🧠:<think>Okay, so I have this problem about two agents, A and B, with utility functions based on their wealth. The economy has a fixed total wealth, W_T, which is the sum of W_A and W_B. The first part asks me to find W_A and W_B when W_A is three times W_B. The second part is about redistributing wealth to maximize the sum of their utilities. Hmm, let me take this step by step.Starting with part 1. I know that W_A = 3W_B, and W_T = W_A + W_B. So if I substitute W_A with 3W_B in the total wealth equation, I get W_T = 3W_B + W_B, which simplifies to W_T = 4W_B. Therefore, W_B must be W_T divided by 4, right? So W_B = W_T / 4. Then, since W_A is three times that, W_A = 3*(W_T / 4) = 3W_T / 4. That seems straightforward. Let me just write that down:1. W_A = (3/4)W_T and W_B = (1/4)W_T.Okay, moving on to part 2. The policy is redistributing wealth such that W'_A = αW_A and W'_B = W_T - W'_A. So, W'_A is a fraction α of the original W_A, and W'_B is the rest. The goal is to find α that maximizes the sum of utilities, which are logarithmic functions.So, the utilities after redistribution are U'_A = ln(W'_A) and U'_B = ln(W'_B). The total utility is U'_A + U'_B = ln(αW_A) + ln(W_T - αW_A). I need to maximize this with respect to α.Let me write that function out:Total Utility (TU) = ln(αW_A) + ln(W_T - αW_A)But I know from part 1 that W_A is (3/4)W_T. So substituting that in:TU = ln(α*(3/4)W_T) + ln(W_T - α*(3/4)W_T)Simplify each term:First term: ln( (3/4)αW_T ) = ln(3/4) + ln(α) + ln(W_T)Second term: ln( W_T - (3/4)αW_T ) = ln( W_T(1 - (3/4)α) ) = ln(W_T) + ln(1 - (3/4)α)So, combining both terms:TU = [ln(3/4) + ln(α) + ln(W_T)] + [ln(W_T) + ln(1 - (3/4)α)]Simplify:TU = ln(3/4) + ln(α) + 2ln(W_T) + ln(1 - (3/4)α)But since W_T is a constant, the terms involving ln(W_T) are constants with respect to α. So, when maximizing TU with respect to α, we can ignore the constant terms because they don't affect the location of the maximum. So, effectively, we can consider the function:f(α) = ln(α) + ln(1 - (3/4)α)We need to find α that maximizes f(α). Let's compute the derivative of f(α) with respect to α and set it to zero.f'(α) = d/dα [ln(α) + ln(1 - (3/4)α)] = (1/α) + [ ( -3/4 ) / (1 - (3/4)α) ]Set derivative equal to zero:(1/α) - (3/4)/(1 - (3/4)α) = 0So,1/α = (3/4)/(1 - (3/4)α)Cross-multiplying:1*(1 - (3/4)α) = (3/4)*αSimplify:1 - (3/4)α = (3/4)αBring terms with α to one side:1 = (3/4)α + (3/4)α = (6/4)α = (3/2)αSo,α = 1 / (3/2) = 2/3Wait, is that correct? Let me check my steps.Starting from the derivative:f'(α) = 1/α - (3/4)/(1 - (3/4)α) = 0So,1/α = (3/4)/(1 - (3/4)α)Cross-multiplying:1*(1 - (3/4)α) = (3/4)*αWhich is:1 - (3/4)α = (3/4)αAdding (3/4)α to both sides:1 = (3/4)α + (3/4)α = (6/4)α = (3/2)αSo,α = 1 / (3/2) = 2/3Yes, that seems correct. So, α is 2/3.But let me think about this. If α is 2/3, then W'_A = (2/3)W_A. Since W_A was 3/4 W_T, then W'_A = (2/3)*(3/4)W_T = (1/2)W_T. Similarly, W'_B = W_T - W'_A = (1/2)W_T. So, the redistribution would make both agents have equal wealth, each with half of W_T.Is that the maximum? Let me verify by plugging back into the total utility.Original utilities:U_A = ln(3/4 W_T), U_B = ln(1/4 W_T). So total utility is ln(3/4) + ln(1/4) + 2ln(W_T). After redistribution, both have W_T/2, so utilities are ln(W_T/2) each. Total utility is 2ln(W_T/2) = 2ln(W_T) - 2ln(2). Comparing to original total utility, which was ln(3/4) + ln(1/4) + 2ln(W_T) = ln(3/16) + 2ln(W_T). So, ln(3/16) is approximately ln(0.1875) ≈ -1.673, while 2ln(W_T/2) is 2ln(W_T) - 2ln(2). The difference is in the constants, but in terms of maximizing the sum, since we're dealing with logarithms, the maximum occurs when the product of their wealths is maximized.Wait, actually, the sum of logs is the log of the product. So, maximizing U_A + U_B is equivalent to maximizing W_A * W_B. So, in this case, the product W_A * W_B is maximized when W_A = W_B, given that W_A + W_B is fixed. That's a standard result from optimization: for a fixed sum, the product is maximized when the numbers are equal. So, indeed, the maximum occurs when W'_A = W'_B = W_T / 2, which corresponds to α = 2/3.Therefore, the value of α that maximizes the total utility is 2/3.Let me just recap:1. Calculated initial wealth distribution: W_A = 3/4 W_T, W_B = 1/4 W_T.2. For redistribution, set up the total utility function, took its derivative, found α = 2/3, which equalizes the wealth. Checked that this makes sense because the product W_A * W_B is maximized when they are equal, given fixed total wealth.I think that's solid. I don't see any mistakes in the reasoning.Final Answer1. ( W_A = boxed{dfrac{3}{4}W_T} ) and ( W_B = boxed{dfrac{1}{4}W_T} ).2. The value of ( alpha ) that maximizes the sum of utilities is ( boxed{dfrac{2}{3}} ).</think>

🧠:<think>Alright, so I have this problem about Alex, a long-haul truck driver, who needs to make a round trip between three cities: A, B, and C. The cities form a triangle with distances AB = 300 miles, BC = 400 miles, and CA = 500 miles. There are two parts to the problem: the first is calculating the time it takes for Alex to complete the round trip at a constant speed of 60 mph, and the second is determining the optimal speed to minimize fuel consumption given a specific fuel consumption function.Starting with the first part. I need to find the total time for the round trip. Since it's a round trip starting and ending at City A, Alex will drive from A to B to C and back to A. So, the total distance is the sum of the distances from A to B, B to C, and C back to A. That would be 300 + 400 + 500 miles. Let me calculate that: 300 + 400 is 700, plus 500 is 1200 miles. So, the total distance Alex needs to cover is 1200 miles.Now, if Alex is driving at a constant speed of 60 miles per hour, the time taken can be found by dividing the total distance by the speed. So, time = distance / speed. Plugging in the numbers, that's 1200 miles divided by 60 mph. Let me compute that: 1200 / 60 is 20. So, the time it will take is 20 hours. That seems straightforward.Moving on to the second part, which is a bit more complex. Alex wants to minimize fuel consumption by adjusting their speed. The fuel consumption function is given as F(v) = a/v + bv, where a and b are constants. Here, a is 0.01 and b is 0.0001. So, the function becomes F(v) = 0.01/v + 0.0001v.First, I need to find the optimal speed v that minimizes fuel consumption. Since fuel consumption is given per mile, I think we need to consider the total fuel consumed over the entire trip. The total fuel consumed would be the fuel consumption per mile multiplied by the total distance. So, total fuel = F(v) * total distance.But before that, maybe I should just focus on minimizing F(v) first because the total fuel will be proportional to F(v). So, if I minimize F(v), the total fuel will be minimized as well. So, let's consider F(v) = 0.01/v + 0.0001v. To find the minimum, we can take the derivative of F(v) with respect to v, set it equal to zero, and solve for v.Calculating the derivative: F'(v) = d/dv [0.01/v + 0.0001v]. The derivative of 0.01/v is -0.01 / v², and the derivative of 0.0001v is 0.0001. So, F'(v) = -0.01 / v² + 0.0001.Setting F'(v) equal to zero: -0.01 / v² + 0.0001 = 0. Let's solve for v.First, move the second term to the other side: -0.01 / v² = -0.0001. Wait, actually, let me write it as 0.0001 = 0.01 / v². So, 0.0001 = 0.01 / v².To solve for v², we can rearrange: v² = 0.01 / 0.0001. Calculating that: 0.01 divided by 0.0001 is 100. So, v² = 100. Taking the square root of both sides, v = sqrt(100) which is 10. So, the optimal speed is 10 mph? Wait, that seems really slow for a truck. Hmm, maybe I made a mistake in my calculation.Let me double-check. The derivative was F'(v) = -0.01 / v² + 0.0001. Setting equal to zero: -0.01 / v² + 0.0001 = 0. So, moving terms: 0.0001 = 0.01 / v². Then, v² = 0.01 / 0.0001. 0.01 divided by 0.0001 is 100. So, v² = 100, so v = 10. Hmm, that's correct mathematically, but in reality, a truck driver driving at 10 mph seems impractical. Maybe the units are different? Wait, the problem says the speed is in miles per hour, and the fuel consumption is in gallons per mile. So, perhaps the constants a and b are given in such a way that the optimal speed comes out to 10 mph. Maybe in the context of the problem, it's acceptable. Alternatively, perhaps I misread the constants. Let me check: a is 0.01 and b is 0.0001. Yes, that's correct.Alternatively, maybe the fuel consumption function is in gallons per hour instead of per mile? Wait, the problem says F(v) is in gallons per mile. So, it's gallons per mile. So, to get total fuel consumption, we need to multiply by the total miles. So, perhaps 10 mph is correct. Let me think about the units. If F(v) is in gallons per mile, then it's the amount of fuel used per mile driven. So, if you drive slower, you might use less fuel per mile, but take more time. But in this case, the function is F(v) = a/v + bv. So, as v increases, a/v decreases but bv increases. So, there's a trade-off.Wait, but in our calculation, the optimal speed is 10 mph, which seems too slow. Maybe I should check the derivative again. Let's see: F(v) = 0.01 / v + 0.0001v. The derivative is F'(v) = -0.01 / v² + 0.0001. Setting equal to zero: -0.01 / v² + 0.0001 = 0. So, 0.0001 = 0.01 / v². Multiply both sides by v²: 0.0001 v² = 0.01. Then, v² = 0.01 / 0.0001 = 100. So, v = 10. Yeah, that's correct. So, mathematically, the optimal speed is 10 mph. Maybe in the context of the problem, it's acceptable, even though in real life, trucks don't drive that slow.Alternatively, perhaps the units of a and b are different? Wait, the problem says a = 0.01 and b = 0.0001, but it doesn't specify units. It just says F(v) is in gallons per mile. So, if a is 0.01 and b is 0.0001, then the units must be consistent with v in mph. So, 0.01 is in gallons per mile per mph, and 0.0001 is in gallons per mile per mph? Wait, no, let me think about the units.F(v) is in gallons per mile. So, a is in gallons per mile per mph, because it's divided by v (mph). Similarly, b is in gallons per mile per mph, because it's multiplied by v (mph). So, a is 0.01 gallons/(mile·mph), and b is 0.0001 gallons/(mile·mph). So, when you compute a/v, it becomes (gallons/(mile·mph)) / (mph) = gallons/(mile·(mph)^2). Wait, that doesn't make sense. Maybe I need to reconsider.Alternatively, perhaps a is in gallons per hour, and b is in hours per gallon? Wait, no, because F(v) is in gallons per mile. So, let's think about the units:F(v) = a / v + b v.If F(v) is gallons per mile, then a / v must be in gallons per mile, and b v must also be in gallons per mile.So, a / v: a must have units of gallons per mile per mph, because when divided by mph (v), it becomes gallons per mile.Similarly, b v: b must have units of gallons per mile per mph, because when multiplied by mph, it becomes gallons per mile.So, a is 0.01 gallons/(mile·mph), and b is 0.0001 gallons/(mile·mph). So, that's consistent.So, the calculation is correct, leading to v = 10 mph. So, maybe in this problem, the optimal speed is indeed 10 mph. It might seem slow, but perhaps the fuel consumption model is such that lower speeds are more efficient, but with a quadratic increase in fuel consumption at higher speeds.So, moving forward, the optimal speed is 10 mph. Now, we need to calculate the total fuel consumed at this speed for the entire round trip.First, let's compute the fuel consumption per mile at v = 10 mph. F(10) = 0.01 / 10 + 0.0001 * 10. Calculating that: 0.01 / 10 is 0.001, and 0.0001 * 10 is 0.001. So, F(10) = 0.001 + 0.001 = 0.002 gallons per mile.Now, the total distance is 1200 miles, as calculated earlier. So, total fuel consumed is 0.002 gallons/mile * 1200 miles. Let me compute that: 0.002 * 1200 = 2.4 gallons. So, Alex would consume 2.4 gallons of fuel for the entire round trip at the optimal speed of 10 mph.Wait, that seems extremely low. 2.4 gallons for 1200 miles? That would be an average of 500 miles per gallon, which is unrealistic for a truck. Hmm, perhaps there's a mistake in the calculation. Let me check.F(v) = 0.01 / v + 0.0001v. At v = 10, F(10) = 0.01 / 10 + 0.0001 * 10 = 0.001 + 0.001 = 0.002 gallons per mile. So, 0.002 gallons per mile times 1200 miles is indeed 2.4 gallons. That seems too low, but mathematically, it's correct based on the given function.Alternatively, maybe the fuel consumption function is meant to be in gallons per hour, not per mile. Let me check the problem statement again. It says F(v) is in gallons per mile. So, it's correct. So, perhaps in this problem, the fuel consumption is extremely efficient, allowing for such low fuel usage. Alternatively, maybe the constants a and b are given in different units, but the problem states a = 0.01 and b = 0.0001 without specifying units beyond gallons per mile.So, unless I'm missing something, the calculation seems correct. Therefore, the optimal speed is 10 mph, and the total fuel consumed is 2.4 gallons.Wait, but let me think again. If the optimal speed is 10 mph, which is very slow, and the fuel consumption is 0.002 gallons per mile, which is 500 miles per gallon, that's an extremely efficient vehicle. Maybe the problem is designed this way, or perhaps I made a mistake in interpreting the fuel consumption function.Alternatively, perhaps the fuel consumption function is in gallons per hour, and we need to convert it to gallons per mile by dividing by speed. Let me consider that possibility.If F(v) is in gallons per hour, then to get gallons per mile, we would divide by speed. So, fuel consumption per mile would be F(v) / v. So, maybe the function is given as F(v) = a / v + b v, where F(v) is in gallons per hour. Then, to get gallons per mile, we would have (a / v + b v) / v = a / v² + b. So, in that case, the fuel consumption per mile would be a / v² + b.But the problem states that F(v) is in gallons per mile, so that interpretation might not be correct. Alternatively, perhaps the function is given as gallons per hour, and we need to convert it to gallons per mile by dividing by speed. So, if F(v) is gallons per hour, then gallons per mile would be F(v) / v. So, in that case, the fuel consumption per mile would be (a / v + b v) / v = a / v² + b.But the problem says F(v) is in gallons per mile, so that might not be necessary. Hmm, I'm a bit confused now. Let me try to clarify.The problem states: "The fuel consumption function F(v) in gallons per mile is given by F(v) = a / v + b v." So, F(v) is already in gallons per mile. Therefore, the calculation is correct as is. So, at 10 mph, the fuel consumption is 0.002 gallons per mile, leading to 2.4 gallons total.Alternatively, maybe the problem expects us to consider the total time and then compute fuel consumption based on that. Wait, no, because fuel consumption is given per mile, so it's independent of time. So, the total fuel is just F(v) multiplied by total distance.So, unless I'm missing a step, I think the calculation is correct. Therefore, the optimal speed is 10 mph, and the total fuel consumed is 2.4 gallons.But just to be thorough, let me consider if the fuel consumption function is in gallons per hour. If that's the case, then to get gallons per mile, we would divide by speed. So, fuel consumption per mile would be (a / v + b v) / v = a / v² + b. Then, the total fuel would be (a / v² + b) * total distance.But since the problem states F(v) is in gallons per mile, I think that's not necessary. So, I think my initial calculation is correct.Therefore, summarizing:1. Total time for the round trip at 60 mph is 20 hours.2. Optimal speed is 10 mph, and total fuel consumed is 2.4 gallons.But just to make sure, let me check the units again. If F(v) is gallons per mile, then at 10 mph, F(v) = 0.002 gallons per mile. So, 1200 miles would be 2.4 gallons. That seems correct.Alternatively, maybe the problem expects us to consider the total time and fuel consumption over that time. But since F(v) is given per mile, it's independent of time. So, I think the calculation is fine.So, I think I've got the answers. The time is 20 hours, and the optimal speed is 10 mph with 2.4 gallons consumed.</think>

🧠:<think>Okay, so I have this problem about Mr. Thompson analyzing the impact of labor union activities on his factory's productivity and costs. There are two functions given: one for productivity, P(t), and one for additional costs, C(t). I need to solve two sub-problems: calculating the total productivity loss over 24 hours and determining the total additional cost over the same period.Starting with Sub-problem 1: Calculate the total productivity loss over 24 hours. The productivity function is P(t) = 100e^{-0.05t}. I need to find the total units produced over 24 hours and compare it to the initial productivity rate maintained without any decrease. So, first, what's the initial productivity rate? At t=0, P(0) = 100e^{0} = 100 units per hour. If there was no decrease, over 24 hours, the total production would be 100 units/hour * 24 hours = 2400 units.But with the union activities, productivity decreases over time. So, the total production is the integral of P(t) from t=0 to t=24. So, I need to compute the integral of 100e^{-0.05t} dt from 0 to 24. Then, subtract this value from 2400 to get the productivity loss.Let me set that up. The integral of 100e^{-0.05t} dt. The integral of e^{kt} dt is (1/k)e^{kt} + C, so here, k is -0.05. So, the integral becomes 100 * (1/(-0.05)) e^{-0.05t} evaluated from 0 to 24.Calculating the integral:Integral = 100 * (-20) [e^{-0.05*24} - e^{0}]= -2000 [e^{-1.2} - 1]= -2000 [ (e^{-1.2}) - 1 ]But since we're calculating total production, which is positive, we can ignore the negative sign because the integral will give a negative value, but we take the absolute value. Alternatively, maybe I should have set it up as 100 * integral e^{-0.05t} dt, which is 100 * (-20)e^{-0.05t} from 0 to 24, so that's 100*(-20)[e^{-1.2} - 1] = -2000[e^{-1.2} - 1]. So, the total production is -2000[e^{-1.2} - 1] = 2000[1 - e^{-1.2}]Compute 1 - e^{-1.2}. Let me calculate e^{-1.2}. I know that e^{-1} is approximately 0.3679, and e^{-1.2} is a bit less. Let me use a calculator: e^{-1.2} ≈ 0.3012. So, 1 - 0.3012 = 0.6988. Therefore, total production is 2000 * 0.6988 ≈ 1397.6 units.So, the initial production without any decrease would be 2400 units, and with the decrease, it's approximately 1397.6 units. Therefore, the productivity loss is 2400 - 1397.6 = 1002.4 units. So, approximately 1002.4 units lost over 24 hours.Wait, let me double-check the integral calculation. The integral of 100e^{-0.05t} dt from 0 to 24 is:100 * ∫e^{-0.05t} dt from 0 to24= 100 * [ (-1/0.05) e^{-0.05t} ] from 0 to24= 100 * (-20) [e^{-1.2} - e^{0}]= -2000 [e^{-1.2} - 1]= 2000 [1 - e^{-1.2}]Yes, that's correct. So, 2000*(1 - e^{-1.2}) ≈ 2000*(1 - 0.3012) = 2000*0.6988 ≈ 1397.6. So, total production is 1397.6, initial would be 2400, so loss is 2400 - 1397.6 = 1002.4. So, about 1002.4 units lost.Alternatively, maybe the problem expects an exact expression rather than a decimal approximation. So, 2000(1 - e^{-1.2}) is the exact total production, so the productivity loss is 2400 - 2000(1 - e^{-1.2}) = 2400 - 2000 + 2000e^{-1.2} = 400 + 2000e^{-1.2}. Wait, that seems different. Wait, no, 2400 - 2000(1 - e^{-1.2}) = 2400 - 2000 + 2000e^{-1.2} = 400 + 2000e^{-1.2}. Hmm, that's another way to write it, but I think 2400 - 1397.6 is more straightforward.Alternatively, maybe I should express the productivity loss as the integral of the decrease in productivity over time. The initial productivity is 100 units per hour, but it's decreasing as P(t) = 100e^{-0.05t}. So, the decrease at time t is 100 - P(t) = 100(1 - e^{-0.05t}). Therefore, the total productivity loss is the integral from 0 to24 of 100(1 - e^{-0.05t}) dt.Which is 100∫(1 - e^{-0.05t}) dt from 0 to24.= 100 [ ∫1 dt - ∫e^{-0.05t} dt ] from 0 to24= 100 [ t + (20)e^{-0.05t} ] from 0 to24Wait, hold on, the integral of e^{-0.05t} is (-1/0.05)e^{-0.05t} = -20e^{-0.05t}. So, the integral becomes:100 [ t - (-20)e^{-0.05t} ] from 0 to24= 100 [ t + 20e^{-0.05t} ] from 0 to24= 100 [ (24 + 20e^{-1.2}) - (0 + 20e^{0}) ]= 100 [24 + 20e^{-1.2} - 20]= 100 [4 + 20e^{-1.2}]= 100*4 + 100*20e^{-1.2}= 400 + 2000e^{-1.2}Which is the same as before: 400 + 2000e^{-1.2} ≈ 400 + 2000*0.3012 ≈ 400 + 602.4 ≈ 1002.4. So, same result. So, that's consistent.Therefore, the total productivity loss is approximately 1002.4 units over 24 hours.Moving on to Sub-problem 2: Determine the total additional cost incurred due to labor union activities over 24 hours. The cost function is C(t) = 50 + 10sin(πt/12). So, the total additional cost is the integral of C(t) from t=0 to t=24.So, integral of 50 + 10sin(πt/12) dt from 0 to24.Let me break this into two integrals: integral of 50 dt + integral of 10sin(πt/12) dt from 0 to24.First integral: ∫50 dt from 0 to24 is 50t evaluated from 0 to24, which is 50*24 - 50*0 = 1200.Second integral: ∫10sin(πt/12) dt from 0 to24. Let me compute that.Let u = πt/12, so du/dt = π/12, so dt = (12/π) du.So, the integral becomes 10 ∫sin(u) * (12/π) du = (120/π) ∫sin(u) du = (120/π)(-cos(u)) + C.So, evaluating from t=0 to t=24, which corresponds to u=0 to u=π*24/12=2π.So, the integral is (120/π)[ -cos(2π) + cos(0) ].But cos(2π) = 1 and cos(0) = 1, so it becomes (120/π)[ -1 + 1 ] = (120/π)(0) = 0.Therefore, the integral of the sine function over one full period (which 24 hours is, since the period of sin(πt/12) is 2π/(π/12) = 24) is zero.Therefore, the total additional cost is 1200 + 0 = 1200.So, the total additional cost over 24 hours is 1200 units of cost.Wait, let me confirm that. The integral of sin over a full period is indeed zero because it's symmetric. So, the positive and negative areas cancel out. So, yes, the integral of 10sin(πt/12) over 0 to24 is zero. So, the total additional cost is just the integral of 50, which is 1200.Therefore, the total additional cost is 1200.So, summarizing:Sub-problem 1: Productivity loss is approximately 1002.4 units.Sub-problem 2: Total additional cost is 1200.But wait, for Sub-problem 1, should I present the exact expression or the approximate value? The problem says to use the integral to find the total units produced and compare it to the initial rate. So, perhaps it's better to present both the exact expression and the approximate value.So, exact productivity loss is 2400 - 2000(1 - e^{-1.2}) = 400 + 2000e^{-1.2}. Alternatively, as 2400 - ∫P(t) dt from 0 to24, which is 2400 - 2000(1 - e^{-1.2}) = 400 + 2000e^{-1.2}.But 2000e^{-1.2} is approximately 2000*0.3012 ≈ 602.4, so 400 + 602.4 ≈ 1002.4.So, I think both are acceptable, but since the problem says to provide necessary steps and justifications, perhaps I should write both.Similarly, for Sub-problem 2, the integral is straightforward, giving 1200.So, I think that's it.Final AnswerSub-problem 1: The total productivity loss is boxed{1002.4} units.Sub-problem 2: The total additional cost is boxed{1200} units of cost.</think>

🧠:<think>Okay, so I have this problem about a department store purchaser who wants to stock two new lines of skincare products: Organic Line (O) and Cruelty-Free Line (C). The store has some policies about the minimum percentages each line must take up in the total stock. The profits per unit are different for each line, and I need to figure out how to maximize the total profit given these constraints.Let me try to break this down step by step.Problem 1: Formulating Constraints and Profit FunctionFirst, I need to define the variables. Let ( x ) be the number of units of Organic Line products, and ( y ) be the number of units of Cruelty-Free Line products. The store's policy says that at least 40% of the total skincare products must be from the Organic Line. So, the number of Organic Line products, ( x ), must be at least 40% of the total stock. Similarly, at least 60% must be from the Cruelty-Free Line, so ( y ) must be at least 60% of the total stock.Wait, hold on. If both ( x ) and ( y ) have to be at least 40% and 60% respectively, what does that mean for the total stock? Because 40% + 60% is 100%, but they can't both be exactly 40% and 60% unless the total stock is such that both can be satisfied.But actually, the total stock is the sum of ( x ) and ( y ). So, if ( x ) is at least 40% of ( x + y ), and ( y ) is at least 60% of ( x + y ), that might lead to some constraints.Let me write that down.The total number of products is ( x + y ). 1. The Organic Line must be at least 40% of the total, so:[ x geq 0.4(x + y) ]2. The Cruelty-Free Line must be at least 60% of the total, so:[ y geq 0.6(x + y) ]Also, since we can't have negative products, we have:[ x geq 0 ][ y geq 0 ]Now, let me simplify these inequalities.Starting with the first constraint:[ x geq 0.4(x + y) ]Let's subtract ( 0.4x ) from both sides:[ x - 0.4x geq 0.4y ][ 0.6x geq 0.4y ]Divide both sides by 0.2 to simplify:[ 3x geq 2y ]So, ( 3x - 2y geq 0 )Similarly, for the second constraint:[ y geq 0.6(x + y) ]Subtract ( 0.6y ) from both sides:[ y - 0.6y geq 0.6x ][ 0.4y geq 0.6x ]Divide both sides by 0.2:[ 2y geq 3x ]Which simplifies to:[ 3x - 2y leq 0 ]Wait, hold on. From the first constraint, we have ( 3x - 2y geq 0 ), and from the second constraint, we have ( 3x - 2y leq 0 ). So combining these two, we get:[ 3x - 2y = 0 ]Which implies:[ 3x = 2y ]So, ( y = frac{3}{2}x )Hmm, that's interesting. So, the constraints force ( y ) to be exactly ( frac{3}{2}x ). That seems a bit restrictive, but let me check my math.Starting with the first constraint:[ x geq 0.4(x + y) ]Multiply both sides by 10 to eliminate decimals:[ 10x geq 4x + 4y ]Subtract ( 4x ) from both sides:[ 6x geq 4y ]Divide both sides by 2:[ 3x geq 2y ]Which is the same as before.Second constraint:[ y geq 0.6(x + y) ]Multiply both sides by 10:[ 10y geq 6x + 6y ]Subtract ( 6y ) from both sides:[ 4y geq 6x ]Divide both sides by 2:[ 2y geq 3x ]Which is the same as before.So, yes, combining these two, we have ( 3x = 2y ). Therefore, ( y = frac{3}{2}x ). So, this is a hard constraint. So, the two lines must be in a ratio of 2:3, with Organic Line being 2 parts and Cruelty-Free Line being 3 parts.But wait, if ( y = frac{3}{2}x ), then substituting back into the total stock, which is ( x + y ), we get:[ x + frac{3}{2}x = frac{5}{2}x ]So, the total stock is ( frac{5}{2}x ). Therefore, if we know the total stock, we can find ( x ) and ( y ).But in Problem 1, we are just asked to formulate the constraints, not necessarily to solve for specific values yet. So, perhaps I need to express the constraints without combining them yet.Alternatively, maybe I misinterpreted the problem. Let me read it again."A department store purchaser is planning to stock two new lines of skincare products: Organic Line (O) and Cruelty-Free Line (C). The store's policy requires that at least 40% of the total skincare products stocked must be from the Organic Line and at least 60% must be from the Cruelty-Free Line."Wait, hold on. So, both lines have to be at least 40% and 60% respectively? That seems impossible because 40% + 60% = 100%, so the total would have to be exactly 100%, meaning that both constraints are tight. So, in that case, the total stock would have to be exactly 40% Organic and 60% Cruelty-Free.But wait, that can't be because 40% + 60% is 100%, so the total stock is fixed in terms of percentages. So, actually, the total stock is fixed in terms of the ratio between x and y.So, if the total stock is ( x + y ), then:- ( x geq 0.4(x + y) )- ( y geq 0.6(x + y) )But adding these two inequalities:( x + y geq 0.4(x + y) + 0.6(x + y) = (0.4 + 0.6)(x + y) = x + y )Which simplifies to ( x + y geq x + y ), which is always true. So, the only way both inequalities can hold is if both are equalities. Therefore, ( x = 0.4(x + y) ) and ( y = 0.6(x + y) ). Therefore, the total stock must be exactly 40% Organic and 60% Cruelty-Free.So, in that case, ( x = 0.4T ) and ( y = 0.6T ), where ( T = x + y ) is the total stock.But in Problem 1, we are not given a specific total stock, just to formulate the constraints. So, perhaps the constraints are:1. ( x geq 0.4(x + y) )2. ( y geq 0.6(x + y) )3. ( x geq 0 )4. ( y geq 0 )But as we saw, these constraints force ( x = 0.4(x + y) ) and ( y = 0.6(x + y) ), so effectively, ( x ) and ( y ) must be in a 2:3 ratio.Alternatively, maybe I need to represent it differently. Let me think.Alternatively, perhaps the store's policy is that at least 40% of the total must be Organic, and separately, at least 60% must be Cruelty-Free. So, these are separate constraints, not necessarily both being tight.But wait, if the total is ( x + y ), then:- ( x geq 0.4(x + y) )- ( y geq 0.6(x + y) )But if both are true, then adding them:( x + y geq 0.4(x + y) + 0.6(x + y) = x + y )Which again gives ( x + y geq x + y ), meaning equality must hold. So, both constraints must be equalities.Therefore, ( x = 0.4(x + y) ) and ( y = 0.6(x + y) ). Therefore, the total stock is fixed in terms of x and y, with x and y in a 2:3 ratio.So, in that case, the constraints are:1. ( x = 0.4(x + y) )2. ( y = 0.6(x + y) )3. ( x geq 0 )4. ( y geq 0 )But since ( x ) and ( y ) are related by ( y = 1.5x ), we can express the total profit in terms of one variable.But in Problem 1, we are just to formulate the constraints, so perhaps I should leave it as:1. ( x geq 0.4(x + y) )2. ( y geq 0.6(x + y) )3. ( x geq 0 )4. ( y geq 0 )But since these two inequalities force ( x = 0.4(x + y) ) and ( y = 0.6(x + y) ), perhaps it's better to express it as equalities.Alternatively, maybe the store's policy is that each line individually must meet their respective percentages, but not necessarily both at the same time. But that seems contradictory because if the total is ( x + y ), and both ( x ) and ( y ) have to be at least 40% and 60% respectively, then their sum would have to be at least 100%, which is only possible if both are exactly 40% and 60%.Therefore, I think the constraints are:1. ( x geq 0.4(x + y) )2. ( y geq 0.6(x + y) )3. ( x geq 0 )4. ( y geq 0 )But these lead to ( x = 0.4(x + y) ) and ( y = 0.6(x + y) ), so effectively, ( x = 0.4T ) and ( y = 0.6T ), where ( T = x + y ).But in Problem 1, we are not given a specific total, so perhaps we can just express the constraints as:1. ( x geq 0.4(x + y) )2. ( y geq 0.6(x + y) )3. ( x geq 0 )4. ( y geq 0 )And the total profit ( P ) is given by the sum of profits from each line. Since the profit per unit for Organic is 5 and for Cruelty-Free is 8, then:[ P = 5x + 8y ]So, that's the profit function.Problem 2: Maximizing Profit with Total Stock Fixed at 500 UnitsNow, moving on to Problem 2. The total number of skincare products to be stocked is fixed at 500 units. So, ( x + y = 500 ).We need to determine the values of ( x ) and ( y ) that maximize the total profit ( P = 5x + 8y ), given the constraints from Problem 1 and the total stock constraint.From Problem 1, we have the constraints:1. ( x geq 0.4(x + y) )2. ( y geq 0.6(x + y) )3. ( x geq 0 )4. ( y geq 0 )But since ( x + y = 500 ), let's substitute that into the constraints.First constraint:[ x geq 0.4(500) ][ x geq 200 ]Second constraint:[ y geq 0.6(500) ][ y geq 300 ]So, now we have:1. ( x geq 200 )2. ( y geq 300 )3. ( x + y = 500 )4. ( x geq 0 )5. ( y geq 0 )But since ( x + y = 500 ), and ( x geq 200 ), ( y geq 300 ), we can see that the minimums for ( x ) and ( y ) add up exactly to 500. Therefore, the only solution that satisfies both ( x geq 200 ) and ( y geq 300 ) with ( x + y = 500 ) is ( x = 200 ) and ( y = 300 ).Therefore, the values of ( x ) and ( y ) that satisfy all constraints are ( x = 200 ) and ( y = 300 ).But wait, let me check. If ( x = 200 ), then ( y = 300 ). Does this satisfy all constraints?- ( x = 200 geq 0.4(500) = 200 ) ✔️- ( y = 300 geq 0.6(500) = 300 ) ✔️- ( x + y = 500 ) ✔️- ( x geq 0 ) ✔️- ( y geq 0 ) ✔️Yes, it does. So, this is the only feasible solution.But wait, the problem says "determine the values of ( x ) and ( y ) that maximize the total profit ( P )". So, is this the only feasible solution? If so, then that's the maximum.But let me think again. If the total stock is fixed at 500, and the constraints require that ( x geq 200 ) and ( y geq 300 ), then ( x ) cannot be more than 200 because ( y ) cannot be less than 300. Similarly, ( y ) cannot be more than 300 because ( x ) cannot be less than 200.Wait, actually, no. If ( x ) is more than 200, then ( y ) would have to be less than 300 to keep the total at 500. But ( y ) has to be at least 300, so ( y ) cannot be less than 300. Therefore, ( x ) cannot be more than 200. Similarly, if ( y ) is more than 300, ( x ) would have to be less than 200, which is not allowed. Therefore, the only feasible solution is ( x = 200 ) and ( y = 300 ).Therefore, the maximum profit is achieved at this point, because any deviation from this would violate the constraints. So, ( P = 5(200) + 8(300) = 1000 + 2400 = 3400 ).But wait, let me confirm if this is indeed the maximum. Since the profit per unit for Cruelty-Free is higher (8) than for Organic (5), wouldn't we want to maximize ( y ) as much as possible? But in this case, ( y ) is already at its minimum required by the constraints, which is 300. So, if we could increase ( y ) beyond 300, we could increase profit, but we can't because ( x ) would have to decrease below 200, which is not allowed.Alternatively, if we could decrease ( y ) below 300, we could increase ( x ), but since ( y ) has a minimum, we can't do that either. Therefore, the only feasible point is ( x = 200 ), ( y = 300 ).So, in conclusion, the maximum profit is achieved when ( x = 200 ) and ( y = 300 ), giving a total profit of 3,400.But let me think again. Maybe I made a mistake in interpreting the constraints. Let me re-examine.The store's policy requires that at least 40% of the total must be Organic and at least 60% must be Cruelty-Free. So, if the total is 500, then:- Organic must be at least 200- Cruelty-Free must be at least 300But 200 + 300 = 500, so the total is exactly 500. Therefore, there is no room for any other combination. So, the only possible solution is ( x = 200 ), ( y = 300 ).Therefore, the maximum profit is 3,400.Wait, but let me think about this again. If the constraints were that each line individually must meet their percentages, but not necessarily both at the same time, then perhaps there could be other solutions. But in this case, since both constraints must be satisfied simultaneously, and their sum is exactly 100% of the total, there is no other solution.Therefore, the answer is ( x = 200 ), ( y = 300 ).Final Answer1. The constraints are ( x geq 0.4(x + y) ), ( y geq 0.6(x + y) ), ( x geq 0 ), and ( y geq 0 ). The total profit is ( P = 5x + 8y ).2. The values that maximize the total profit are ( x = boxed{200} ) and ( y = boxed{300} ).</think>