🧠:<think>Alright, so I have this problem where a philanthropist wants to donate 2,000,000 to three organizations, A, B, and C. Each organization has a different efficacy in terms of how many refugees they can help per 1,000 donated. The goal is to maximize the total number of refugees getting legal aid. Then, in part two, the donations are invested and grow at different rates, and we need to figure out how many additional refugees can be helped after one year.Starting with part 1. Let me think about how to model this. So, we have three variables: x, y, z, which are the amounts donated to A, B, and C respectively. The total donation is x + y + z = 2,000,000. Each organization helps a certain number of refugees per 1,000. So, for every 1,000 given to A, they help 5 refugees. Similarly, B helps 3, and C helps 2. So, the total number of refugees helped would be (x / 1000)*5 + (y / 1000)*3 + (z / 1000)*2. To make it simpler, maybe I can express this as (5x + 3y + 2z)/1000. So, the objective is to maximize this expression subject to x + y + z = 2,000,000, and x, y, z >= 0.Hmm, okay, so this is a linear optimization problem. The objective function is linear, and the constraint is also linear. So, the maximum will occur at one of the vertices of the feasible region.In such cases, the optimal solution is usually achieved by putting as much as possible into the most efficient option. So, since A is the most efficient (5 refugees per 1,000), followed by B (3), then C (2), the optimal strategy should be to donate all the money to A, right? Because that would maximize the number of refugees helped.Wait, let me verify that. If I donate all to A, then x = 2,000,000, y = z = 0. The total refugees would be (5*2,000,000)/1000 = 10,000 refugees.If I donate some to B or C, the number would be less. For example, if I donate 1,000,000 to A and 1,000,000 to B, the total refugees would be (5*1,000,000 + 3*1,000,000)/1000 = (5,000,000 + 3,000,000)/1000 = 8,000 refugees, which is less than 10,000. Similarly, donating to C would only decrease the total further.So, yes, the maximum number of refugees is achieved by donating all the money to organization A. Therefore, x = 2,000,000, y = 0, z = 0.But wait, let me think again. Is there any constraint that requires the philanthropist to donate to all three organizations? The problem doesn't specify any such requirement, so it's okay to donate all to A.Alright, so part 1 is solved. Now, moving on to part 2.After one year, the donations are invested and grow at respective rates. So, the amount donated to A grows at 5%, B at 4%, and C at 3%. So, the new amounts after one year would be:For A: x * 1.05For B: y * 1.04For C: z * 1.03But in part 1, we donated all the money to A, so x = 2,000,000, y = z = 0. Therefore, after one year, the amount from A would be 2,000,000 * 1.05 = 2,100,000. The amounts from B and C would still be 0, since we didn't donate anything there.So, the new total amount available is 2,100,000. Now, we need to calculate how many additional refugees can be supported with this new total amount, using the same efficacy rates.Wait, is the question asking for the additional refugees beyond the original 10,000? Or is it asking for the total number of refugees after one year? Hmm, let me check the wording."Calculate the new total amount available after one year. How many additional refugees can be supported with the new total amount if the efficacy rates remain the same?"So, it's asking for the additional refugees, meaning beyond the original 10,000. So, we need to calculate how many more refugees can be helped with the new total amount, which is 2,100,000.But wait, actually, the new total amount is 2,100,000. But the original donation was 2,000,000, which helped 10,000 refugees. So, the additional amount is 100,000. But is that correct?Wait, no. Because the original 2,000,000 was invested and grew to 2,100,000. So, the total amount available after one year is 2,100,000. But the original 2,000,000 was already used to help 10,000 refugees. So, the additional amount is 100,000, which can be used to help more refugees.But wait, actually, no. Because the donations are invested, so the 2,000,000 was given to A, which then grew to 2,100,000. So, the total amount after one year is 2,100,000, which can be used to help refugees. But does that mean we can help more refugees with the same amount? Or is the 2,100,000 the new total to be donated?Wait, I think the wording is that the donations are invested, so after one year, the philanthropist revisits the donations. So, the total amount is now 2,100,000, which can be redistributed to the organizations. But the question is, how many additional refugees can be supported with the new total amount.Wait, maybe I need to think differently. The original 2,000,000 was used to help 10,000 refugees. After one year, the amount grows to 2,100,000. So, the additional amount is 100,000. So, with this 100,000, how many additional refugees can be helped?But the problem is, the donations are invested, so the 2,000,000 was given to A, which then grew to 2,100,000. So, now, the philanthropist has 2,100,000 to donate. So, the total amount available is 2,100,000, which can be used to help refugees. But the original 2,000,000 was already used to help 10,000 refugees. So, is the additional amount 100,000, which can be used to help more refugees?Wait, but the problem says "the new total amount available after one year." So, the total amount is 2,100,000. How many additional refugees can be supported with this new total amount, using the same efficacy rates.Wait, perhaps it's interpreted as, the total amount now is 2,100,000, which can be used to help refugees. So, how many refugees can be helped with 2,100,000, given the same efficacy rates. Then, subtract the original 10,000 to get the additional refugees.But that might not be correct because the original 2,000,000 was already used. So, perhaps the 2,100,000 is the new total, and the philanthropist can redistribute it to maximize the number of refugees, which would again be all to A, giving 2,100,000 / 1000 *5 = 10,500 refugees. So, the additional refugees would be 10,500 - 10,000 = 500.Alternatively, maybe the 2,100,000 is the total amount, and the additional refugees are calculated based on the growth. So, the growth is 100,000, which can be used to help 100,000 / 1000 *5 = 500 refugees. So, that would be the additional refugees.Yes, that makes sense. So, the additional amount is 100,000, which can be used to help 500 more refugees.But let me make sure. The initial donation was 2,000,000, which helped 10,000 refugees. After one year, the amount grows to 2,100,000. So, the philanthropist can now use this 2,100,000 to help refugees. But does that mean they can help 10,500 refugees in total, which is 500 more than before? Or is it that the additional 100,000 can be used to help 500 more?I think it's the latter. Because the original 2,000,000 was already used to help 10,000. The growth is 100,000, which can be used to help an additional 500 refugees. So, the answer would be 500 additional refugees.Alternatively, if we consider that the total amount now is 2,100,000, and the philanthropist can redistribute all of it, then the maximum number of refugees would be 2,100,000 / 1000 *5 = 10,500. So, the additional refugees would be 10,500 - 10,000 = 500.Either way, the additional refugees are 500.But let me think again. If the philanthropist is revisiting the donations, does that mean they can redistribute the entire 2,100,000, or just the growth? The problem says "the donations to each organization are invested in a fund that grows annually at respective rates... After one year, the philanthropist revisits the donations. Calculate the new total amount available after one year. How many additional refugees can be supported with the new total amount if the efficacy rates remain the same?"So, the new total amount is 2,100,000. So, the philanthropist can use this entire amount to support refugees. So, the total number of refugees would be 2,100,000 / 1000 *5 = 10,500. So, the additional refugees are 500.Yes, that seems correct.Alternatively, if the philanthropist had kept the original donations, and now has an additional 100,000, then the additional refugees would be 500. But in this case, the total amount is 2,100,000, so the total refugees are 10,500, which is 500 more than the original 10,000.So, the answer is 500 additional refugees.Wait, but let me make sure. If the philanthropist had donated all to A initially, and then after one year, the amount is 2,100,000. So, if they redistribute all 2,100,000 to A, they can help 10,500 refugees. So, the additional refugees are 500.Alternatively, if they don't redistribute, but just use the growth, then they have 100,000 more, which can help 500 more refugees.Either way, the additional refugees are 500.So, I think that's the answer.But let me think about another angle. Suppose the philanthropist doesn't have to keep the same distribution. So, after one year, they can redistribute the entire 2,100,000 to maximize the number of refugees. So, they would again donate all to A, getting 10,500 refugees. So, the additional refugees are 500.Alternatively, if they kept the same distribution, meaning x was 2,000,000, y and z were 0, then after one year, the amount is 2,100,000, but they might still donate all to A, so the additional refugees are 500.Either way, the answer is 500.So, to summarize:1. Donate all to A: x=2,000,000, y=0, z=0. Total refugees: 10,000.2. After one year, the amount grows to 2,100,000. Donate all to A again: total refugees 10,500. Additional refugees: 500.Therefore, the answers are:1. x=2,000,000, y=0, z=0.2. Additional refugees: 500.I think that's it.</think>

🧠:<think>Okay, so I have this problem where a Sensei wants to prove his martial arts style is superior by optimizing the energy efficiency and impact force of a strike. There are two parts to this problem. Let me tackle them one by one.Starting with the first part: finding the angle θ that maximizes the energy efficiency E(x). The function given is E(x) = 2 sin(x) + cos²(x) + ln(1 + x), and we need to find θ in [0, 2π] where E(x) is maximized.Alright, to find the maximum of a continuous function on a closed interval, I remember from calculus that we need to check critical points and endpoints. Critical points occur where the derivative is zero or undefined. Since E(x) is composed of sine, cosine, and logarithmic functions, which are all differentiable on [0, 2π], the only critical points will be where the derivative is zero.So, first step: find the derivative E’(x).Let me compute E’(x):E(x) = 2 sin(x) + cos²(x) + ln(1 + x)Differentiating term by term:- The derivative of 2 sin(x) is 2 cos(x).- The derivative of cos²(x) can be found using the chain rule: 2 cos(x) * (-sin(x)) = -2 sin(x) cos(x).- The derivative of ln(1 + x) is 1/(1 + x).So putting it all together:E’(x) = 2 cos(x) - 2 sin(x) cos(x) + 1/(1 + x)Simplify E’(x):I can factor out 2 cos(x) from the first two terms:E’(x) = 2 cos(x)(1 - sin(x)) + 1/(1 + x)Now, to find critical points, set E’(x) = 0:2 cos(x)(1 - sin(x)) + 1/(1 + x) = 0Hmm, this equation looks a bit complicated. It's a transcendental equation because it involves both trigonometric and algebraic terms. I might not be able to solve this analytically, so maybe I need to use numerical methods or graphing to approximate the solution.But before jumping into that, let me see if I can simplify or analyze the equation a bit more.First, let's denote:2 cos(x)(1 - sin(x)) + 1/(1 + x) = 0Let me rearrange it:2 cos(x)(1 - sin(x)) = -1/(1 + x)Since 1/(1 + x) is always positive for x > -1, which it is in our interval [0, 2π], the right-hand side is negative. So, 2 cos(x)(1 - sin(x)) must be negative.Therefore, cos(x)(1 - sin(x)) < 0Let me analyze the sign of cos(x) and (1 - sin(x)).First, (1 - sin(x)) is always non-negative because sin(x) ranges between -1 and 1, so 1 - sin(x) is between 0 and 2. It's zero when sin(x) = 1, which occurs at x = π/2 + 2πk.So, (1 - sin(x)) ≥ 0 for all x.Therefore, for cos(x)(1 - sin(x)) < 0, since (1 - sin(x)) is non-negative, cos(x) must be negative.So, cos(x) < 0.Which implies that x is in the interval (π/2, 3π/2) within [0, 2π].So, our critical points, if any, must lie in (π/2, 3π/2).Alright, so now we know that any critical points must be in the second and third quadrants.Now, let's consider the equation again:2 cos(x)(1 - sin(x)) = -1/(1 + x)Let me write it as:cos(x)(1 - sin(x)) = -1/(2(1 + x))Given that cos(x) is negative in (π/2, 3π/2), and (1 - sin(x)) is positive, their product is negative, which matches the right-hand side.So, we can write:cos(x)(1 - sin(x)) = -1/(2(1 + x))Let me denote y = x for simplicity.So, cos(y)(1 - sin(y)) = -1/(2(1 + y))This is still a transcendental equation, so I think the best approach is to use numerical methods to approximate the solution.Alternatively, maybe I can graph both sides and see where they intersect.But since I don't have graphing tools right now, perhaps I can estimate the solution by testing values in the interval (π/2, 3π/2).First, let's note that π is approximately 3.1416, so π/2 ≈ 1.5708, 3π/2 ≈ 4.7124.So, our interval is approximately (1.5708, 4.7124).Let me pick some test points in this interval and compute both sides.First, let's try x = π (≈3.1416):Left-hand side (LHS): cos(π)(1 - sin(π)) = (-1)(1 - 0) = -1Right-hand side (RHS): -1/(2(1 + π)) ≈ -1/(2*4.1416) ≈ -1/8.2832 ≈ -0.1207So, LHS = -1, RHS ≈ -0.1207. So, LHS < RHS.So, at x = π, LHS < RHS.Now, let's try x = 3π/4 (≈2.3562):LHS: cos(3π/4)(1 - sin(3π/4)) = (-√2/2)(1 - √2/2) ≈ (-0.7071)(1 - 0.7071) ≈ (-0.7071)(0.2929) ≈ -0.2071RHS: -1/(2(1 + 3π/4)) ≈ -1/(2*(1 + 2.3562)) ≈ -1/(2*3.3562) ≈ -1/6.7124 ≈ -0.1490So, LHS ≈ -0.2071, RHS ≈ -0.1490. So, LHS < RHS.So, still LHS < RHS.Now, let's try x = 5π/4 (≈3.9270):LHS: cos(5π/4)(1 - sin(5π/4)) = (-√2/2)(1 - (-√2/2)) = (-√2/2)(1 + √2/2) ≈ (-0.7071)(1 + 0.7071) ≈ (-0.7071)(1.7071) ≈ -1.2071RHS: -1/(2(1 + 5π/4)) ≈ -1/(2*(1 + 3.9270)) ≈ -1/(2*4.9270) ≈ -1/9.854 ≈ -0.1015So, LHS ≈ -1.2071, RHS ≈ -0.1015. So, LHS < RHS.Hmm, so at x = 5π/4, LHS is way less than RHS.Wait, maybe I need to check a point closer to 3π/2.Let me try x = 2π - π/6 = 11π/6 ≈ 5.7596, but that's beyond 3π/2, which is 4.7124. So, let's pick x = 4π/3 ≈4.1888.Wait, 4π/3 is approximately 4.1888, which is still less than 3π/2 ≈4.7124.So, x = 4π/3:LHS: cos(4π/3)(1 - sin(4π/3)) = (-1/2)(1 - (-√3/2)) = (-1/2)(1 + √3/2) ≈ (-0.5)(1 + 0.8660) ≈ (-0.5)(1.8660) ≈ -0.9330RHS: -1/(2(1 + 4π/3)) ≈ -1/(2*(1 + 4.1888)) ≈ -1/(2*5.1888) ≈ -1/10.3776 ≈ -0.0964So, LHS ≈ -0.9330, RHS ≈ -0.0964. Still LHS < RHS.Hmm, so LHS is more negative than RHS in all these points. Maybe I need to check if the function crosses somewhere else.Wait, let me think again. At x = π/2, which is the boundary, cos(π/2) = 0, so LHS = 0*(1 - sin(π/2)) = 0, RHS = -1/(2(1 + π/2)) ≈ -1/(2*2.5708) ≈ -0.1942. So, at x = π/2, LHS = 0, RHS ≈ -0.1942, so LHS > RHS.Similarly, at x approaching π/2 from the right, cos(x) approaches 0 from the negative side, so LHS approaches 0 from the negative side, while RHS is negative.Wait, maybe I need to check between π/2 and π.Wait, earlier I thought that cos(x) is negative in (π/2, 3π/2), which is correct, but let me check at x slightly greater than π/2, say x = 1.6 (which is just above π/2 ≈1.5708).Compute LHS and RHS:x = 1.6cos(1.6) ≈ cos(91.67 degrees) ≈ -0.0292sin(1.6) ≈ 0.9996So, LHS = (-0.0292)(1 - 0.9996) ≈ (-0.0292)(0.0004) ≈ -0.00001168RHS = -1/(2(1 + 1.6)) = -1/(2*2.6) ≈ -1/5.2 ≈ -0.1923So, LHS ≈ -0.00001168, RHS ≈ -0.1923. So, LHS > RHS.So, at x = 1.6, LHS > RHS.At x = π ≈3.1416, LHS = -1, RHS ≈ -0.1207, so LHS < RHS.Therefore, somewhere between x = 1.6 and x = π, the function crosses from LHS > RHS to LHS < RHS, meaning there must be a solution in (1.6, π).Similarly, let's check at x = 2:x = 2 radians ≈114.59 degreescos(2) ≈ -0.4161sin(2) ≈0.9093So, LHS = (-0.4161)(1 - 0.9093) ≈ (-0.4161)(0.0907) ≈ -0.0377RHS = -1/(2(1 + 2)) = -1/(2*3) = -1/6 ≈ -0.1667So, LHS ≈ -0.0377, RHS ≈ -0.1667. So, LHS > RHS.So, at x = 2, LHS > RHS.At x = 3:x = 3 radians ≈171.89 degreescos(3) ≈ -0.98999sin(3) ≈0.1411So, LHS = (-0.98999)(1 - 0.1411) ≈ (-0.98999)(0.8589) ≈ -0.850RHS = -1/(2(1 + 3)) = -1/(2*4) = -1/8 = -0.125So, LHS ≈ -0.850, RHS ≈ -0.125. So, LHS < RHS.So, between x = 2 and x = 3, the function crosses from LHS > RHS to LHS < RHS.Therefore, there is a solution between x = 2 and x = 3.Wait, but earlier, at x = π ≈3.1416, LHS = -1, RHS ≈ -0.1207, so LHS < RHS.Wait, but x = 3 is less than π, so actually, x = 3 is approximately 3 radians, which is about 171.89 degrees, still less than π (180 degrees). So, perhaps the crossing is between x = 2 and x = 3.Wait, but at x = 2, LHS ≈ -0.0377, RHS ≈ -0.1667, so LHS > RHS.At x = 3, LHS ≈ -0.850, RHS ≈ -0.125, so LHS < RHS.Therefore, the crossing is between x = 2 and x = 3.Let me try x = 2.5:x = 2.5 radians ≈143.24 degreescos(2.5) ≈ -0.8011sin(2.5) ≈0.5985So, LHS = (-0.8011)(1 - 0.5985) ≈ (-0.8011)(0.4015) ≈ -0.3217RHS = -1/(2(1 + 2.5)) = -1/(2*3.5) = -1/7 ≈ -0.1429So, LHS ≈ -0.3217, RHS ≈ -0.1429. So, LHS < RHS.So, at x = 2.5, LHS < RHS.So, the crossing is between x = 2 and x = 2.5.Let me try x = 2.25:x = 2.25 radians ≈128.9 degreescos(2.25) ≈ -0.6276sin(2.25) ≈0.7781So, LHS = (-0.6276)(1 - 0.7781) ≈ (-0.6276)(0.2219) ≈ -0.1392RHS = -1/(2(1 + 2.25)) = -1/(2*3.25) = -1/6.5 ≈ -0.1538So, LHS ≈ -0.1392, RHS ≈ -0.1538. So, LHS > RHS.So, at x = 2.25, LHS > RHS.At x = 2.5, LHS ≈ -0.3217 < RHS ≈ -0.1429.So, crossing between x = 2.25 and x = 2.5.Let me try x = 2.375:x = 2.375 radians ≈136.0 degreescos(2.375) ≈ -0.7481sin(2.375) ≈0.6636So, LHS = (-0.7481)(1 - 0.6636) ≈ (-0.7481)(0.3364) ≈ -0.2512RHS = -1/(2(1 + 2.375)) = -1/(2*3.375) = -1/6.75 ≈ -0.1481So, LHS ≈ -0.2512 < RHS ≈ -0.1481.So, LHS < RHS at x = 2.375.So, crossing between x = 2.25 and x = 2.375.Let me try x = 2.3125:x = 2.3125 radians ≈132.5 degreescos(2.3125) ≈ -0.7000sin(2.3125) ≈0.7141So, LHS = (-0.7000)(1 - 0.7141) ≈ (-0.7000)(0.2859) ≈ -0.1991RHS = -1/(2(1 + 2.3125)) = -1/(2*3.3125) ≈ -1/6.625 ≈ -0.1509So, LHS ≈ -0.1991 < RHS ≈ -0.1509.So, LHS < RHS.Wait, but at x = 2.25, LHS ≈ -0.1392 > RHS ≈ -0.1538.So, crossing between x = 2.25 and x = 2.3125.Let me try x = 2.28125:x ≈2.28125 radians ≈130.7 degreescos(2.28125) ≈ -0.6736sin(2.28125) ≈0.7387So, LHS = (-0.6736)(1 - 0.7387) ≈ (-0.6736)(0.2613) ≈ -0.1763RHS = -1/(2(1 + 2.28125)) = -1/(2*3.28125) ≈ -1/6.5625 ≈ -0.1524So, LHS ≈ -0.1763 < RHS ≈ -0.1524.Still LHS < RHS.Wait, but at x = 2.25, LHS ≈ -0.1392 > RHS ≈ -0.1538.So, crossing between x = 2.25 and x = 2.28125.Let me try x = 2.265625:x ≈2.265625 radians ≈129.8 degreescos(2.265625) ≈ -0.6543sin(2.265625) ≈0.7568So, LHS = (-0.6543)(1 - 0.7568) ≈ (-0.6543)(0.2432) ≈ -0.1593RHS = -1/(2(1 + 2.265625)) = -1/(2*3.265625) ≈ -1/6.53125 ≈ -0.1531So, LHS ≈ -0.1593 < RHS ≈ -0.1531.Still LHS < RHS.Wait, but at x = 2.25, LHS ≈ -0.1392 > RHS ≈ -0.1538.So, crossing is between x = 2.25 and x = 2.265625.Let me try x = 2.2578125:x ≈2.2578125 radians ≈129.3 degreescos(2.2578125) ≈ -0.6435sin(2.2578125) ≈0.7650So, LHS = (-0.6435)(1 - 0.7650) ≈ (-0.6435)(0.235) ≈ -0.1512RHS = -1/(2(1 + 2.2578125)) = -1/(2*3.2578125) ≈ -1/6.515625 ≈ -0.1535So, LHS ≈ -0.1512 > RHS ≈ -0.1535.So, LHS > RHS at x ≈2.2578.So, between x = 2.2578 and x = 2.2656, the function crosses from LHS > RHS to LHS < RHS.So, let's try x = 2.26171875 (midpoint between 2.2578 and 2.2656):x ≈2.26171875 radians ≈129.6 degreescos(x) ≈ -0.6488sin(x) ≈0.7616So, LHS = (-0.6488)(1 - 0.7616) ≈ (-0.6488)(0.2384) ≈ -0.1548RHS = -1/(2(1 + 2.26171875)) = -1/(2*3.26171875) ≈ -1/6.5234375 ≈ -0.1533So, LHS ≈ -0.1548 < RHS ≈ -0.1533.So, LHS < RHS.Therefore, crossing is between x = 2.2578 and x = 2.2617.Let me compute at x = 2.259765625 (midpoint):x ≈2.259765625 radians ≈129.5 degreescos(x) ≈ -0.6462sin(x) ≈0.7634LHS = (-0.6462)(1 - 0.7634) ≈ (-0.6462)(0.2366) ≈ -0.1527RHS = -1/(2(1 + 2.259765625)) = -1/(2*3.259765625) ≈ -1/6.51953125 ≈ -0.1534So, LHS ≈ -0.1527 > RHS ≈ -0.1534.So, LHS > RHS.So, crossing is between x = 2.259765625 and x = 2.26171875.Let me take the midpoint: x ≈2.2607421875x ≈2.2607421875 radians ≈129.5 degreescos(x) ≈ -0.6475sin(x) ≈0.7625LHS = (-0.6475)(1 - 0.7625) ≈ (-0.6475)(0.2375) ≈ -0.1536RHS = -1/(2(1 + 2.2607421875)) = -1/(2*3.2607421875) ≈ -1/6.521484375 ≈ -0.1534So, LHS ≈ -0.1536 < RHS ≈ -0.1534.So, LHS < RHS.Therefore, the root is between x ≈2.259765625 and x ≈2.2607421875.Let me compute at x = 2.26025390625 (midpoint):x ≈2.26025390625 radians ≈129.5 degreescos(x) ≈ -0.6470sin(x) ≈0.7629LHS = (-0.6470)(1 - 0.7629) ≈ (-0.6470)(0.2371) ≈ -0.1533RHS = -1/(2(1 + 2.26025390625)) = -1/(2*3.26025390625) ≈ -1/6.5205078125 ≈ -0.1534So, LHS ≈ -0.1533 > RHS ≈ -0.1534.So, LHS > RHS.Therefore, the root is between x ≈2.26025390625 and x ≈2.2607421875.This is getting quite precise, but maybe I can stop here and approximate the root as x ≈2.2605 radians.So, approximately 2.2605 radians.To check, let me compute E’(2.2605):cos(2.2605) ≈ -0.6472sin(2.2605) ≈0.7626So, LHS = (-0.6472)(1 - 0.7626) ≈ (-0.6472)(0.2374) ≈ -0.1535RHS = -1/(2(1 + 2.2605)) ≈ -1/(2*3.2605) ≈ -1/6.521 ≈ -0.1534So, LHS ≈ -0.1535 ≈ RHS ≈ -0.1534. Close enough.So, the critical point is approximately x ≈2.2605 radians.Now, we need to check if this is a maximum.Since E(x) is continuous on [0, 2π], and we have only one critical point in (π/2, 3π/2), which is approximately 2.2605, we can check the second derivative or use test points to see if this is a maximum.Alternatively, since the function E(x) is smooth, and we found a critical point, we can compute E''(x) to check concavity.But maybe it's easier to compute E(x) at this critical point and at the endpoints to see which is the maximum.So, let's compute E(x) at x ≈2.2605, x =0, x =2π.First, E(0):E(0) = 2 sin(0) + cos²(0) + ln(1 + 0) = 0 + 1 + 0 =1.E(2π):E(2π) = 2 sin(2π) + cos²(2π) + ln(1 + 2π) = 0 + 1 + ln(1 + 6.2832) ≈0 +1 + ln(7.2832) ≈1 + 1.986 ≈2.986.Now, E(2.2605):Compute each term:sin(2.2605) ≈0.7626cos(2.2605) ≈-0.6472So,2 sin(x) ≈2*0.7626 ≈1.5252cos²(x) ≈(-0.6472)^2 ≈0.4189ln(1 + x) ≈ln(1 + 2.2605) ≈ln(3.2605) ≈1.182So, E(x) ≈1.5252 + 0.4189 +1.182 ≈3.1261.So, E(2.2605) ≈3.1261.Compare with E(2π) ≈2.986.So, E(2.2605) is higher than E(2π). Also, E(0) is 1, which is much lower.Therefore, the maximum occurs at x ≈2.2605 radians.But let me check if there are any other critical points.Wait, earlier we saw that in (π/2, 3π/2), the only critical point is around 2.2605. Let me check near x = 3π/2 ≈4.7124.Compute E’(4.7124):cos(4.7124) ≈0 (since cos(3π/2)=0)sin(4.7124) ≈-1So, LHS = 0*(1 - (-1)) =0RHS = -1/(2(1 + 4.7124)) ≈-1/(2*5.7124) ≈-0.0875So, E’(4.7124) =0 + (-0.0875) ≈-0.0875 <0.So, the derivative is negative at x =3π/2.But since cos(x) is zero there, the LHS is zero, so RHS is negative.So, the derivative is negative there.So, no critical point at x =3π/2.Therefore, the only critical point in (π/2, 3π/2) is at x ≈2.2605.Hence, the maximum occurs at x ≈2.2605 radians.But let me see if there are any other critical points outside (π/2, 3π/2). Wait, earlier we saw that cos(x)(1 - sin(x)) must be negative, which only occurs in (π/2, 3π/2). So, no other critical points.Therefore, the maximum occurs at x ≈2.2605 radians.But let me convert this to exact terms if possible.Wait, 2.2605 radians is approximately 129.5 degrees.But is there an exact value? Maybe not, since it's a transcendental equation.Therefore, the angle θ that maximizes E(x) is approximately 2.2605 radians.But maybe we can express it in terms of inverse functions or something, but I think it's fine to leave it as an approximate value.Alternatively, perhaps we can write it as θ = arccos(...) but it's complicated.Alternatively, maybe we can write it as the solution to the equation 2 cos(x)(1 - sin(x)) + 1/(1 + x) =0, but I think the problem expects a numerical value.So, θ ≈2.2605 radians.But let me check if I can get a more precise value.Alternatively, maybe I can use Newton-Raphson method for better approximation.Let me try that.Given f(x) = 2 cos(x)(1 - sin(x)) + 1/(1 + x)We need to solve f(x) =0.We have an approximate root at x ≈2.2605.Let me compute f(2.2605):cos(2.2605) ≈-0.6472sin(2.2605) ≈0.7626So,2 cos(x)(1 - sin(x)) ≈2*(-0.6472)*(1 -0.7626) ≈2*(-0.6472)*(0.2374) ≈2*(-0.1535) ≈-0.3071/(1 + x) ≈1/(3.2605) ≈0.3067So, f(x) ≈-0.307 +0.3067 ≈-0.0003So, f(2.2605) ≈-0.0003.Almost zero.Compute f'(x):f(x) =2 cos(x)(1 - sin(x)) + 1/(1 + x)f’(x) = derivative of first term + derivative of second term.First term: 2 cos(x)(1 - sin(x))Derivative: 2[-sin(x)(1 - sin(x)) + cos(x)(-cos(x))] = 2[-sin(x) + sin²(x) - cos²(x)]Second term: 1/(1 + x)Derivative: -1/(1 + x)^2So, f’(x) = 2[-sin(x) + sin²(x) - cos²(x)] -1/(1 + x)^2At x ≈2.2605:sin(x) ≈0.7626cos(x) ≈-0.6472So,First part: 2[-0.7626 + (0.7626)^2 - (-0.6472)^2]Compute:-0.7626 + (0.5816) - (0.4189) ≈-0.7626 +0.5816 -0.4189 ≈-0.7626 +0.1627 ≈-0.5999Multiply by 2: ≈-1.1998Second part: -1/(1 +2.2605)^2 ≈-1/(3.2605)^2 ≈-1/10.629 ≈-0.0941So, f’(x) ≈-1.1998 -0.0941 ≈-1.2939So, Newton-Raphson update:x_new = x - f(x)/f’(x) ≈2.2605 - (-0.0003)/(-1.2939) ≈2.2605 - (0.0003/1.2939) ≈2.2605 -0.000232 ≈2.260268So, x ≈2.260268Compute f(2.260268):cos(2.260268) ≈-0.6472sin(2.260268) ≈0.7626So,2 cos(x)(1 - sin(x)) ≈2*(-0.6472)*(1 -0.7626) ≈2*(-0.6472)*(0.2374) ≈-0.3071/(1 + x) ≈1/(3.260268) ≈0.3067So, f(x) ≈-0.307 +0.3067 ≈-0.0003Wait, same as before. Maybe my approximation is not precise enough.Alternatively, perhaps I need to use more precise values for sin and cos.Alternatively, maybe I can accept that x ≈2.2605 is accurate enough.So, θ ≈2.2605 radians.But to express it more precisely, maybe I can write it as θ ≈2.26 radians.Alternatively, perhaps the exact value is π - something, but I don't think so.So, for the first part, the angle θ that maximizes E(x) is approximately 2.26 radians.Now, moving on to the second part:The impact force F(t) is given by F(t) = a t^b e^{-c t}, where a, b, c are positive constants. Given that the maximum occurs at t = 2/c, we need to express b in terms of c.Alright, to find the maximum of F(t), we take the derivative with respect to t, set it equal to zero, and solve for t. Then, set that t equal to 2/c and solve for b.So, let's compute F’(t):F(t) = a t^b e^{-c t}Using product rule:F’(t) = a [d/dt (t^b) * e^{-c t} + t^b * d/dt (e^{-c t})]= a [b t^{b -1} e^{-c t} + t^b (-c) e^{-c t}]Factor out a t^{b -1} e^{-c t}:= a t^{b -1} e^{-c t} [b - c t]Set F’(t) =0:a t^{b -1} e^{-c t} [b - c t] =0Since a, t^{b -1}, and e^{-c t} are all positive for t >0, the only solution is when [b - c t] =0.So, b - c t =0 => t = b/c.But the problem states that the maximum occurs at t =2/c.Therefore, t = b/c =2/c => b =2.Wait, that seems straightforward.Wait, let me double-check.Given F(t) = a t^b e^{-c t}F’(t) = a [b t^{b -1} e^{-c t} - c t^b e^{-c t}] = a t^{b -1} e^{-c t} (b - c t)Set F’(t)=0:b - c t =0 => t = b/c.Given that maximum occurs at t =2/c, so b/c =2/c => b=2.Yes, that's correct.So, b=2.Therefore, the constant b is equal to 2, regardless of c.So, b=2.So, summarizing:1. The angle θ that maximizes E(x) is approximately 2.26 radians.2. The constant b is equal to 2.Final Answer1. The angle that maximizes energy efficiency is (boxed{theta approx 2.26}) radians.2. The constant (b) is expressed as (boxed{2}).</think>

🧠:<think>Okay, so I need to figure out the break-even point for this new robotic system. Let me start by understanding the problem.The factory currently produces 1,000 units per day with a 5% defect rate. That means out of 1,000 units, 5% are defective. Let me calculate how many defective units that is. 5% of 1,000 is 0.05 * 1,000 = 50 units. So, currently, they have 50 defective units each day.Each defective unit costs them 50 in rework and lost profit. So, the daily cost from defects is 50 units * 50 = 2,500 per day.Now, the new system promises to reduce the defect rate to 1%. Let me compute the number of defective units with the new system. 1% of 1,000 is 0.01 * 1,000 = 10 units. So, with the new system, they would have 10 defective units each day.The daily cost from defects with the new system would be 10 units * 50 = 500 per day.So, the savings per day from the reduced defects would be the current daily defect cost minus the new daily defect cost. That's 2,500 - 500 = 2,000 per day.But wait, the new system also has an operational cost of 2,000 per day. Hmm, so the savings from defects is exactly equal to the operational cost. That means, in terms of daily cash flow, the savings offset the operational cost. So, the net savings per day would be 2,000 - 2,000 = 0. So, the company isn't saving any money each day after considering the operational cost.But the installation cost is 500,000. So, the company needs to recoup this 500,000 investment. Since the net daily savings are 0, how will they ever break even? That doesn't make sense. Maybe I made a mistake.Wait, let me double-check. Current defect cost: 50 units * 50 = 2,500. New defect cost: 10 units * 50 = 500. So, savings: 2,500 - 500 = 2,000. But the new system costs 2,000 per day to operate. So, the 2,000 savings are exactly offset by the 2,000 operational cost. Therefore, the net cash flow is zero each day.But the installation cost is a one-time expense of 500,000. So, if the net daily cash flow is zero, the company will never recover the installation cost. That can't be right because the problem is asking for a break-even point. Maybe I'm misunderstanding something.Wait, perhaps the 2,000 per day is an additional operational cost on top of the savings. So, the total cost of the new system is the installation plus the operational cost. But the savings are only from the reduced defects.So, the net daily benefit is savings minus operational cost: 2,000 - 2,000 = 0. So, again, same result.But that would mean the break-even point is never, which doesn't make sense. Maybe I need to consider that the installation cost is a fixed cost, and the operational cost is variable. So, the total cost of the new system is 500,000 + (2,000 * number of days). The savings are 2,000 * number of days. So, to break even, the total savings should equal total costs.Wait, let me set up an equation.Let x be the number of days.Total cost = 500,000 + 2,000xTotal savings = 2,000xBreak-even occurs when total savings = total cost:2,000x = 500,000 + 2,000xSubtract 2,000x from both sides:0 = 500,000That can't be. So, this suggests that the break-even point is never achieved because the savings exactly offset the operational cost, but the installation cost is a sunk cost that isn't being offset. Therefore, the company will never recover the 500,000 investment through daily savings.But the problem is asking for the break-even point, so maybe I'm missing something else. Perhaps the 500,000 is a one-time cost, and the operational cost is separate. So, the daily savings are 2,000, but the daily operational cost is 2,000, so net zero. Therefore, the installation cost can't be recovered through daily operations because the net is zero. So, the break-even point is infinity, which doesn't make sense.Wait, maybe the operational cost is in addition to the savings. So, the net cash flow per day is savings minus operational cost: 2,000 - 2,000 = 0. Therefore, the company isn't making any money back each day, so the installation cost will never be recovered. Therefore, the break-even point is never.But the problem says to consider only the savings from reduced defects. So, maybe I shouldn't subtract the operational cost when calculating the break-even point. Let me read the question again."Determine the break-even point, in days, for the investment in the new robotic system, considering only the savings from reduced defects."Oh, okay, so maybe I should only consider the savings from defects and not subtract the operational cost. That is, the savings per day are 2,000, and the installation cost is 500,000. So, the break-even point is when the total savings equal the installation cost.So, total savings = 2,000 * x = 500,000Therefore, x = 500,000 / 2,000 = 250 days.But wait, the problem mentions the operational cost. It says, "considering only the savings from reduced defects." So, maybe the operational cost isn't considered in the savings. So, the break-even is based solely on the defect savings, ignoring the operational cost.Therefore, the break-even point is 250 days.But that seems contradictory because the operational cost is a cost of the system. So, if we're only considering savings from defects, we don't subtract the operational cost when calculating break-even. So, the break-even is when the savings cover the installation cost.So, 250 days.Okay, that makes sense. So, the answer to part 1 is 250 days.Now, moving on to part 2.The company's profit margin per non-defective unit is 20% after the cost of production, which is 100 per unit. So, the cost of production is 100 per unit, and the profit margin is 20% on that.So, the selling price per unit would be cost plus profit. 20% of 100 is 20, so selling price is 120 per unit.But wait, the profit margin is 20% after the cost of production. So, profit margin is 20% of the selling price, or 20% of the cost?Wait, profit margin can be expressed in different ways. If it's a 20% profit margin on cost, that would mean selling price is cost plus 20% of cost, which is 120. If it's a 20% profit margin on selling price, that would mean selling price is cost divided by (1 - 0.20) = 125.But the problem says "profit margin per non-defective unit is 20% after the cost of production." Hmm, that wording is a bit ambiguous. But in business terms, profit margin is usually expressed as a percentage of sales. So, if it's 20% profit margin on sales, then selling price is 100 / (1 - 0.20) = 125.But let me think again. The problem says "profit margin per non-defective unit is 20% after the cost of production, which is 100 per unit." So, maybe it's 20% of the cost, meaning 20 profit per unit. So, selling price is 120.But let's verify. If the profit margin is 20% after cost, that could mean that the profit is 20% of the cost. So, profit = 0.20 * 100 = 20. Therefore, selling price is 100 + 20 = 120.Alternatively, if it's a 20% profit margin on sales, then profit = 0.20 * selling price, so selling price = profit / 0.20. But since profit is selling price - cost, we have:profit = selling price - 100 = 0.20 * selling priceTherefore, selling price - 100 = 0.20 * selling priceSo, selling price - 0.20 * selling price = 1000.80 * selling price = 100selling price = 100 / 0.80 = 125So, profit is 25 per unit.But the problem says "profit margin per non-defective unit is 20% after the cost of production." Hmm, the wording is a bit unclear. It could be interpreted as 20% of the selling price is profit, which would make selling price 125, or 20% of the cost is profit, making selling price 120.But in business, profit margin is typically expressed as a percentage of sales. So, if it's a 20% profit margin, that usually means 20% of sales is profit.Therefore, selling price is 125, cost is 100, profit is 25.But let me check the exact wording: "profit margin per non-defective unit is 20% after the cost of production, which is 100 per unit."So, "after the cost of production" might mean that the profit margin is calculated after subtracting the cost of production. So, profit margin = (selling price - cost) / selling price = 20%.Therefore, (selling price - 100) / selling price = 0.20So, same as before, selling price = 125, profit per unit is 25.Alternatively, if it's profit margin on cost, then profit is 20% of 100, which is 20, selling price is 120.But since the problem says "profit margin per non-defective unit is 20% after the cost of production," it's more likely referring to profit margin on sales, which is 20%. So, selling price is 125, profit is 25.But to be safe, let's consider both scenarios.First, assuming profit margin is 20% on sales:Profit per unit = 0.20 * selling priceBut selling price = cost + profit = 100 + 0.20 * selling priceSo, selling price - 0.20 * selling price = 1000.80 * selling price = 100selling price = 125Profit per unit = 25Alternatively, if profit margin is 20% on cost:Profit per unit = 0.20 * 100 = 20Selling price = 100 + 20 = 120But the problem says "profit margin per non-defective unit is 20% after the cost of production." The phrase "after the cost of production" might mean that the profit is calculated after the cost, so it's profit margin on cost. So, profit margin = profit / cost = 20%.Therefore, profit = 0.20 * 100 = 20Selling price = 100 + 20 = 120But I'm not entirely sure. Let me think about the wording again."Profit margin per non-defective unit is 20% after the cost of production, which is 100 per unit."So, "after the cost of production" might mean that the profit margin is calculated after subtracting the cost of production, which would imply that profit margin is profit / (selling price - cost). But that's not standard.Alternatively, it could mean that the profit margin is 20% of the selling price, which is after the cost. Hmm.Alternatively, perhaps the profit margin is 20% of the selling price, which is after the cost. So, profit = 0.20 * selling price, and selling price = cost + profit = 100 + 0.20 * selling price, leading to selling price = 125.I think that's the more standard interpretation.So, assuming selling price is 125, profit is 25 per unit.Now, the company currently has 1,000 units per day, with 50 defective. So, non-defective units are 950.With the new system, defect rate is 1%, so defective units are 10, non-defective are 990.So, the increase in non-defective units is 990 - 950 = 40 units per day.Each non-defective unit gives a profit of 25, so increased profit per day is 40 * 25 = 1,000.But wait, let me verify.Current non-defective units: 950New non-defective units: 990Difference: 40Profit per non-defective unit: 25So, increased profit: 40 * 25 = 1,000 per day.Alternatively, if profit per unit was 20, then increased profit would be 40 * 20 = 800.But since I think the profit margin is 20% on sales, leading to 25 profit per unit, the increased profit is 1,000 per day.But let me double-check.If selling price is 125, cost is 100, profit is 25.Current non-defective: 950 units, so profit: 950 * 25 = 23,750New non-defective: 990 units, profit: 990 * 25 = 24,750Difference: 24,750 - 23,750 = 1,000Yes, so increased profit is 1,000 per day.Alternatively, if profit was 20 per unit:Current profit: 950 * 20 = 19,000New profit: 990 * 20 = 19,800Difference: 800But since the problem says "profit margin per non-defective unit is 20% after the cost of production," I think the correct interpretation is that the profit margin is 20% of the selling price, so profit per unit is 25, leading to increased profit of 1,000 per day.Therefore, the answer to part 2 is 1,000 per day.But wait, let me make sure I didn't miss anything else. The problem says "calculate the increased profit per day as a result of implementing the new system." So, it's the difference in profit between the new system and the old system.Yes, that's what I calculated: 40 more non-defective units, each giving 25 profit, so 1,000 more per day.So, final answers:1. Break-even point is 250 days.2. Increased profit per day is 1,000.Final Answer1. The break-even point is boxed{250} days.2. The increased profit per day is boxed{1000} dollars.</think>

🧠:<think>Alright, so I have this problem about a professor analyzing socioeconomic factors' effects on educational outcomes. The dataset has 1,000 students with variables like family income (X1), parental education (X2), student attendance (X3), and a composite score Y for educational outcomes. The professor thinks these predictors have both linear and interaction effects on Y. The first question asks me to formulate a multiple regression model that includes main effects of X1, X2, X3 and their two-way interaction effects. I need to write the regression equation and specify the number of parameters to estimate.Okay, so in multiple regression, the general form is Y = β0 + β1X1 + β2X2 + β3X3 + ... + ε, where ε is the error term. But since we need to include two-way interactions, I have to add terms for each pair of variables multiplied together.So, the main effects are X1, X2, X3. The two-way interactions would be X1*X2, X1*X3, and X2*X3. Each interaction term will have its own coefficient. Therefore, the regression equation should look like:Y = β0 + β1X1 + β2X2 + β3X3 + β4X1X2 + β5X1X3 + β6X2X3 + εNow, counting the parameters: β0 is the intercept, then β1, β2, β3 for the main effects, and β4, β5, β6 for the interactions. So that's 1 + 3 + 3 = 7 parameters in total.Wait, but the question mentions "the number of parameters that need to be estimated." So, yeah, it's 7.Moving on to the second question. The professor wants to test if the interaction between family income (X1) and parental education (X2) has a significant effect on Y. Using an F-test, I need to derive the null hypothesis, the test statistic, and the degrees of freedom.Hmm, so the null hypothesis is that the coefficient for the interaction term X1X2 is zero. But wait, in the context of an F-test for a subset of coefficients, it's about whether a set of coefficients are jointly zero. Since it's just one interaction term, maybe it's a t-test? But the question specifies an F-test, so perhaps it's considering the interaction as part of a larger model.Wait, the professor's model includes all possible two-way interactions, so the total number of parameters is k. Let me think. The full model includes all main effects and all two-way interactions. So, the number of parameters is 1 (intercept) + 3 (main) + 3 (interactions) = 7, as before.But when testing the significance of a particular interaction, say X1X2, we can set up a nested model where we exclude that interaction. So, the restricted model would have one fewer parameter. Therefore, the null hypothesis is that the coefficient for X1X2 is zero, i.e., H0: β4 = 0.But in an F-test, we compare the full model (with X1X2) and the restricted model (without X1X2). The F-test statistic is calculated as [ (SSE_restricted - SSE_full) / (df_restricted - df_full) ] / [ SSE_full / df_full ]Where SSE is the sum of squared errors, and df is the degrees of freedom.In this case, the full model has k parameters, which is 7. The restricted model would have k - 1 = 6 parameters because we're removing one interaction term.The degrees of freedom for the numerator is the difference in the number of parameters, which is 1. The denominator degrees of freedom is the degrees of freedom for the full model, which is n - k. Since n is 1000, df_full = 1000 - 7 = 993.So, the F-test statistic would be [ (SSE_restricted - SSE_full) / 1 ] / [ SSE_full / 993 ]Therefore, the null hypothesis is H0: β4 = 0, the test statistic is F = [ (SSE_restricted - SSE_full) / 1 ] / [ SSE_full / 993 ], and the degrees of freedom are 1 and 993.Wait, but sometimes in F-tests for multiple coefficients, the numerator df is the number of restrictions, which here is 1. So, that makes sense.Let me double-check. The full model includes all interactions, so when testing a single interaction, we're comparing it to a model without that interaction. So, yes, the numerator df is 1, and the denominator is the error degrees of freedom from the full model, which is n - k = 1000 - 7 = 993.So, I think that's correct.</think>