🧠:<think>Alright, so I've got these two problems to solve. Let me start with the first one about the baker mixing flour from two suppliers. Hmm, okay, the baker wants to create a 100 kg mixture with exactly 13% protein. Supplier A has 12.5% protein, and Supplier B has 14%. I need to figure out how much flour to take from each supplier.I remember that these kinds of problems are usually solved using systems of equations. Let me define variables first. Let's say x is the amount of flour from Supplier A, and y is the amount from Supplier B. Since the total mixture is 100 kg, I can write the equation:x + y = 100That's straightforward. Now, for the protein content. The total protein from Supplier A would be 12.5% of x, which is 0.125x. Similarly, the protein from Supplier B is 14% of y, which is 0.14y. The total protein in the mixture should be 13% of 100 kg, which is 13 kg. So, the second equation is:0.125x + 0.14y = 13Now, I have two equations:1. x + y = 1002. 0.125x + 0.14y = 13I can solve this system using substitution or elimination. Maybe substitution is easier here. From the first equation, I can express y in terms of x:y = 100 - xNow, substitute this into the second equation:0.125x + 0.14(100 - x) = 13Let me compute that step by step. First, expand the equation:0.125x + 0.14*100 - 0.14x = 13Calculate 0.14*100, which is 14:0.125x + 14 - 0.14x = 13Combine like terms. 0.125x - 0.14x is -0.015x:-0.015x + 14 = 13Now, subtract 14 from both sides:-0.015x = -1Divide both sides by -0.015:x = (-1)/(-0.015) = 1/0.015Calculating that, 1 divided by 0.015. Hmm, 0.015 is 1.5%, so 1 divided by 0.015 is the same as 1000 divided by 15, which is approximately 66.666... So, x is approximately 66.666 kg.Since x is the amount from Supplier A, and y is 100 - x, y would be 100 - 66.666 = 33.333 kg.Let me double-check the calculations to make sure I didn't make a mistake. So, 66.666 kg from A at 12.5% protein would contribute 66.666 * 0.125 = 8.333 kg of protein. From B, 33.333 kg at 14% protein would contribute 33.333 * 0.14 = 4.666 kg of protein. Adding those together, 8.333 + 4.666 = 12.999 kg, which is approximately 13 kg. That checks out.So, the baker should use about 66.67 kg from Supplier A and 33.33 kg from Supplier B.Moving on to the second problem. The baker is printing a limited edition of 1000 books, each with a unique recipe variation. The cost of printing varies inversely with the number of books printed. So, if more books are printed, the cost per book decreases. The cost of printing 500 books is 15 per book. Now, we need to find the cost per book if 1000 books are printed.Inverse variation means that the cost per book (C) is inversely proportional to the number of books (N). So, mathematically, that can be written as:C = k / NWhere k is the constant of proportionality.Given that when N = 500, C = 15. So, we can find k:15 = k / 500Multiply both sides by 500:k = 15 * 500 = 7500So, the constant k is 7500. Now, we can find the cost per book when N = 1000:C = 7500 / 1000 = 7.5So, the cost per book would be 7.50.Let me verify that. If printing 500 books costs 15 each, the total cost is 500 * 15 = 7500. If we print 1000 books, the total cost should still be 7500 (since the cost varies inversely, meaning the total cost remains the same, right? Wait, hold on. Is the total cost the same, or is the cost per book inversely proportional?Wait, the problem says the cost of printing a book varies inversely with the number of books printed. So, it's the cost per book that varies inversely, not the total cost. So, if more books are printed, each book becomes cheaper.So, in that case, the total cost would actually be C * N, which would be k, a constant. So, when N = 500, C = 15, so total cost is 500 * 15 = 7500. If N = 1000, then total cost is still 7500, so C = 7500 / 1000 = 7.5.Yes, that makes sense. So, the cost per book decreases as the number of books increases, keeping the total cost the same.So, the cost per book for 1000 copies is 7.50.Final Answer1. The baker should use boxed{66.67} kg from Supplier A and boxed{33.33} kg from Supplier B.2. The cost per book if 1000 books are printed is boxed{7.50} dollars.</think>

🧠:<think>Alright, so I've got this problem about a sociologist analyzing a community with 100 households. The resources are collectively owned and allocated based on a voting system. Each household has a different voting power, which depends on their contribution to the collective resources. The first part of the problem asks me to model the voting power ( v_i ) of the ( i )-th household as ( v_i = alpha cdot c_i^{beta} ). They give me that the total contribution ( sum_{i=1}^{100} c_i = 500 ) and the total voting power ( sum_{i=1}^{100} v_i = 2000 ). I need to determine the constants ( alpha ) and ( beta ).Hmm, okay. So I have two equations here involving ( alpha ) and ( beta ). The first equation is the sum of all contributions, which is 500, and the second is the sum of all voting powers, which is 2000. Let me write down what I know:1. ( sum_{i=1}^{100} c_i = 500 )2. ( sum_{i=1}^{100} v_i = 2000 )3. ( v_i = alpha cdot c_i^{beta} )So, substituting the third equation into the second, I get:( sum_{i=1}^{100} alpha cdot c_i^{beta} = 2000 )Which simplifies to:( alpha cdot sum_{i=1}^{100} c_i^{beta} = 2000 )So, if I can express ( sum_{i=1}^{100} c_i^{beta} ) in terms of the total contribution, which is 500, I can solve for ( alpha ) and ( beta ). But wait, I only have two equations and two unknowns, so maybe I can set up a system of equations.But here's the thing: without knowing the individual ( c_i ) values, it's tricky because ( sum c_i^{beta} ) isn't directly related to ( sum c_i ) unless we have more information about the distribution of ( c_i ). Wait, the problem says each household has a different amount of voting power based on their contribution. It doesn't specify whether the contributions are equally distributed or follow a particular distribution. Hmm, maybe I need to make an assumption here. If all ( c_i ) are the same, then ( c_i = 5 ) for each household since 100 households contribute 500 total. But the problem says each household has a different contribution, so they can't all be 5. So, perhaps the contributions are spread out in some way. But without knowing the exact distribution, how can I compute ( sum c_i^{beta} )?Wait, maybe the problem is designed such that the sum ( sum c_i^{beta} ) can be expressed in terms of ( sum c_i ) if we assume a specific value for ( beta ). For example, if ( beta = 1 ), then ( sum c_i^{beta} = sum c_i = 500 ), which would make ( alpha = 2000 / 500 = 4 ). But is ( beta = 1 ) the only possibility?Alternatively, if ( beta = 0 ), then ( c_i^0 = 1 ) for all ( i ), so ( sum c_i^0 = 100 ), which would make ( alpha = 2000 / 100 = 20 ). But ( beta = 0 ) might not make much sense in the context of voting power based on contribution.Alternatively, if ( beta = 2 ), then ( sum c_i^2 ) would be something else. But without knowing the individual ( c_i ), I can't compute ( sum c_i^2 ). So, perhaps the problem expects me to assume that all ( c_i ) are equal? But the problem states that each household has a different contribution, so that can't be.Wait, maybe the problem is designed such that we can solve for ( alpha ) and ( beta ) without knowing the exact distribution. Let me think.Suppose we take the total voting power equation:( alpha cdot sum c_i^{beta} = 2000 )And we know that ( sum c_i = 500 ). If we can relate ( sum c_i^{beta} ) to ( sum c_i ), perhaps through some exponent relation.Wait, if ( beta = 1 ), then ( sum c_i^{beta} = 500 ), so ( alpha = 4 ). But is there a way to verify if ( beta ) is 1 or another value?Alternatively, maybe the problem is designed so that regardless of the distribution, we can solve for ( alpha ) and ( beta ). But that seems unlikely because ( sum c_i^{beta} ) depends on the distribution.Wait, perhaps the problem is assuming that all ( c_i ) are equal, even though it says different. Maybe it's a translation issue or a misstatement. If I assume all ( c_i = 5 ), then ( v_i = alpha cdot 5^{beta} ), and total voting power is ( 100 cdot alpha cdot 5^{beta} = 2000 ). So, ( alpha cdot 5^{beta} = 20 ).But we also have ( sum c_i = 500 ), which is 100*5=500, so that's consistent. So, if I assume all ( c_i = 5 ), then ( alpha cdot 5^{beta} = 20 ). But we have two variables, ( alpha ) and ( beta ), so we need another equation. But we only have one equation here. Hmm.Wait, maybe the problem is expecting me to recognize that if all ( c_i ) are equal, then ( beta ) can be any value, but ( alpha ) would adjust accordingly. But without another condition, I can't determine both ( alpha ) and ( beta ). So perhaps the problem is missing some information or I'm misinterpreting it.Wait, let me read the problem again:"The sociologist models the voting power ( v_i ) of the ( i )-th household as ( v_i = alpha cdot c_i^{beta} ), where ( c_i ) represents the contribution of the ( i )-th household, and ( alpha ) and ( beta ) are constants. Given that the total contribution ( sum_{i=1}^{100} c_i = 500 ) and the total voting power ( sum_{i=1}^{100} v_i = 2000 ), determine ( alpha ) and ( beta )."So, no, it doesn't specify anything else. So, perhaps the problem is designed such that regardless of the distribution, we can solve for ( alpha ) and ( beta ). But how?Wait, maybe it's assuming that the contributions are such that ( sum c_i^{beta} ) can be expressed in terms of ( sum c_i ). For example, if ( beta = 1 ), then ( sum c_i^{beta} = 500 ), so ( alpha = 4 ). But if ( beta ) is something else, we can't determine it without more info.Alternatively, perhaps the problem is expecting me to recognize that without additional information about the distribution of ( c_i ), we can't uniquely determine ( alpha ) and ( beta ). But that seems unlikely because the problem is asking to determine them.Wait, maybe the problem is assuming that the contributions are such that ( sum c_i^{beta} ) is proportional to ( sum c_i ), but that would require ( beta = 1 ), which would make ( alpha = 4 ). But is that the only possibility?Alternatively, perhaps the problem is expecting me to use logarithms or some other method. Let me think.If I take the total voting power equation:( alpha cdot sum c_i^{beta} = 2000 )And the total contribution:( sum c_i = 500 )If I take the ratio of these two equations:( frac{alpha cdot sum c_i^{beta}}{sum c_i} = frac{2000}{500} = 4 )So,( alpha cdot frac{sum c_i^{beta}}{sum c_i} = 4 )But without knowing ( sum c_i^{beta} ), I can't proceed. Unless I can express ( sum c_i^{beta} ) in terms of ( sum c_i ) somehow.Wait, perhaps the problem is assuming that the contributions are such that ( c_i ) are in a geometric progression or something, but that's a stretch.Alternatively, maybe the problem is expecting me to recognize that if ( beta = 1 ), then ( alpha = 4 ), and that's the only solution. But that might not be the case.Wait, let me think differently. Suppose I consider the case where all ( c_i ) are equal, which is 5 each. Then, as I said earlier, ( v_i = alpha cdot 5^{beta} ), and total voting power is ( 100 cdot alpha cdot 5^{beta} = 2000 ), so ( alpha cdot 5^{beta} = 20 ). But I also have ( sum c_i = 500 ), which is consistent. So, in this case, ( alpha ) and ( beta ) can be any values such that ( alpha cdot 5^{beta} = 20 ). But without another equation, I can't solve for both.Wait, but the problem says each household has a different contribution, so they can't all be 5. So, that approach is invalid.Hmm, maybe I need to think about the problem differently. Perhaps the voting power is proportional to contribution, meaning ( beta = 1 ), so ( v_i = alpha c_i ). Then, total voting power is ( alpha sum c_i = 2000 ), so ( alpha = 2000 / 500 = 4 ). So, ( alpha = 4 ) and ( beta = 1 ).But is that the only possibility? The problem doesn't specify that the voting power is proportional, just that it's a function of contribution. So, maybe the sociologist is assuming a linear relationship, which would make ( beta = 1 ). That seems plausible.Alternatively, maybe the problem is expecting me to assume that the voting power is proportional to the contribution, hence ( beta = 1 ), leading to ( alpha = 4 ).But let me check if that's the only solution. Suppose ( beta ) is not 1. For example, if ( beta = 2 ), then ( sum c_i^2 ) would be some value, but without knowing the individual ( c_i ), I can't compute it. So, unless the problem provides more information, I can't determine ( beta ).Wait, maybe the problem is designed such that regardless of the distribution, ( sum c_i^{beta} ) can be expressed in terms of ( sum c_i ). But that's only possible if ( beta = 1 ), because for other exponents, the sum depends on the distribution.Therefore, perhaps the only way to solve this is to assume ( beta = 1 ), which gives ( alpha = 4 ).So, tentatively, I think ( alpha = 4 ) and ( beta = 1 ).Now, moving on to part 2.The sociologist further investigates the effect of voting power on resource allocation. The allocation ( a_i ) for the ( i )-th household is proportional to their voting power, such that ( a_i = gamma cdot v_i ). The total resources allocated sum to 1000 units. I need to find the constant ( gamma ) and the allocation ( a_i ) for a household whose contribution ( c_i = 5 ).Okay, so first, since ( a_i = gamma cdot v_i ), the total allocation is ( sum a_i = gamma cdot sum v_i = 1000 ).From part 1, we have ( sum v_i = 2000 ). Therefore,( gamma cdot 2000 = 1000 )So, ( gamma = 1000 / 2000 = 0.5 ).Therefore, ( gamma = 0.5 ).Now, for a household with ( c_i = 5 ), their voting power is ( v_i = alpha cdot c_i^{beta} ). From part 1, assuming ( alpha = 4 ) and ( beta = 1 ), then ( v_i = 4 cdot 5 = 20 ).Therefore, their allocation ( a_i = gamma cdot v_i = 0.5 cdot 20 = 10 ).So, the allocation for that household is 10 units.But wait, let me double-check. If ( beta = 1 ), then ( v_i = 4c_i ), so for ( c_i = 5 ), ( v_i = 20 ). Then, ( a_i = 0.5 * 20 = 10 ). That seems correct.But just to be thorough, let me consider if ( beta ) was something else. Suppose ( beta = 2 ), then ( v_i = alpha c_i^2 ). But without knowing ( alpha ), I can't compute it. But since in part 1, I concluded ( beta = 1 ), I think that's the way to go.So, summarizing:1. ( alpha = 4 ), ( beta = 1 )2. ( gamma = 0.5 ), and for ( c_i = 5 ), ( a_i = 10 )</think>

🧠:<think>Okay, so Aisha is a pregnant woman planning her diet during Ramadan. She needs to make sure she gets enough calories and nutrients while fasting from dawn to sunset. Her doctor has given her specific guidelines, so I need to help her figure out if her Suhoor meal is sufficient and then determine what she should eat for Iftar.First, let's tackle the first sub-problem: verifying if her Suhoor meal meets her daily requirements. She's planning to have 150 grams of carbs, 80 grams of proteins, and 60 grams of fats. I remember that carbs and proteins each provide 4 calories per gram, and fats provide 9 calories per gram. So, I can calculate the calories from each macronutrient.For carbs: 150 grams * 4 calories/gram = 600 caloriesFor proteins: 80 grams * 4 calories/gram = 320 caloriesFor fats: 60 grams * 9 calories/gram = 540 caloriesAdding those up: 600 + 320 + 540 = 1460 calories for Suhoor.Now, her doctor said she needs at least 2200 calories per day. So, 1460 calories is her Suhoor. That leaves 2200 - 1460 = 740 calories for Iftar. But wait, the doctor also said that no more than 60% of her total calories should be consumed at Iftar. Let me check what 60% of 2200 is.60% of 2200 = 0.6 * 2200 = 1320 calories. So, Iftar can't exceed 1320 calories. But from the calculation, she only needs 740 calories for Iftar, which is way below 1320. So, in terms of calories, her Suhoor is fine, and she has room for more if needed, but she only needs 740.But wait, maybe I should check the macronutrient distribution for Suhoor. The doctor advised 45% carbs, 30% proteins, and 25% fats. Let's see what percentage each macronutrient contributes in her Suhoor.Total calories from Suhoor: 1460Carbs: 600 / 1460 = ~0.4109 or 41.09%Proteins: 320 / 1460 = ~0.2192 or 21.92%Fats: 540 / 1460 = ~0.3699 or 36.99%Comparing to the advised: 45%, 30%, 25%. So, her Suhoor has slightly less carbs, significantly less proteins, and more fats than recommended. Hmm, that might be a problem. She's not meeting the macronutrient distribution for her meal.But wait, is the macronutrient distribution supposed to be for the entire day or each meal? The problem says her diet should consist of those percentages, so I think it's for the entire day. So, her Suhoor alone doesn't meet the distribution, but maybe when combined with Iftar, it will.But the first sub-problem is just about Suhoor. So, in terms of calories, it's 1460, which is less than 2200, so she needs more. But in terms of macronutrients, she's not meeting the percentages. So, her Suhoor doesn't meet the daily requirements in terms of macronutrient distribution because it's only part of her day.Wait, maybe I misread. The doctor's advice is for her daily intake, so each meal should contribute to that. But the question is about verifying if her Suhoor meets her daily requirements. So, if she only eats Suhoor, it's 1460 calories, which is less than 2200, so it doesn't meet the caloric requirement. Also, the macronutrient percentages are off.But actually, she will have Iftar as well, so maybe the question is whether Suhoor alone meets the daily requirements? That doesn't make sense because she's fasting, so she eats Suhoor and Iftar. So, the question is probably whether Suhoor meets the daily requirements in terms of calories and macronutrients, but since she's only eating Suhoor and Iftar, it's about whether the meal itself meets the distribution, not the total.Wait, the problem says: "Verify if Aisha's planned Suhoor meal meets her daily nutritional requirements in terms of calories and macronutrient distribution." So, does that mean that Suhoor alone should meet the daily requirements? That can't be, because she's also having Iftar. So, maybe it's about whether the meal itself is balanced according to the percentages, regardless of the total calories.But the total calories for Suhoor are 1460, which is less than 2200, so it doesn't meet the caloric requirement. So, the answer is that Suhoor doesn't meet the daily caloric intake, but also the macronutrient distribution is off.Wait, but the doctor's advice is for the entire day, so each meal contributes to that. So, maybe the question is whether the Suhoor meal, as part of the day, contributes appropriately. But the question is a bit ambiguous.Alternatively, maybe the question is whether the Suhoor meal meets the daily requirements in terms of calories and macronutrients, meaning that if she only ate Suhoor, would it meet the requirements? But that doesn't make sense because she's supposed to have two meals.Alternatively, perhaps the question is whether the Suhoor meal meets the daily requirements in terms of the percentages, regardless of the total calories. So, even if the total calories are less, does the meal have the right proportion of carbs, proteins, and fats.In that case, let's calculate the percentages:Carbs: 600 / 1460 ≈ 41.09%Proteins: 320 / 1460 ≈ 21.92%Fats: 540 / 1460 ≈ 36.99%Comparing to 45%, 30%, 25%. So, carbs are a bit low, proteins are significantly low, and fats are high. So, the macronutrient distribution is not met.But also, the total calories are 1460, which is less than 2200, so it doesn't meet the caloric requirement.Therefore, the answer is that Suhoor does not meet the daily caloric intake and the macronutrient distribution is off.Wait, but maybe the question is whether the meal itself is balanced according to the percentages, regardless of the total calories. So, if the meal has the right proportion, even if the total calories are low. But the question says "meets her daily nutritional requirements in terms of calories and macronutrient distribution." So, both.So, since the total calories are 1460, which is less than 2200, it doesn't meet the caloric requirement. And the macronutrient distribution is off as well.So, the answer to the first sub-problem is that Suhoor does not meet the daily caloric intake and the macronutrient distribution is not as advised.Now, moving on to the second sub-problem: determining how much she should consume for Iftar.She needs a total of 2200 calories. She already has 1460 from Suhoor, so she needs 2200 - 1460 = 740 calories from Iftar.But the doctor said no more than 60% of the total calories should be consumed at Iftar. 60% of 2200 is 1320 calories. Since 740 is less than 1320, she's fine.But she also needs to meet the macronutrient distribution for the entire day. So, total carbs should be 45% of 2200 = 0.45 * 2200 = 990 calories. Since carbs are 4 calories per gram, she needs 990 / 4 = 247.5 grams of carbs per day.Similarly, proteins: 30% of 2200 = 660 calories. 660 / 4 = 165 grams.Fats: 25% of 2200 = 550 calories. 550 / 9 ≈ 61.11 grams.She already consumed 150g carbs, 80g proteins, 60g fats in Suhoor. So, for Iftar, she needs:Carbs: 247.5 - 150 = 97.5 gramsProteins: 165 - 80 = 85 gramsFats: 61.11 - 60 ≈ 1.11 gramsBut wait, let's check the calories from these:Carbs: 97.5g * 4 = 390 caloriesProteins: 85g * 4 = 340 caloriesFats: 1.11g * 9 ≈ 10 caloriesTotal: 390 + 340 + 10 = 740 calories, which matches the needed calories for Iftar.But wait, 1.11 grams of fats seems very low. Is that possible? Maybe she can have a bit more, but according to the calculation, she only needs about 1 gram more of fats. Alternatively, maybe I made a mistake.Wait, let's recalculate the total fats needed:Total fats needed: 550 calories / 9 ≈ 61.11 grams. She already has 60 grams, so she needs 1.11 grams more. That seems correct.But in reality, it's hard to consume only 1 gram of fat. Maybe she can have a bit more, but according to the calculation, that's what she needs.Alternatively, maybe she can adjust the macronutrients to make it more practical. But the question is to determine how many grams she should consume, so we'll go with the exact numbers.So, for Iftar, she needs approximately 97.5g carbs, 85g proteins, and 1.11g fats.But let's check if the macronutrient distribution for Iftar is okay. The doctor didn't specify for each meal, just the total day. So, as long as the total day meets the percentages, it's fine.But let's also check the macronutrient distribution for Iftar itself. The calories from Iftar are 740.Carbs: 97.5g *4=390Proteins:85g*4=340Fats:1.11g*9≈10Total:740So, percentages:Carbs: 390/740 ≈52.7%Proteins:340/740≈45.9%Fats:10/740≈1.35%That's way off the advised 45%,30%,25%. But again, the doctor's advice is for the entire day, not each meal. So, as long as the total day meets the percentages, it's okay.But wait, the problem says "her diet should consist of 45% carbohydrates, 30% proteins, and 25% fats." So, it's about her total diet, not each meal. So, as long as the total day meets that, it's fine.Therefore, the answer is that for Iftar, she needs approximately 97.5g carbs, 85g proteins, and 1.11g fats.But let's present it more neatly.Carbs: 97.5gProteins:85gFats:1.11gBut since we can't have fractions of a gram in practice, maybe round to the nearest whole number. So, 98g carbs, 85g proteins, and 1g fats.But let's check the calories again with rounded numbers:98g carbs *4=39285g proteins*4=3401g fats*9=9Total=392+340+9=741 calories, which is 1 calorie over. Alternatively, 97g carbs=388, 85g proteins=340, 1g fats=9, total=388+340+9=737, which is 3 calories under. So, maybe 97.5g is acceptable.Alternatively, maybe she can have 97g carbs, 85g proteins, and 1g fats, totaling 737 calories, and then have a bit more of something else to make up the 740. But the question is to determine the grams needed, so we can present it as 97.5g, 85g, and 1.11g.So, summarizing:1. Suhoor provides 1460 calories, which is less than the required 2200. The macronutrient distribution is 41.09% carbs, 21.92% proteins, and 36.99% fats, which doesn't meet the advised 45%,30%,25%.2. For Iftar, she needs 97.5g carbs, 85g proteins, and 1.11g fats, totaling 740 calories, which is within the 60% limit (1320 calories). The total day will meet the macronutrient distribution.</think>

🧠:<think>Okay, so I have two problems here about Abercrombie & Fitch's sales and pricing strategies over the last decade. Let me try to tackle them one by one. I'll start with the first one.Problem 1: Total Sales Over a DecadeThe sales function is given as ( S(t) = 50 + 10 sinleft(frac{pi t}{2}right) + 0.5t ), where ( t ) is the number of quarters since the start of the decade. I need to calculate the total sales over the entire decade. Hmm, a decade is 10 years, and since each year has 4 quarters, that's 40 quarters in total. So, ( t ) will range from 0 to 40.To find the total sales, I think I need to integrate the sales function over the interval from 0 to 40. That makes sense because integration will sum up all the sales over each quarter. So, the integral expression should be:[int_{0}^{40} S(t) , dt = int_{0}^{40} left(50 + 10 sinleft(frac{pi t}{2}right) + 0.5tright) dt]Alright, now I need to compute this integral. Let me break it down into three separate integrals for easier calculation:1. Integral of 50 with respect to t.2. Integral of ( 10 sinleft(frac{pi t}{2}right) ) with respect to t.3. Integral of 0.5t with respect to t.Starting with the first one:[int 50 , dt = 50t + C]That's straightforward. Now, the second integral:[int 10 sinleft(frac{pi t}{2}right) dt]I remember that the integral of ( sin(ax) ) is ( -frac{1}{a} cos(ax) + C ). So, applying that here:Let ( a = frac{pi}{2} ), so:[int 10 sinleft(frac{pi t}{2}right) dt = 10 times left(-frac{2}{pi}right) cosleft(frac{pi t}{2}right) + C = -frac{20}{pi} cosleft(frac{pi t}{2}right) + C]Okay, that seems right. Now, the third integral:[int 0.5t , dt = 0.5 times frac{t^2}{2} + C = frac{t^2}{4} + C]So, putting it all together, the integral of S(t) from 0 to 40 is:[left[50t - frac{20}{pi} cosleft(frac{pi t}{2}right) + frac{t^2}{4}right]_{0}^{40}]Now, I need to evaluate this from 0 to 40. Let's compute each term at t=40 and t=0.First, at t=40:1. ( 50 times 40 = 2000 )2. ( -frac{20}{pi} cosleft(frac{pi times 40}{2}right) = -frac{20}{pi} cos(20pi) )3. ( frac{40^2}{4} = frac{1600}{4} = 400 )Now, ( cos(20pi) ). Since cosine has a period of ( 2pi ), ( cos(20pi) = cos(0) = 1 ). So, the second term becomes ( -frac{20}{pi} times 1 = -frac{20}{pi} ).So, adding up the terms at t=40:2000 - ( frac{20}{pi} ) + 400 = 2400 - ( frac{20}{pi} )Now, at t=0:1. ( 50 times 0 = 0 )2. ( -frac{20}{pi} cos(0) = -frac{20}{pi} times 1 = -frac{20}{pi} )3. ( frac{0^2}{4} = 0 )So, adding up the terms at t=0:0 - ( frac{20}{pi} ) + 0 = -( frac{20}{pi} )Now, subtracting the lower limit from the upper limit:[2400 - ( frac{20}{pi} )] - [ -( frac{20}{pi} ) ] = 2400 - ( frac{20}{pi} ) + ( frac{20}{pi} ) = 2400Wait, that's nice! The cosine terms cancel out. So, the total sales over the decade are 2400 million dollars. That seems clean, but let me double-check.Wait, the integral of 50 from 0 to 40 is 50*40=2000, correct. The integral of 0.5t is (0.5)*(40^2)/2 = (0.5)*(1600)/2 = 400, correct. The integral of the sine term is -20/pi [cos(20pi) - cos(0)] = -20/pi [1 - 1] = 0. So, yeah, the total is 2000 + 400 = 2400. So, that's correct.Problem 2: Average Price Index Over the DecadeThe price index is given by ( P(t) = P_0 e^{0.03t} ), with ( P_0 = 100 ). So, ( P(t) = 100 e^{0.03t} ). I need to find the average price index over the decade, which is 40 quarters.The average value of a function over an interval [a, b] is given by:[frac{1}{b - a} int_{a}^{b} P(t) dt]So, in this case, a=0, b=40. Therefore, the average price index is:[frac{1}{40} int_{0}^{40} 100 e^{0.03t} dt]Let me write that integral expression:[text{Average Price Index} = frac{1}{40} int_{0}^{40} 100 e^{0.03t} dt]Simplify the constants:[= frac{100}{40} int_{0}^{40} e^{0.03t} dt = frac{5}{2} int_{0}^{40} e^{0.03t} dt]Now, compute the integral of ( e^{0.03t} ) with respect to t. The integral of ( e^{kt} ) is ( frac{1}{k} e^{kt} + C ). So, here k=0.03.Thus,[int e^{0.03t} dt = frac{1}{0.03} e^{0.03t} + C]So, evaluating from 0 to 40:[left[ frac{1}{0.03} e^{0.03t} right]_0^{40} = frac{1}{0.03} left( e^{0.03 times 40} - e^{0} right) = frac{1}{0.03} left( e^{1.2} - 1 right)]Compute ( e^{1.2} ). Let me recall that ( e^{1} approx 2.71828 ), and ( e^{0.2} approx 1.22140. So, ( e^{1.2} = e^{1 + 0.2} = e^1 times e^{0.2} approx 2.71828 times 1.22140 approx 3.3201 ).So, ( e^{1.2} - 1 approx 3.3201 - 1 = 2.3201 ).Therefore, the integral becomes:[frac{1}{0.03} times 2.3201 approx frac{2.3201}{0.03} approx 77.3367]So, going back to the average price index:[frac{5}{2} times 77.3367 approx 2.5 times 77.3367 approx 193.34175]So, approximately 193.34. Let me check if that makes sense.Alternatively, let me compute ( e^{1.2} ) more accurately. Using a calculator, ( e^{1.2} ) is approximately 3.3201169228. So, subtracting 1 gives 2.3201169228. Dividing by 0.03: 2.3201169228 / 0.03 ≈ 77.33723076. Then, multiplying by 2.5: 77.33723076 * 2.5 = 193.3430769.So, approximately 193.34. Rounding to two decimal places, 193.34.But let me think if I did everything correctly. The integral of ( e^{0.03t} ) is indeed ( frac{1}{0.03} e^{0.03t} ). Evaluated from 0 to 40, it's ( frac{1}{0.03}(e^{1.2} - 1) ). Then, multiplying by 100/40, which is 2.5, gives the average. That seems correct.Alternatively, I can compute it step by step:Compute ( e^{1.2} approx 3.3201 )Compute ( e^{1.2} - 1 = 2.3201 )Divide by 0.03: 2.3201 / 0.03 ≈ 77.3367Multiply by 2.5: 77.3367 * 2.5 ≈ 193.34175So, yes, approximately 193.34. So, the average price index over the decade is about 193.34.Wait, but let me think again. The average price index is 193.34, which is higher than the initial 100, which makes sense because the price index is growing exponentially. So, over 40 quarters, it's increasing, so the average should be higher than the starting point.Alternatively, if I compute it more precisely, using more decimal places:Compute ( e^{1.2} ):We know that:( e^{1} = 2.718281828459045 )( e^{0.2} approx 1.221402758 )So, ( e^{1.2} = e^{1} times e^{0.2} ≈ 2.718281828459045 times 1.221402758 ≈ )Let me compute that:2.718281828459045 * 1.221402758First, 2 * 1.221402758 = 2.4428055160.7 * 1.221402758 ≈ 0.85498193060.018281828459045 * 1.221402758 ≈ approximately 0.0223Adding them up: 2.442805516 + 0.8549819306 ≈ 3.2977874466 + 0.0223 ≈ 3.3200874466So, ( e^{1.2} ≈ 3.3200874466 )Thus, ( e^{1.2} - 1 ≈ 2.3200874466 )Divide by 0.03: 2.3200874466 / 0.03 ≈ 77.33624822Multiply by 2.5: 77.33624822 * 2.5 = 193.34062055So, approximately 193.3406, which is about 193.34 when rounded to two decimal places.Therefore, the average price index over the decade is approximately 193.34.Wait, but let me think if I can represent this exactly without approximating. The integral is:[frac{1}{40} times 100 times left( frac{e^{1.2} - 1}{0.03} right) = frac{100}{40 times 0.03} (e^{1.2} - 1) = frac{100}{1.2} (e^{1.2} - 1) ≈ 83.3333 times (3.3201 - 1) ≈ 83.3333 times 2.3201 ≈ 193.34]Wait, hold on. Wait, 100 divided by (40 * 0.03) is 100 / 1.2 ≈ 83.3333. Then, 83.3333 * (e^{1.2} - 1) ≈ 83.3333 * 2.3201 ≈ 193.34. So, that's another way to compute it, same result.So, yes, 193.34 is correct.Summary of Thoughts:For the first problem, integrating the sales function over 40 quarters gives a total of 2400 million dollars. The sine term's integral cancels out over the interval, leaving a straightforward calculation.For the second problem, calculating the average price index involves integrating the exponential function over 40 quarters and then dividing by the interval length. The result is approximately 193.34, which is higher than the initial price index, as expected due to exponential growth.I think I've covered all the steps and double-checked the calculations to ensure accuracy. I didn't see any mistakes in the process, so I feel confident about these results.Final Answer1. The total sales over the decade are boxed{2400} million dollars.2. The average price index over the decade is approximately boxed{193.34}.</think>