 * 0.044 ≈ 48.252Total ≈ 1425.6229 + 48.252 ≈ 1473.8749So, approximately 1473.87Thus, 437.42^1.2 ≈ 1473.872. Compute A^η = 1500^0.8Again, let's compute this.1500^0.8 = e^(0.8 * ln(1500))Compute ln(1500):We know that ln(1000) = 6.907755ln(1500) = ln(1000) + ln(1.5) ≈ 6.907755 + 0.405465 ≈ 7.31322So, ln(1500) ≈ 7.31322Multiply by 0.8: 0.8 * 7.31322 ≈ 5.850576Now, compute e^5.850576We know that e^5 ≈ 148.4132e^0.850576 ≈ ?Compute ln(2.34) ≈ 0.85, so e^0.85 ≈ 2.34Thus, e^5.850576 ≈ e^5 * e^0.850576 ≈ 148.4132 * 2.34 ≈Compute 148.4132 * 2 = 296.8264148.4132 * 0.34 ≈ 50.4565Total ≈ 296.8264 + 50.4565 ≈ 347.2829But let me verify:e^5.850576We can compute e^5 = 148.4132e^0.850576 ≈ e^0.85 ≈ 2.34So, 148.4132 * 2.34 ≈ 148.4132 * 2 + 148.4132 * 0.34 ≈ 296.8264 + 50.4565 ≈ 347.2829Thus, 1500^0.8 ≈ 347.28Now, multiply all together:EB = 0.05 * 1473.87 * 347.28First, compute 1473.87 * 347.28This is a large multiplication. Let me break it down.Compute 1473.87 * 300 = 442,161Compute 1473.87 * 47.28 ≈ ?Compute 1473.87 * 40 = 58,954.81473.87 * 7.28 ≈ ?Compute 1473.87 * 7 = 10,317.091473.87 * 0.28 ≈ 412.6836So, total ≈ 10,317.09 + 412.6836 ≈ 10,729.7736Thus, 1473.87 * 47.28 ≈ 58,954.8 + 10,729.7736 ≈ 69,684.5736So, total 1473.87 * 347.28 ≈ 442,161 + 69,684.5736 ≈ 511,845.5736Now, multiply by 0.05:EB = 0.05 * 511,845.5736 ≈ 25,592.27868So, approximately 25,592.28Wait, let me verify the multiplication steps because that's a lot of numbers.Alternatively, perhaps I can compute 1473.87 * 347.28 more accurately.But given the time, maybe I can accept that approximation.Alternatively, perhaps I can compute 1473.87 * 347.28 as follows:First, note that 1473.87 * 347.28 = 1473.87 * (300 + 40 + 7 + 0.28)Compute each part:1473.87 * 300 = 442,1611473.87 * 40 = 58,954.81473.87 * 7 = 10,317.091473.87 * 0.28 = 412.6836Now, sum all these:442,161 + 58,954.8 = 501,115.8501,115.8 + 10,317.09 = 511,432.89511,432.89 + 412.6836 ≈ 511,845.5736So, same as before.Thus, 1473.87 * 347.28 ≈ 511,845.5736Multiply by 0.05:511,845.5736 * 0.05 = 25,592.27868So, approximately 25,592.28Therefore, the economic benefit (EB) is approximately 25,592.28But let me check if I can compute this more accurately.Alternatively, perhaps I can use logarithms or exponents more accurately.But given the time, I think this is sufficient.So, summarizing:1. CI ≈ 437.422. EB ≈ 25,592.28But let me check if I made any calculation errors.Wait, in the first part, I had CI ≈ 437.42In the second part, EB = 0.05 * CI^1.2 * A^0.8We computed CI^1.2 ≈ 1473.87A^0.8 ≈ 347.28Then, 1473.87 * 347.28 ≈ 511,845.57Multiply by 0.05: 25,592.28Yes, that seems consistent.Alternatively, perhaps I can use more precise exponentiation.But given the time, I think this is acceptable.So, final answers:1. CI ≈ 437.422. EB ≈ 25,592.28But let me check if I can compute CI^1.2 and A^0.8 more accurately.Alternatively, perhaps I can use a calculator for these exponentiations.But since I'm doing this manually, I think the approximations are acceptable.So, I think I'm confident with these results.Final Answer1. The cultural integration index (CI) is boxed{437.42}.2. The economic benefit (EB) is boxed{25592.28}.</think>

🧠:<think>Okay, so I have this problem about a healthcare company developing a chatbot. The first part is about building a logistic regression model, and the second part is about optimizing the interaction time. Let me try to tackle each part step by step.Starting with the first part: Predictive Model Construction. They want to build a logistic regression model to predict whether a new patient will require a diagnosis based on 20 features from 10,000 interactions. The data is given as a matrix X (10000x20) and a vector y (10000x1) with binary outcomes (0 or 1). They want the closed-form solution for the logistic regression weights β using maximum likelihood estimation, assuming X is full rank.Hmm, I remember that logistic regression doesn't have a closed-form solution like linear regression does. In linear regression, we can use the normal equation to find the weights directly. But for logistic regression, the cost function is non-convex, so we usually use iterative methods like gradient descent or Newton-Raphson. Wait, but the question says to derive the closed-form solution. Maybe I'm missing something here.Wait, the question says "using the maximum likelihood estimation method." So, in maximum likelihood estimation for logistic regression, we set the derivative of the log-likelihood function to zero and solve for β. But unlike linear regression, this doesn't result in a straightforward formula because the equation is nonlinear. So, perhaps the question is expecting an expression for the maximum likelihood estimator, even if it's not solvable in closed-form?Alternatively, maybe they are referring to the iterative process, but phrased as a closed-form? Hmm, not sure. Let me think again.In linear regression, the closed-form solution is β = (X^T X)^{-1} X^T y. But for logistic regression, the maximum likelihood estimator doesn't have such a simple form. Instead, we have to solve the equation ∇L(β) = 0, where L is the log-likelihood function. The gradient is given by X^T (y - p), where p is the predicted probabilities. So, setting the gradient to zero gives X^T (y - p) = 0. But p depends on β through the logistic function, so it's a nonlinear equation that can't be solved in closed-form.Therefore, I think the question might be expecting an expression for the maximum likelihood estimator, acknowledging that it doesn't have a closed-form solution. Alternatively, maybe they are referring to the iterative formula for Newton-Raphson? But that's not a closed-form either.Wait, maybe I'm overcomplicating. Perhaps they just want the expression for the maximum likelihood estimator, which is the solution to the equation X^T (y - σ(Xβ)) = 0, where σ is the logistic function. So, in that case, the closed-form solution is the β that satisfies this equation, which can't be written explicitly without iterative methods.Alternatively, maybe they made a mistake and actually meant linear regression? Because linear regression does have a closed-form solution. But the question specifically says logistic regression.Wait, let me check the problem statement again. It says, "derive the closed-form solution for the logistic regression weights β using the maximum likelihood estimation method." Hmm. Maybe they are expecting the formula for the maximum likelihood estimator, which is the solution to the equation I mentioned earlier.So, perhaps the answer is that the closed-form solution is given by solving X^T (y - σ(Xβ)) = 0, where σ(z) = 1 / (1 + e^{-z}). But since this is a nonlinear equation, it doesn't have an explicit solution and must be solved numerically.Alternatively, maybe they are expecting the expression for the maximum likelihood estimator in terms of the data, but I don't think that's possible in closed-form for logistic regression.Wait, maybe I should write down the log-likelihood function and set its derivative to zero. The log-likelihood for logistic regression is:L(β) = Σ [y_i log(p_i) + (1 - y_i) log(1 - p_i)], where p_i = σ(β^T x_i).Taking the derivative with respect to β gives:∇L(β) = Σ (y_i - p_i) x_i = 0.So, in matrix form, this is X^T (y - p) = 0, where p is the vector of predicted probabilities.So, the maximum likelihood estimator is the β that satisfies this equation. Since p depends on β, it's a fixed point equation that can't be solved in closed-form. Therefore, the closed-form solution doesn't exist, and we have to use iterative methods.But the question says to derive the closed-form solution. Maybe they are expecting the expression for the maximum likelihood estimator, even if it's not solvable explicitly.Alternatively, perhaps they are referring to the fact that under certain conditions, logistic regression can be approximated or transformed into a linear model, but I don't think that's the case here.Wait, another thought: in some cases, if the data is separable, the maximum likelihood estimate doesn't exist because the coefficients go to infinity. But the question says X is full rank, so maybe it's not separable? Not sure.Alternatively, maybe they are considering the case where the logistic regression reduces to linear regression under some transformation, but that's not standard.Hmm, I'm a bit stuck here. Maybe I should proceed to the second part and come back to this.The second part is about optimizing interaction time. The chatbot needs to minimize average interaction time while maintaining at least 95% accuracy. Interaction time T is a linear function of the number of questions Q: T = aQ + b, with a=2, b=3. The accuracy A(Q) follows a logistic function: A(Q) = 1 / (1 + e^{-c(Q - d)}), where c=0.5, d=10.They want the minimum Q such that A(Q) >= 0.95, then find the minimum T.Alright, let's solve this part first. It seems more straightforward.So, set A(Q) = 0.95 and solve for Q.0.95 = 1 / (1 + e^{-0.5(Q - 10)}).Let me solve for Q.First, take reciprocals:1 / 0.95 = 1 + e^{-0.5(Q - 10)}.1 / 0.95 ≈ 1.0526.So, 1.0526 = 1 + e^{-0.5(Q - 10)}.Subtract 1:0.0526 = e^{-0.5(Q - 10)}.Take natural logarithm:ln(0.0526) = -0.5(Q - 10).Compute ln(0.0526). Let's see, ln(0.05) is about -2.9957, ln(0.0526) is slightly higher. Let me calculate it more accurately.0.0526 is approximately 1/19. So, ln(1/19) = -ln(19) ≈ -2.9444.So, -2.9444 ≈ -0.5(Q - 10).Multiply both sides by -1:2.9444 ≈ 0.5(Q - 10).Multiply both sides by 2:5.8888 ≈ Q - 10.So, Q ≈ 10 + 5.8888 ≈ 15.8888.Since Q must be an integer (number of questions), we round up to 16.So, the minimum number of questions is 16.Then, compute T = 2*16 + 3 = 32 + 3 = 35 minutes.Wait, but let me double-check the calculation.Starting from A(Q) = 0.95:0.95 = 1 / (1 + e^{-0.5(Q - 10)}).So, 1 + e^{-0.5(Q - 10)} = 1 / 0.95 ≈ 1.0526315789.Therefore, e^{-0.5(Q - 10)} = 1.0526315789 - 1 = 0.0526315789.Take natural log:-0.5(Q - 10) = ln(0.0526315789).Compute ln(0.0526315789):Since 0.0526315789 is 1/19, ln(1/19) = -ln(19) ≈ -2.944438979.So,-0.5(Q - 10) = -2.944438979.Multiply both sides by -1:0.5(Q - 10) = 2.944438979.Multiply both sides by 2:Q - 10 = 5.888877958.Therefore, Q = 10 + 5.888877958 ≈ 15.888877958.Since Q must be an integer, we round up to 16.Thus, minimum Q is 16, and T = 2*16 + 3 = 35 minutes.Okay, that seems solid.Now, going back to the first part. Since I'm stuck, maybe I should look up if there's a closed-form solution for logistic regression. From what I recall, there isn't one, but perhaps under certain conditions or approximations, there might be.Wait, maybe the question is referring to the iterative formula, like Newton-Raphson, which can be written in closed-form steps, but not a single formula for β. Alternatively, maybe they are expecting the expression for the maximum likelihood estimator, which is the solution to the equation X^T (y - p) = 0, but that's not a closed-form solution.Alternatively, perhaps they are considering the case where the logistic function is approximated by a linear function, but that's not standard.Wait, another thought: in some cases, when the number of features is large, or when the data is separable, but the question says X is full rank, which is 20 features with 10000 samples, so it's likely not rank-deficient.Alternatively, maybe they are expecting the formula for the maximum likelihood estimator in terms of the data, but as I said, it's not possible in closed-form.Wait, perhaps the question is actually about linear regression, not logistic regression? Because linear regression does have a closed-form solution. Maybe a misstatement in the problem.But the problem clearly states logistic regression, so I have to stick with that.Alternatively, maybe they are referring to the fact that the maximum likelihood estimator can be expressed as the solution to a system of equations, which is a closed-form in the sense of a system, but not an explicit formula.So, perhaps the answer is that the closed-form solution is given by solving the equation X^T (y - σ(Xβ)) = 0, where σ is the logistic function.But since this is a nonlinear equation, it doesn't have an explicit solution, so we have to use numerical methods.Alternatively, maybe they are expecting the expression for the maximum likelihood estimator in terms of the data, but I don't think that's possible.Wait, another angle: in some cases, when the number of features is small, you can write the equations explicitly, but with 20 features, that's impractical.Alternatively, maybe they are referring to the fact that the maximum likelihood estimator can be written as the solution to a convex optimization problem, but that's more of a characterization than a closed-form solution.Hmm, I'm not sure. Maybe I should just state that there is no closed-form solution for logistic regression and that iterative methods are required, but the problem says to derive it, so perhaps I'm missing something.Wait, maybe they are considering the case where the logistic function is linearized, but that's not standard.Alternatively, maybe they are referring to the fact that the maximum likelihood estimator can be expressed as the solution to a weighted least squares problem in an iterative manner, but again, that's not a closed-form.Alternatively, perhaps they are expecting the formula for the maximum likelihood estimator in terms of the data, but as I said, it's not possible.Wait, maybe I should write down the maximum likelihood equations and see if I can manipulate them into a closed-form.The log-likelihood is:L(β) = Σ [y_i log(σ(β^T x_i)) + (1 - y_i) log(1 - σ(β^T x_i))].Taking derivative with respect to β:∇L(β) = Σ (y_i - σ(β^T x_i)) x_i = 0.So, in matrix form:X^T (y - σ(Xβ)) = 0.This is a system of nonlinear equations because σ(Xβ) is nonlinear in β. Therefore, there's no closed-form solution, and we have to solve this numerically.Therefore, the closed-form solution doesn't exist, and iterative methods are required.But the question says to derive the closed-form solution, so maybe they are expecting the expression for the maximum likelihood estimator, which is the solution to this equation.Alternatively, perhaps they are expecting the formula for the maximum likelihood estimator in terms of the data, but as I said, it's not possible.Wait, maybe I should write the maximum likelihood estimator as:β = (X^T W X)^{-1} X^T W z,where W is a diagonal matrix of weights and z is the working response, but that's from the iteratively reweighted least squares (IRLS) algorithm, which is an iterative method. So, that's not a closed-form solution either.Hmm, I'm stuck. Maybe I should just state that there is no closed-form solution for logistic regression and that iterative methods are required, but the problem says to derive it, so perhaps I'm missing something.Wait, maybe they are considering the case where the logistic function is approximated by a linear function, but that's not standard.Alternatively, maybe they are referring to the fact that the maximum likelihood estimator can be expressed as the solution to a system of equations, which is a closed-form in the sense of a system, but not an explicit formula.Alternatively, perhaps they are expecting the expression for the maximum likelihood estimator in terms of the data, but as I said, it's not possible.Wait, another thought: in some cases, when the number of features is large, or when the data is separable, but the question says X is full rank, which is 20 features with 10000 samples, so it's likely not rank-deficient.Alternatively, maybe they are expecting the formula for the maximum likelihood estimator in terms of the data, but I don't think that's possible.Wait, maybe I should just write that the closed-form solution is given by solving X^T (y - σ(Xβ)) = 0, where σ is the logistic function, and that this requires iterative methods.But the question says to derive the closed-form solution, so perhaps they are expecting the expression for β in terms of X and y, but as I said, it's not possible.Alternatively, maybe they are referring to the fact that the maximum likelihood estimator can be expressed as the solution to a convex optimization problem, but that's more of a characterization than a closed-form solution.Hmm, I think I have to conclude that there is no closed-form solution for logistic regression, and that iterative methods are required. Therefore, the answer is that the closed-form solution does not exist, and one must use numerical methods like gradient descent or Newton-Raphson to find the weights β.But the problem says to derive the closed-form solution, so maybe I'm misunderstanding the question. Perhaps they are referring to the case where the logistic regression reduces to linear regression under some transformation, but I don't think that's the case here.Alternatively, maybe they are considering the case where the outcome is rare, but that's not indicated here.Wait, another angle: if the number of features is small, say 1, then maybe we can solve for β explicitly, but with 20 features, that's not feasible.Alternatively, maybe they are expecting the formula for the maximum likelihood estimator in terms of the data, but as I said, it's not possible.Wait, perhaps they are referring to the fact that the maximum likelihood estimator can be written as the solution to a system of equations, which is a closed-form in the sense of a system, but not an explicit formula.Alternatively, maybe they are expecting the expression for the maximum likelihood estimator in terms of the data, but I don't think that's possible.Wait, maybe I should just write down the maximum likelihood equations and say that's the closed-form solution, even though it's not solvable explicitly.So, in conclusion, for the first part, the closed-form solution for logistic regression weights β is given by solving the equation X^T (y - σ(Xβ)) = 0, where σ(z) = 1 / (1 + e^{-z}). Since this is a nonlinear equation, it doesn't have an explicit closed-form solution and must be solved numerically.For the second part, the minimum number of questions Q is 16, leading to a minimum interaction time T of 35 minutes.I think that's the best I can do for now.</think>

🧠:<think>Okay, so I have this problem about a fashion designer creating garments using two materials, A and B. The goal is to maximize the aesthetic appeal while considering some constraints. Let me try to break this down step by step.First, the problem is divided into two parts. Part 1 is about maximizing the aesthetic appeal without any additional constraints except that the sum of the proportions of materials A and B is 1. Part 2 introduces a sustainability constraint where the environmental impact must not exceed a certain value. I need to handle each part separately.Starting with Part 1:1. Reformulate the aesthetic appeal function as a single-variable function.Given that ( x + y = 1 ), I can express one variable in terms of the other. Let me solve for y: ( y = 1 - x ). So, substituting ( y ) into the aesthetic function ( A(x, y) = 5x^2 + 4xy + 3y^2 ), we get:( A(x) = 5x^2 + 4x(1 - x) + 3(1 - x)^2 )Let me expand this step by step.First, expand ( 4x(1 - x) ):( 4x - 4x^2 )Next, expand ( 3(1 - x)^2 ):( 3(1 - 2x + x^2) = 3 - 6x + 3x^2 )Now, combine all the terms:( 5x^2 + (4x - 4x^2) + (3 - 6x + 3x^2) )Let me combine like terms:- ( x^2 ) terms: ( 5x^2 - 4x^2 + 3x^2 = 4x^2 )- ( x ) terms: ( 4x - 6x = -2x )- Constant term: ( 3 )So, the function simplifies to:( A(x) = 4x^2 - 2x + 3 )2. Determine the proportion of Material A and B that maximizes A(x).Since this is a quadratic function in terms of x, and the coefficient of ( x^2 ) is positive (4), the parabola opens upwards, meaning it has a minimum point, not a maximum. Wait, that can't be right because we are supposed to maximize the aesthetic appeal. Hmm, maybe I made a mistake in simplifying.Wait, let me double-check my expansion:Original function:( A(x, y) = 5x^2 + 4xy + 3y^2 )Substitute ( y = 1 - x ):( 5x^2 + 4x(1 - x) + 3(1 - x)^2 )Compute each term:- ( 5x^2 ) stays as is.- ( 4x(1 - x) = 4x - 4x^2 )- ( 3(1 - 2x + x^2) = 3 - 6x + 3x^2 )Combine all terms:( 5x^2 + 4x - 4x^2 + 3 - 6x + 3x^2 )Now, combining like terms:- ( x^2 ): 5x^2 - 4x^2 + 3x^2 = 4x^2- ( x ): 4x - 6x = -2x- Constants: 3So, yes, ( A(x) = 4x^2 - 2x + 3 ). Since the coefficient of ( x^2 ) is positive, it's a convex function, meaning it has a minimum, not a maximum. But the problem says to maximize the aesthetic appeal. That seems contradictory.Wait, maybe I did something wrong in substitution. Let me check again.Original function: ( 5x^2 + 4xy + 3y^2 )With ( y = 1 - x ), so:( 5x^2 + 4x(1 - x) + 3(1 - x)^2 )= ( 5x^2 + 4x - 4x^2 + 3(1 - 2x + x^2) )= ( 5x^2 + 4x - 4x^2 + 3 - 6x + 3x^2 )= ( (5x^2 - 4x^2 + 3x^2) + (4x - 6x) + 3 )= ( 4x^2 - 2x + 3 )Yes, that's correct. So, the function is indeed ( 4x^2 - 2x + 3 ), which is a quadratic with a minimum. So, does that mean the maximum occurs at the endpoints of the domain?Since ( x ) and ( y ) are proportions, they must be between 0 and 1. So, the domain of ( x ) is [0,1]. Therefore, the maximum of ( A(x) ) would occur either at ( x = 0 ) or ( x = 1 ).Let me compute ( A(0) ) and ( A(1) ):- ( A(0) = 4(0)^2 - 2(0) + 3 = 3 )- ( A(1) = 4(1)^2 - 2(1) + 3 = 4 - 2 + 3 = 5 )So, the maximum occurs at ( x = 1 ), which means Material A is 100% and Material B is 0%. But that seems a bit extreme. Let me think again.Wait, maybe I should have considered that the function is quadratic, but perhaps I need to check if it's concave or convex. Since the coefficient of ( x^2 ) is positive, it's convex, so the minimum is at the vertex, and the maximum is at the endpoints.So, yes, the maximum occurs at the endpoints. Therefore, the maximum aesthetic appeal is 5 when ( x = 1 ) and ( y = 0 ).But is that the case? Let me plug in some other values to see.For example, at ( x = 0.5 ), ( y = 0.5 ):( A(0.5, 0.5) = 5*(0.25) + 4*(0.25) + 3*(0.25) = 1.25 + 1 + 0.75 = 3 )Which is less than 5. At ( x = 0.25 ), ( y = 0.75 ):( A(0.25, 0.75) = 5*(0.0625) + 4*(0.1875) + 3*(0.5625) )= 0.3125 + 0.75 + 1.6875 = 2.75Still less than 5. Hmm, so it seems that the maximum is indeed at x=1.But wait, maybe I should have taken the derivative and found the critical point, even though it's a minimum.Let me compute the derivative of ( A(x) ):( A'(x) = 8x - 2 )Set derivative equal to zero to find critical points:( 8x - 2 = 0 )( 8x = 2 )( x = 2/8 = 1/4 )So, the critical point is at ( x = 1/4 ). Since the function is convex, this is a minimum. Therefore, the maximum must be at the endpoints.So, at ( x = 0 ), ( A = 3 ); at ( x = 1 ), ( A = 5 ). Therefore, the maximum occurs at ( x = 1 ).So, the proportions are Material A: 1, Material B: 0.But wait, in the original function, is that correct? Let me plug ( x = 1 ), ( y = 0 ) into ( A(x, y) ):( A(1, 0) = 5*(1)^2 + 4*(1)*(0) + 3*(0)^2 = 5 + 0 + 0 = 5 ). Yes, that's correct.So, the first part answer is x=1, y=0.Moving on to Part 2:2. Introduce a sustainability constraint where the environmental impact ( E(x, y) = 2x^2 + 3xy + y^2 ) must not exceed 2. Find the new proportions of A and B that maximize A while keeping E ≤ 2.So, now we have two constraints:1. ( x + y = 1 )2. ( E(x, y) = 2x^2 + 3xy + y^2 ≤ 2 )We need to maximize ( A(x, y) ) under these constraints.First, let me express E in terms of a single variable, using ( y = 1 - x ):( E(x) = 2x^2 + 3x(1 - x) + (1 - x)^2 )Let me expand this:First, expand each term:- ( 2x^2 ) stays as is.- ( 3x(1 - x) = 3x - 3x^2 )- ( (1 - x)^2 = 1 - 2x + x^2 )Combine all terms:( 2x^2 + 3x - 3x^2 + 1 - 2x + x^2 )Combine like terms:- ( x^2 ): 2x^2 - 3x^2 + x^2 = 0x^2- ( x ): 3x - 2x = x- Constants: 1So, ( E(x) = x + 1 )Wait, that's interesting. So, ( E(x) = x + 1 ). Therefore, the constraint ( E(x) ≤ 2 ) becomes:( x + 1 ≤ 2 )( x ≤ 1 )But since ( x ) is already between 0 and 1 (because ( x + y = 1 ) and both x and y are proportions), the constraint ( x ≤ 1 ) is automatically satisfied. Therefore, the environmental impact constraint doesn't restrict x any further because even at x=1, E=2, which is exactly the limit.Wait, let me verify:At x=1, E=1 + 1=2. At x=0, E=0 +1=1. So, E ranges from 1 to 2 as x goes from 0 to1. Therefore, the constraint E ≤2 is always satisfied because E can't exceed 2 in this setup.Therefore, the sustainability constraint doesn't actually impose any new restrictions because the maximum E is 2, which is exactly the limit. So, the feasible region is still x ∈ [0,1].But wait, that seems odd. Maybe I made a mistake in simplifying E(x).Let me recompute E(x):( E(x) = 2x^2 + 3x(1 - x) + (1 - x)^2 )Compute each term:- ( 2x^2 )- ( 3x(1 - x) = 3x - 3x^2 )- ( (1 - x)^2 = 1 - 2x + x^2 )Combine all terms:( 2x^2 + 3x - 3x^2 + 1 - 2x + x^2 )Combine like terms:- ( x^2 ): 2x^2 - 3x^2 + x^2 = 0x^2- ( x ): 3x - 2x = x- Constants: 1So, yes, ( E(x) = x + 1 ). Therefore, the constraint E ≤2 is equivalent to x ≤1, which is already true because x + y =1 and x ≥0.Therefore, the sustainability constraint doesn't affect the feasible region. So, the maximum of A(x) is still at x=1, y=0.But wait, that seems counterintuitive. If the environmental impact is E=2 when x=1, which is exactly the limit, then technically, it's allowed. So, the proportions remain the same.But let me think again. Maybe I should consider if there's a case where E=2 and x <1, but that would require E(x)=2, which is x +1=2, so x=1. So, the only point where E=2 is x=1. Therefore, the constraint doesn't restrict any other points because E is always ≤2 in the domain x ∈ [0,1].Therefore, the maximum of A(x) is still at x=1, y=0.But wait, let me check if there's a possibility that when considering both A and E, maybe the maximum A is achieved at a point where E=2, which is x=1. So, the answer is the same as part 1.But perhaps I should approach this using Lagrange multipliers to be thorough, even though the constraint is redundant.Let me set up the Lagrangian for part 2:We need to maximize ( A(x, y) = 5x^2 + 4xy + 3y^2 ) subject to:1. ( x + y = 1 )2. ( 2x^2 + 3xy + y^2 ≤ 2 )But since the second constraint is always satisfied, the maximum is still at x=1, y=0.Alternatively, if I consider that maybe the constraint could be binding, but in this case, it's not because E(x) = x +1, which is always ≤2 in the domain.Therefore, the proportions remain x=1, y=0.But wait, let me think again. Maybe I should consider if the maximum of A(x) is achieved at a point where E=2, which is x=1, but perhaps there's a higher A(x) beyond x=1, but since x can't exceed 1, it's not possible.Therefore, the answer for part 2 is the same as part 1.But that seems odd because usually, constraints would change the result. Maybe I made a mistake in simplifying E(x).Wait, let me double-check the expansion of E(x):( E(x) = 2x^2 + 3x(1 - x) + (1 - x)^2 )= ( 2x^2 + 3x - 3x^2 + 1 - 2x + x^2 )= ( (2x^2 - 3x^2 + x^2) + (3x - 2x) + 1 )= ( 0x^2 + x + 1 )= ( x + 1 )Yes, that's correct. So, E(x) = x +1, which is linear. Therefore, the maximum E is at x=1, which is 2, exactly the constraint. So, the constraint is binding only at x=1.Therefore, the feasible region is x ∈ [0,1], and the maximum of A(x) is at x=1, y=0.So, the answer for both parts is x=1, y=0.But wait, let me think again. Maybe I should consider that the constraint E ≤2 is not redundant because E(x) = x +1, so for x >1, E would exceed 2, but since x cannot exceed 1, the constraint is automatically satisfied. Therefore, the maximum of A(x) is still at x=1.Alternatively, if the constraint were E ≤1.5, then x would have to be ≤0.5, and we would have to maximize A(x) in [0,0.5]. But in this case, the constraint is E ≤2, which is always true, so no change.Therefore, the proportions remain x=1, y=0 for both parts.But let me check if there's a mistake in the problem statement. The environmental impact function is E(x,y)=2x² +3xy + y². When x=1, y=0, E=2(1)+0+0=2, which is exactly the limit. So, it's allowed.Therefore, the answer is the same for both parts.But wait, maybe I should consider that the maximum of A(x) is at x=1, but perhaps there's a higher A(x) beyond x=1, but since x can't exceed 1, it's not possible.Alternatively, maybe I should use Lagrange multipliers for part 2, considering both constraints.Let me try that.We need to maximize A(x,y) subject to:1. ( x + y = 1 )2. ( 2x^2 + 3xy + y^2 ≤ 2 )But since the second constraint is always satisfied, the maximum is still at x=1, y=0.Alternatively, if I set up the Lagrangian with both constraints, but since the second constraint is not binding except at x=1, the solution remains the same.Therefore, the answer is x=1, y=0 for both parts.But wait, let me think again. Maybe I should consider that the maximum of A(x) is achieved at x=1, but perhaps the constraint E=2 is only satisfied at x=1, so that's the only point where E=2, and since A(x) is maximized there, that's the solution.Yes, that makes sense.So, in conclusion:Part 1: x=1, y=0Part 2: x=1, y=0But wait, that seems too straightforward. Maybe I should check if there's a higher A(x) when E=2, but since E=2 only at x=1, and A(x) is maximized there, it's correct.Alternatively, if E were a binding constraint elsewhere, but in this case, it's not.Therefore, the final answers are:1. x=1, y=02. x=1, y=0But let me write the answers in the required format.</think>