🧠:<think>Okay, so I've got these two questions about voter turnout in Wolverhampton. Let me try to figure them out step by step.Starting with the first question. It says that there's a linear regression equation relating the average age of voters, A, to the voter turnout percentage, T. The equation is T = 0.5A + 20. The total number of registered voters in the city is 200,000, and the average age for the entire city is 45 years. I need to estimate the total number of votes cast in the election.Hmm, okay. So, the regression equation gives us the expected voter turnout percentage based on the average age. Since the average age for the entire city is 45, I can plug that into the equation to find the average voter turnout percentage.Let me write that down:T = 0.5 * A + 20Given A = 45, so:T = 0.5 * 45 + 20Calculating that:0.5 * 45 is 22.5, and 22.5 + 20 is 42.5. So, the average voter turnout percentage is 42.5%.Wait, but hold on. Is this the average for the entire city or per ward? The regression equation was derived from data across different wards, each with their own average age and voter turnout. So, if the city-wide average age is 45, does that mean the city-wide voter turnout is 42.5%?I think so, because the regression equation models the relationship between average age and turnout. So, if the city's average age is 45, then the expected city-wide turnout would be 42.5%.But let me make sure. The total number of registered voters is 200,000. So, the total votes cast would be 42.5% of 200,000.Calculating that:42.5% is 0.425 in decimal. So, 0.425 * 200,000.Let me compute that:200,000 * 0.425. Hmm, 200,000 * 0.4 is 80,000, and 200,000 * 0.025 is 5,000. So, adding them together, 80,000 + 5,000 = 85,000.So, the estimated total number of votes cast is 85,000.Wait, but let me think again. Is the regression equation giving the average turnout for the entire city? Or is it per ward? Because each ward has a different number of registered voters. So, if I just take the city-wide average age and plug it into the regression, does that give me the city-wide turnout?I think it does, because the regression equation is derived from the wards, but if the city's average age is 45, then on average, the turnout should be 42.5%. So, multiplying that by the total registered voters should give the total votes.Alternatively, if I had to do it per ward, I would need the number of registered voters in each ward and their respective average ages, but since I don't have that data, I have to use the city-wide average.So, I think 85,000 is the right estimate.Moving on to the second question. The blogger is looking at the variance in voter turnout among the 20 wards. The voter turnout percentages are T1, T2, ..., T20. The mean voter turnout percentage is 55%, and the sum of the squared deviations from the mean is 800. I need to calculate the standard deviation.Alright, standard deviation is the square root of the variance. The variance is the average of the squared deviations from the mean. Since we have 20 wards, the sample variance would be the sum of squared deviations divided by (n-1), which is 19. But sometimes, depending on the context, it might be divided by n. I need to figure out which one to use here.The question says "the sum of the squared deviations from the mean is 800." It doesn't specify whether it's sample variance or population variance. Since the data given is for all 20 wards, which is the entire population of wards in Wolverhampton, I think we should treat this as a population, not a sample. Therefore, the variance would be the sum of squared deviations divided by N, which is 20.So, variance = 800 / 20 = 40.Then, standard deviation is the square root of variance, so sqrt(40).Calculating sqrt(40). Well, sqrt(36) is 6, sqrt(49) is 7, so sqrt(40) is approximately 6.32455532.But since the question doesn't specify rounding, I can either leave it as sqrt(40) or approximate it. However, in statistical contexts, standard deviation is often reported to two decimal places.So, sqrt(40) ≈ 6.32.Wait, but let me double-check. If the sum of squared deviations is 800, and we have 20 wards, then yes, variance is 800 / 20 = 40, so standard deviation is sqrt(40) ≈ 6.32.Alternatively, if it were a sample, variance would be 800 / 19 ≈ 42.105, and standard deviation would be sqrt(42.105) ≈ 6.49. But since it's the entire population of wards, I think we use N, so 6.32 is correct.Therefore, the standard deviation is approximately 6.32%.Wait, hold on, the mean is 55%, so the standard deviation is in percentage points. So, it's 6.32 percentage points.But let me make sure about the variance calculation. The formula for population variance is:σ² = Σ(Ti - μ)² / NWhere Σ(Ti - μ)² is 800, N is 20.So, σ² = 800 / 20 = 40.Then, σ = sqrt(40) ≈ 6.32.Yes, that seems right.So, summarizing:1. The total number of votes cast is 85,000.2. The standard deviation of voter turnout percentages is approximately 6.32%.I think that's it.Final Answer1. The estimated total number of votes cast is boxed{85000}.2. The standard deviation of the voter turnout percentages is boxed{6.32}.</think>

🧠:<think>Alright, so I've got this problem about Dr. Smith speaking at a corporate law event. There are 150 executives, each attending at least one but no more than three sessions. The sessions are on Mergers and Acquisitions (M&A), Intellectual Property (IP), and Regulatory Compliance (RC). The first part asks me to find the number of executives who attend all three sessions. They've given me the numbers for each individual session and the overlaps between two sessions. Let me jot down the numbers:- Total executives: 150- M&A attendees: 80- IP attendees: 70- RC attendees: 60- Both M&A and IP: 30- Both M&A and RC: 25- Both IP and RC: 20I remember something about the principle of inclusion-exclusion for sets. It's a formula that helps calculate the total number of elements in the union of multiple sets by adding the sizes of the sets and then subtracting the sizes of all possible intersections. Since we have three sets here, the formula should be:Total = |M&A| + |IP| + |RC| - |M&A ∩ IP| - |M&A ∩ RC| - |IP ∩ RC| + |M&A ∩ IP ∩ RC|But wait, in this case, the total number of executives is 150, and each executive attends at least one session. So, the union of all three sessions should be 150. Let me plug in the numbers:150 = 80 + 70 + 60 - 30 - 25 - 20 + xWhere x is the number of executives attending all three sessions. Let me compute the right side step by step.First, add up the individual sessions: 80 + 70 + 60 = 210.Then, subtract the pairwise overlaps: 210 - 30 - 25 - 20. Let's compute that:210 - 30 = 180180 - 25 = 155155 - 20 = 135So now, 135 + x = 150Therefore, x = 150 - 135 = 15.Hmm, so 15 executives attend all three sessions. Let me double-check that because sometimes inclusion-exclusion can be tricky.If 15 attend all three, then the number attending only M&A and IP would be 30 - 15 = 15. Similarly, only M&A and RC would be 25 - 15 = 10, and only IP and RC would be 20 - 15 = 5.Now, let's compute the number attending only M&A: 80 - (15 + 10 + 15) = 80 - 40 = 40.Only IP: 70 - (15 + 5 + 15) = 70 - 35 = 35.Only RC: 60 - (10 + 5 + 15) = 60 - 30 = 30.Now, let's add up all these:Only M&A: 40Only IP: 35Only RC: 30M&A and IP only: 15M&A and RC only: 10IP and RC only: 5All three: 15Total = 40 + 35 + 30 + 15 + 10 + 5 + 15 = Let's compute:40 + 35 = 7575 + 30 = 105105 + 15 = 120120 + 10 = 130130 + 5 = 135135 + 15 = 150Perfect, that adds up to 150. So, the number of executives attending all three sessions is indeed 15.Moving on to the second part. It says that the probability of an executive attending M&A is independent of attending IP or RC. I need to find the probability that a randomly selected executive attends at least one of the three sessions.Wait, hold on. If the probabilities are independent, does that mean that attending M&A doesn't affect the probability of attending IP or RC? So, the probability of attending M&A, IP, and RC are independent events.But in reality, the numbers given are counts, not probabilities. So, maybe I need to convert these counts into probabilities first.Let me think. The total number of executives is 150. So, the probability of attending M&A is 80/150, IP is 70/150, and RC is 60/150.But if these are independent, then the probability of attending at least one session is 1 minus the probability of attending none. Since each executive attends at least one session, the probability of attending none is zero. Wait, that can't be right because the problem says each executive attends at least one session. So, the probability of attending at least one session is 1.But that seems contradictory because the second part is presented as a separate question, implying that maybe the independence is a different scenario. Maybe the first part is based on actual data, and the second part is a theoretical probability assuming independence.Wait, let me read it again: "Given that the probability of an executive attending a session on M&A is independent of the probability of attending a session on IP or RC, calculate the probability that a randomly selected executive will attend at least one of the three sessions."So, it's assuming independence between attending M&A, IP, and RC. So, the counts in the first part are not necessarily reflecting independence, but the second part is a separate calculation assuming independence.So, in the second part, I need to compute the probability that an executive attends at least one session, assuming that attending M&A, IP, and RC are independent events.So, first, I need to find the probabilities of attending each session. But wait, in the first part, the counts are given, but in the second part, we're assuming independence, so perhaps the probabilities are based on the counts?Wait, maybe not. Maybe the probabilities are given as independent, but we don't know the exact probabilities. Hmm, the problem is a bit unclear.Wait, actually, the problem says "the probability of an executive attending a session on M&A is independent of the probability of attending a session on IP or RC." So, perhaps the probability of attending M&A is independent of attending IP, and also independent of attending RC. But does that mean that attending IP and RC are also independent? Or just that M&A is independent of both IP and RC?I think it means that attending M&A is independent of attending IP, and attending M&A is independent of attending RC. But it doesn't necessarily say that attending IP and RC are independent. Hmm, that complicates things.But maybe for simplicity, since it's given that M&A is independent of IP and RC, perhaps we can assume that all three are independent. Or maybe not. The wording is a bit ambiguous.Wait, the problem says: "the probability of an executive attending a session on M&A is independent of the probability of attending a session on IP or RC." So, attending M&A is independent of attending IP or RC. So, that could mean that attending M&A is independent of attending IP, and attending M&A is independent of attending RC. But it doesn't necessarily say that attending IP and RC are independent.So, maybe only M&A is independent of the others, but IP and RC could be dependent.But in probability, if we have multiple independent events, sometimes we can compute the joint probabilities as products. But since only M&A is independent, perhaps we can model it as such.Wait, maybe I need to think differently. Since each executive attends at least one session, the probability of attending at least one is 1. But that can't be, because the second part is asking for the probability, implying it's less than 1.Wait, hold on. In the first part, we have specific counts, but in the second part, it's a different scenario where the probabilities are independent. So, maybe in the second part, we don't have the same counts, but instead, we have probabilities that are independent.So, perhaps in the second part, we need to compute the probability that an executive attends at least one session, given that attending M&A, IP, and RC are independent events. But we don't have the exact probabilities, so maybe we need to use the counts from the first part to estimate the probabilities and then compute the probability under independence.Wait, that might make sense. So, first, compute the probabilities of attending each session based on the first part, then assume independence, and compute the probability of attending at least one.So, let's try that.From the first part, we have:- P(M&A) = 80/150 = 16/30 ≈ 0.5333- P(IP) = 70/150 = 14/30 ≈ 0.4667- P(RC) = 60/150 = 12/30 = 0.4Assuming independence between M&A, IP, and RC, the probability of attending none would be:P(not M&A) * P(not IP) * P(not RC) = (1 - 16/30) * (1 - 14/30) * (1 - 12/30)Compute each term:1 - 16/30 = 14/301 - 14/30 = 16/301 - 12/30 = 18/30So, multiply them together:(14/30) * (16/30) * (18/30)Let me compute that step by step.First, 14 * 16 = 224224 * 18 = Let's compute 224 * 10 = 2240, 224 * 8 = 1792, so total 2240 + 1792 = 4032Denominator: 30 * 30 * 30 = 27000So, 4032 / 27000Simplify that fraction:Divide numerator and denominator by 12: 4032 ÷12=336, 27000 ÷12=2250336 / 2250Divide numerator and denominator by 6: 336 ÷6=56, 2250 ÷6=37556 / 375Can't simplify further. So, 56/375 ≈ 0.1493Therefore, the probability of attending none is approximately 0.1493, so the probability of attending at least one is 1 - 0.1493 ≈ 0.8507.But let me express it as a fraction. 56/375 is the probability of attending none, so 1 - 56/375 = (375 - 56)/375 = 319/375.Simplify 319/375. Let's see if 319 and 375 have any common factors. 319 divided by 11 is 29, because 11*29=319. 375 divided by 11 is not an integer. So, 319/375 is the simplified fraction.So, the probability is 319/375, which is approximately 0.8507 or 85.07%.Wait, but in the first part, all executives attend at least one session, so the probability is 1. But in the second part, we're assuming independence, which might lead to some executives not attending any, hence the probability less than 1.So, that seems consistent.But let me double-check my calculations.First, probabilities:P(M&A) = 80/150 = 16/30P(IP) = 70/150 = 14/30P(RC) = 60/150 = 12/30Assuming independence, P(not M&A) = 14/30, P(not IP)=16/30, P(not RC)=18/30Multiplying these: (14/30)*(16/30)*(18/30) = (14*16*18)/(30^3) = 4032/27000Simplify 4032/27000:Divide numerator and denominator by 12: 336/2250Divide by 6: 56/375Yes, that's correct.So, 1 - 56/375 = 319/375 ≈ 0.8507So, the probability is 319/375.Alternatively, as a decimal, approximately 0.8507.So, to express it as a fraction, 319/375 is the exact value.Alternatively, if we want to write it in simplest terms, since 319 is a prime number? Wait, 319 divided by 11 is 29, as I thought earlier. 11*29=319. 375 divided by 11 is 34.09, which is not an integer, so 319 and 375 share no common factors besides 1. So, 319/375 is the simplest form.Therefore, the probability is 319/375.So, summarizing:1. The number of executives attending all three sessions is 15.2. The probability that a randomly selected executive attends at least one session is 319/375.I think that's it. Let me just make sure I didn't make any calculation errors.In the first part, inclusion-exclusion gave me 15, and the breakdown of only, two, and all three added up correctly.In the second part, assuming independence, I converted counts to probabilities, computed the probability of attending none, subtracted from 1, and got 319/375. That seems correct.Yeah, I think I'm confident with these answers.</think>

🧠:<think>Alright, so I have this problem about a local producer who wants to ensure the quality of their eco-friendly products before supplying them to online platforms. There are two parts to this problem. Let me try to tackle them one by one.Starting with the first part: The producer makes two types of products, Type A and Type B. Their durability follows normal distributions. For Type A, the mean durability is 50 days with a standard deviation of 5 days. For Type B, it's 60 days with a standard deviation of 8 days. The online platform requires that at least 95% of the products must last more than a certain threshold T. I need to find this threshold T that both types must exceed.Hmm, okay. So, since both products need to meet this requirement, I think I need to find a T such that for both Type A and Type B, the probability that a product lasts more than T is at least 95%. That means, for each type, P(Durability > T) >= 0.95.But wait, actually, the wording says "at least 95% of the products must last more than a certain threshold T." So, that translates to P(Durability > T) >= 0.95. So, for both Type A and Type B, we need to find T such that this probability holds.Since both types need to meet this, I think the threshold T must be such that both probabilities are satisfied. So, I need to find T_A for Type A and T_B for Type B such that P(A > T_A) = 0.95 and P(B > T_B) = 0.95. But the question says "the threshold T that both types of products must exceed." So, maybe T is the same for both? Or perhaps, T is the minimum of the two thresholds? Hmm, I need to clarify.Wait, the wording is a bit ambiguous. It says "the threshold T that both types of products must exceed." So, perhaps T is a single value such that both Type A and Type B have at least 95% of their products lasting more than T. So, T must be a value that is less than or equal to both T_A and T_B? Or maybe it's the maximum of the two thresholds?Wait, let me think again. If we have two distributions, each with their own thresholds T_A and T_B such that 95% of each distribution is above their respective T. But the platform requires that both types must exceed T. So, T must be a value that is less than or equal to both T_A and T_B? Or perhaps, T is the minimum of T_A and T_B?Wait, no. Let me think in terms of the problem. The platform requires that at least 95% of the products must last more than T. So, for each type, individually, 95% must exceed T. So, T must be such that for Type A, P(A > T) >= 0.95, and for Type B, P(B > T) >= 0.95. So, T must be a value that is less than or equal to both the 5th percentile of Type A and the 5th percentile of Type B? Wait, no, because if T is too low, then P(A > T) would be more than 0.95, but we need the maximum T such that both P(A > T) and P(B > T) are at least 0.95.Wait, actually, to find the threshold T, we need to find the maximum T such that both P(A > T) >= 0.95 and P(B > T) >= 0.95. So, T is the minimum of the two thresholds T_A and T_B? Because if T is higher than T_A, then P(A > T) would be less than 0.95, which doesn't satisfy the requirement. Similarly, if T is higher than T_B, then P(B > T) would be less than 0.95. So, to satisfy both, T must be less than or equal to both T_A and T_B. But since we want the maximum T that satisfies both, T would be the minimum of T_A and T_B.Wait, let me make sure. Suppose T_A is the threshold for Type A, so P(A > T_A) = 0.95. Similarly, T_B for Type B. If T is set to T_A, then for Type B, P(B > T_A) would be greater than 0.95 because T_A is lower than T_B (since Type B has a higher mean). Similarly, if T is set to T_B, then for Type A, P(A > T_B) would be less than 0.95 because T_B is higher than T_A. So, to satisfy both, T must be set to T_A, because if we set it higher than T_A, Type A won't meet the 95% requirement. Therefore, T must be T_A, which is the lower of the two thresholds.Wait, but let me think about the distributions. Type A has a mean of 50 and Type B has a mean of 60. So, Type B is more durable on average. Therefore, the threshold T_A for Type A will be lower than T_B for Type B. So, if we set T to T_A, then for Type B, since T_A < T_B, P(B > T_A) will be higher than 0.95, which is acceptable. But for Type A, P(A > T_A) = 0.95, which is exactly the requirement. So, setting T to T_A would satisfy both types, with Type A meeting exactly 95% and Type B exceeding it.Alternatively, if we set T to a higher value, say somewhere between T_A and T_B, then for Type A, P(A > T) would be less than 0.95, which doesn't meet the requirement. Therefore, the maximum T that satisfies both is T_A.But wait, let me verify this with calculations.First, let's find T_A such that P(A > T_A) = 0.95. Since A ~ N(50, 5^2), we can standardize it.Z = (T_A - μ_A) / σ_AWe need P(A > T_A) = 0.95, which is equivalent to P(Z > z) = 0.95. So, the z-score corresponding to the 5th percentile (since 1 - 0.95 = 0.05) is z = -1.645 (since it's the lower tail). Wait, no, actually, for P(Z > z) = 0.95, z is the value such that the area to the right is 0.95, which is the same as the area to the left is 0.05. So, z = -1.645.Wait, no, hold on. If we have P(Z > z) = 0.95, that means z is the value where 95% of the distribution is to the right. So, the cumulative probability up to z is 0.05. So, z is the 5th percentile of the standard normal distribution, which is indeed -1.645.Therefore, T_A = μ_A + z * σ_A = 50 + (-1.645)*5 = 50 - 8.225 = 41.775 days.Similarly, for Type B, T_B is such that P(B > T_B) = 0.95.B ~ N(60, 8^2). So, standardizing:Z = (T_B - 60)/8We need P(B > T_B) = 0.95, so again, z = -1.645.Thus, T_B = 60 + (-1.645)*8 = 60 - 13.16 = 46.84 days.So, T_A is approximately 41.775 days, and T_B is approximately 46.84 days.Since the platform requires that both types must exceed T, and T must be such that both have at least 95% exceeding it. So, if we set T to 41.775, then for Type A, exactly 95% exceed it, and for Type B, since 41.775 < 46.84, the probability that Type B exceeds 41.775 is higher than 95%. Similarly, if we set T to 46.84, then Type A would have a probability less than 95% of exceeding it, which doesn't meet the requirement.Therefore, the maximum T that satisfies both is T_A, which is approximately 41.775 days. So, the threshold T is 41.775 days.But wait, let me think again. If T is set to 41.775, then Type A meets exactly 95%, and Type B exceeds it. So, both types meet the requirement. If T is set higher, say 46.84, then Type A would have less than 95% exceeding it, which doesn't satisfy the requirement. Therefore, T must be 41.775 days.Alternatively, perhaps the question is asking for a T such that both types individually have at least 95% exceeding T. So, T must be less than or equal to both T_A and T_B. But since T_A < T_B, the maximum T that is less than or equal to both is T_A. So, yes, T is 41.775 days.Wait, but let me check the calculation for T_A again. Z-score for 0.05 is indeed -1.645. So, T_A = 50 + (-1.645)*5 = 50 - 8.225 = 41.775. Correct.Similarly, T_B = 60 + (-1.645)*8 = 60 - 13.16 = 46.84. Correct.So, the threshold T is 41.775 days. But wait, the question says "the threshold T that both types of products must exceed." So, does that mean T is a single value that both must exceed? So, T must be such that for both types, P(Durability > T) >= 0.95. So, T must be the minimum of T_A and T_B, which is 41.775. Because if T is higher than 41.775, then for Type A, P(A > T) < 0.95, which doesn't meet the requirement. Therefore, T must be 41.775 days.Okay, that seems to make sense.Now, moving on to the second part: The producer has a constraint on the biodegradability rate. Type A follows an exponential distribution with an average rate λ_A = 0.02 per day, and Type B follows an exponential distribution with λ_B = 0.015 per day. I need to calculate the probability that a randomly selected product from Type A will biodegrade within 30 days, and compare it to the probability for Type B.So, for an exponential distribution, the probability that a product biodegrades within time t is given by P(X <= t) = 1 - e^(-λ*t).So, for Type A, P(A <= 30) = 1 - e^(-0.02*30).Similarly, for Type B, P(B <= 30) = 1 - e^(-0.015*30).Let me compute these.First, for Type A:λ_A = 0.02 per day, t = 30 days.P(A <= 30) = 1 - e^(-0.02*30) = 1 - e^(-0.6).Calculating e^(-0.6): e^0.6 is approximately 1.8221, so e^(-0.6) is approximately 0.5488.Therefore, P(A <= 30) ≈ 1 - 0.5488 = 0.4512, or 45.12%.For Type B:λ_B = 0.015 per day, t = 30 days.P(B <= 30) = 1 - e^(-0.015*30) = 1 - e^(-0.45).Calculating e^(-0.45): e^0.45 is approximately 1.5683, so e^(-0.45) is approximately 0.6376.Therefore, P(B <= 30) ≈ 1 - 0.6376 = 0.3624, or 36.24%.So, comparing the two, Type A has a higher probability of biodegrading within 30 days (45.12%) compared to Type B (36.24%). This makes sense because Type A has a higher decay rate (λ_A > λ_B), meaning it degrades faster on average.Wait, let me double-check the calculations.For Type A:0.02 * 30 = 0.6e^(-0.6) ≈ 0.54881 - 0.5488 ≈ 0.4512, correct.For Type B:0.015 * 30 = 0.45e^(-0.45) ≈ 0.63761 - 0.6376 ≈ 0.3624, correct.Yes, that seems right.So, summarizing:1. The threshold T is approximately 41.775 days.2. The probability for Type A is approximately 45.12%, and for Type B, approximately 36.24%. Therefore, Type A is more likely to biodegrade within 30 days than Type B.I think that's it. Let me just write down the final answers clearly.</think>

🧠:<think>Okay, so I have this problem where a political science student is analyzing the economic impact of a new public administration policy at Anglia Ruskin University. There are two parts to this problem. Let me try to tackle them one by one.Starting with the first part: The university's annual budget allocation over the next 5 years is modeled by the function ( B(t) = 2t^3 - 15t^2 + 36t + 50 ) (in millions of pounds), where ( t ) is the number of years from the present. I need to calculate the total budget allocation over this 5-year period.Hmm, so the function ( B(t) ) gives the budget for each year, right? So, to find the total budget over 5 years, I think I need to sum the budget for each year from t=0 to t=4, because t=0 is the current year, and t=5 would be the sixth year, which is beyond the 5-year period. Wait, actually, the problem says "over the next 5 years," so does that include t=0? Let me think.If t=0 is the current year, then the next 5 years would be t=1 to t=5. But the function is defined for t=0 as well. The question is a bit ambiguous. Let me check the wording again: "the next 5 years is modeled by the function... where t is the number of years from the present (i.e., t=0 corresponds to the current year)." So, if t=0 is the current year, then t=1 is the first year of the next 5 years, t=2 the second, and so on up to t=5, which would be the fifth year. So, the total budget over the next 5 years would be from t=1 to t=5.Wait, but sometimes in these problems, they include the current year as part of the period. Hmm. Let me think again. If t=0 is the current year, then the next 5 years would be t=1 to t=5. So, the total budget would be the sum from t=1 to t=5. Alternatively, if they consider t=0 to t=4 as the next 5 years, that's another interpretation. I need to clarify this.Looking at the wording: "the next 5 years is modeled by the function... where t is the number of years from the present (i.e., t=0 corresponds to the current year)." So, t=0 is the current year, and the next 5 years would be t=1, t=2, t=3, t=4, t=5. So, the total budget would be the sum of B(1) + B(2) + B(3) + B(4) + B(5). Alternatively, if they consider the current year as part of the 5-year period, it would be t=0 to t=4. But the wording says "the next 5 years," which suggests starting from t=1. Hmm, this is a bit confusing.Wait, perhaps the function is defined for t=0 to t=5, and the total over the next 5 years would be from t=0 to t=4, because t=5 would be the sixth year. No, that might not make sense. Alternatively, maybe the total budget over the 5-year period is the integral of B(t) from t=0 to t=5. But the function is given as a discrete function, not a continuous one. Wait, no, the function is given as a polynomial in t, which is continuous, but the budget is annual, so it's discrete. Hmm, this is a bit tricky.Wait, the problem says "the annual budget allocation over the next 5 years is modeled by the function B(t)." So, B(t) gives the budget for each year t, where t is the number of years from the present. So, t=0 is the current year, t=1 is the first year of the next 5 years, and so on up to t=5, which is the fifth year. Therefore, the total budget over the next 5 years would be the sum of B(1) + B(2) + B(3) + B(4) + B(5). Alternatively, if they consider t=0 as the first year of the period, then it would be t=0 to t=4. But the wording says "the next 5 years," which usually means starting from the next year, so t=1 to t=5.Wait, but let me check the function at t=0: B(0) = 2(0)^3 -15(0)^2 +36(0) +50 = 50 million pounds. So, that's the current year. If the next 5 years are t=1 to t=5, then the total would be B(1) + B(2) + B(3) + B(4) + B(5). Alternatively, if they include t=0, it's 5 years from t=0 to t=4. Hmm, I think the correct interpretation is that the next 5 years are t=1 to t=5, so the total budget is the sum from t=1 to t=5.But to be thorough, let me compute both and see which makes more sense. Let's compute B(0) to B(5):B(0) = 50B(1) = 2(1)^3 -15(1)^2 +36(1) +50 = 2 -15 +36 +50 = 73B(2) = 2(8) -15(4) +36(2) +50 = 16 -60 +72 +50 = 78B(3) = 2(27) -15(9) +36(3) +50 = 54 -135 +108 +50 = 77B(4) = 2(64) -15(16) +36(4) +50 = 128 -240 +144 +50 = 82B(5) = 2(125) -15(25) +36(5) +50 = 250 -375 +180 +50 = 105So, if we sum from t=1 to t=5: 73 +78 +77 +82 +105 = let's compute:73 +78 = 151151 +77 = 228228 +82 = 310310 +105 = 415So, total budget would be 415 million pounds.Alternatively, if we include t=0, it would be 50 +73 +78 +77 +82 = let's compute:50 +73 = 123123 +78 = 201201 +77 = 278278 +82 = 360So, 360 million pounds.But the question says "the next 5 years," so I think it's more likely to be t=1 to t=5, giving 415 million pounds.Wait, but let me think again. If t=0 is the current year, then the next 5 years would be t=1 to t=5, so the total budget over the next 5 years would be the sum from t=1 to t=5, which is 415 million pounds.Alternatively, if the problem is considering t=0 as the first year of the 5-year period, then it would be t=0 to t=4, which is 50 +73 +78 +77 +82 = 360 million pounds.Hmm, I think the correct interpretation is that the next 5 years are t=1 to t=5, so the total is 415 million pounds.But to be safe, maybe the problem expects the integral of B(t) from t=0 to t=5, treating it as a continuous function. Let me check that approach.If we model the budget as a continuous function, the total budget over 5 years would be the integral from t=0 to t=5 of B(t) dt.So, let's compute that:Integral of B(t) = Integral of (2t^3 -15t^2 +36t +50) dt from 0 to 5.The integral is:(2/4)t^4 - (15/3)t^3 + (36/2)t^2 +50t evaluated from 0 to 5.Simplify:(0.5)t^4 -5t^3 +18t^2 +50t.Now, plug in t=5:0.5*(625) -5*(125) +18*(25) +50*(5)= 312.5 -625 +450 +250Compute step by step:312.5 -625 = -312.5-312.5 +450 = 137.5137.5 +250 = 387.5Now, plug in t=0: all terms are 0, so the integral from 0 to 5 is 387.5 million pounds.But wait, this is a continuous integral, which might not be the correct approach if the budget is annual and discrete. The problem says "annual budget allocation," so it's discrete, meaning we should sum the values at each integer t from 1 to 5, which we did earlier as 415 million pounds.Therefore, I think the correct approach is to sum B(1) to B(5), giving 415 million pounds.Wait, but let me think again. The problem says "the annual budget allocation over the next 5 years is modeled by the function B(t)." So, B(t) is the budget for year t, where t=0 is the current year. So, the next 5 years would be t=1 to t=5, so the total budget is the sum of B(1) to B(5), which is 73 +78 +77 +82 +105 = 415 million pounds.Yes, that makes sense.Now, moving on to the second part: The projected economic growth rate G(t) (in percentage) due to the new policy is given by the differential equation ( frac{dG}{dt} = frac{2G}{t+1} - 3 ), with the initial condition ( G(0) = 5 ). I need to solve this differential equation to find G(t) and determine the projected economic growth rate at the end of the 5-year period, i.e., G(5).Alright, so this is a first-order linear ordinary differential equation. The standard form is ( frac{dG}{dt} + P(t)G = Q(t) ). Let me rewrite the given equation in this form.Given: ( frac{dG}{dt} = frac{2G}{t+1} - 3 )Let me rearrange terms:( frac{dG}{dt} - frac{2}{t+1}G = -3 )So, comparing to the standard form, we have:P(t) = -2/(t+1)Q(t) = -3To solve this, we can use an integrating factor. The integrating factor μ(t) is given by:μ(t) = exp(∫ P(t) dt) = exp(∫ -2/(t+1) dt)Compute the integral:∫ -2/(t+1) dt = -2 ln|t+1| + C = ln| (t+1)^{-2} | + CSo, μ(t) = exp(ln| (t+1)^{-2} | ) = (t+1)^{-2} = 1/(t+1)^2Now, multiply both sides of the differential equation by μ(t):1/(t+1)^2 * dG/dt - 2/(t+1)^3 * G = -3/(t+1)^2The left side is the derivative of [G * μ(t)] with respect to t. So:d/dt [G/(t+1)^2] = -3/(t+1)^2Now, integrate both sides with respect to t:∫ d/dt [G/(t+1)^2] dt = ∫ -3/(t+1)^2 dtSo, G/(t+1)^2 = ∫ -3/(t+1)^2 dtCompute the integral on the right:∫ -3/(t+1)^2 dt = -3 ∫ (t+1)^{-2} dt = -3 [ -1/(t+1) ] + C = 3/(t+1) + CTherefore, G/(t+1)^2 = 3/(t+1) + CMultiply both sides by (t+1)^2:G(t) = 3(t+1) + C(t+1)^2Now, apply the initial condition G(0) = 5:G(0) = 3(0+1) + C(0+1)^2 = 3 + C = 5So, 3 + C = 5 => C = 2Therefore, the solution is:G(t) = 3(t+1) + 2(t+1)^2Simplify:G(t) = 3t + 3 + 2(t^2 + 2t +1) = 3t +3 + 2t^2 +4t +2 = 2t^2 +7t +5So, G(t) = 2t^2 +7t +5Now, we need to find the projected economic growth rate at the end of the 5-year period, which is G(5):G(5) = 2*(25) +7*5 +5 = 50 +35 +5 = 90Wait, that seems high. Let me double-check the calculations.Wait, G(t) = 2t^2 +7t +5So, G(5) = 2*(5)^2 +7*(5) +5 = 2*25 +35 +5 = 50 +35 +5 = 90. Yes, that's correct.But let me check the differential equation solution again to make sure I didn't make a mistake.We had:dG/dt = 2G/(t+1) -3We rearranged to:dG/dt - 2G/(t+1) = -3Integrating factor μ(t) = exp(∫ -2/(t+1) dt) = (t+1)^{-2}Multiply through:G/(t+1)^2 = ∫ -3/(t+1)^2 dt = 3/(t+1) + CSo, G(t) = 3(t+1) + C(t+1)^2At t=0, G(0)=5:5 = 3(1) + C(1)^2 => 5 =3 + C => C=2Thus, G(t)=3(t+1)+2(t+1)^2Expanding:3t +3 +2t^2 +4t +2 = 2t^2 +7t +5Yes, that's correct.So, G(5)=2*(25)+7*5+5=50+35+5=90%.Wait, that seems quite high for an economic growth rate. Maybe I made a mistake in interpreting the differential equation.Wait, let me check the steps again.Starting from:dG/dt = 2G/(t+1) -3Rearranged to:dG/dt - 2G/(t+1) = -3Integrating factor μ(t) = exp(∫ -2/(t+1) dt) = (t+1)^{-2}Multiply both sides:(t+1)^{-2} dG/dt - 2(t+1)^{-3} G = -3(t+1)^{-2}Left side is d/dt [G/(t+1)^2] = -3/(t+1)^2Integrate both sides:G/(t+1)^2 = ∫ -3/(t+1)^2 dt = 3/(t+1) + CMultiply by (t+1)^2:G(t) = 3(t+1) + C(t+1)^2At t=0, G(0)=5:5 = 3(1) + C(1)^2 => C=2Thus, G(t)=3(t+1)+2(t+1)^2Which simplifies to G(t)=2t^2 +7t +5So, G(5)=2*(25)+7*5+5=50+35+5=90Hmm, 90% growth rate seems high, but perhaps it's correct given the differential equation. Let me check if plugging G(t)=2t^2 +7t +5 into the original DE satisfies it.Compute dG/dt:dG/dt = 4t +7Now, compute RHS: 2G/(t+1) -3G=2t^2 +7t +5So, 2G/(t+1) = 2*(2t^2 +7t +5)/(t+1)Let me perform polynomial division or factor the numerator:2t^2 +7t +5 divided by t+1.Let me try to factor 2t^2 +7t +5.Looking for factors of 2*5=10 that add up to 7. 5 and 2: 5*2=10, 5+2=7.So, 2t^2 +5t +2t +5 = t(2t +5) +1(2t +5) = (t+1)(2t +5)So, 2t^2 +7t +5 = (t+1)(2t +5)Therefore, 2G/(t+1) = 2*(t+1)(2t +5)/(t+1) = 2*(2t +5) =4t +10Thus, RHS =4t +10 -3=4t +7Which equals dG/dt=4t +7So, yes, the solution satisfies the differential equation. Therefore, G(t)=2t^2 +7t +5 is correct, and G(5)=90%.So, the projected economic growth rate at the end of the 5-year period is 90%.Wait, that seems extremely high for an economic growth rate. Typically, growth rates are in single digits, so 90% seems unrealistic. Maybe I made a mistake in the integration factor or the solution.Wait, let me check the integrating factor again.Given:dG/dt - (2/(t+1))G = -3So, P(t) = -2/(t+1)Thus, integrating factor μ(t) = exp(∫ P(t) dt) = exp(∫ -2/(t+1) dt) = exp(-2 ln(t+1)) = (t+1)^{-2}Yes, that's correct.Then, multiplying through:(t+1)^{-2} dG/dt - 2(t+1)^{-3} G = -3(t+1)^{-2}Left side is d/dt [G/(t+1)^2] = -3/(t+1)^2Integrate both sides:G/(t+1)^2 = ∫ -3/(t+1)^2 dt = 3/(t+1) + CMultiply by (t+1)^2:G(t) = 3(t+1) + C(t+1)^2At t=0, G(0)=5:5 =3(1) + C(1)^2 => C=2Thus, G(t)=3(t+1)+2(t+1)^2Which is G(t)=2t^2 +7t +5Yes, that's correct. So, G(5)=90%.Hmm, perhaps the model is assuming something else, or maybe it's a hypothetical scenario. So, I'll go with that.So, summarizing:1. Total budget allocation over the next 5 years is 415 million pounds.2. The projected economic growth rate at the end of the 5-year period is 90%.Wait, but let me just confirm the total budget again. Earlier, I calculated B(1)=73, B(2)=78, B(3)=77, B(4)=82, B(5)=105. Summing these gives 73+78=151, 151+77=228, 228+82=310, 310+105=415. Yes, that's correct.Alternatively, if I had included t=0, it would have been 50+73+78+77+82=360, but as per the problem statement, it's the next 5 years, so t=1 to t=5, giving 415 million pounds.So, final answers:1. Total budget allocation over the next 5 years: 415 million pounds.2. Projected economic growth rate at the end of 5 years: 90%.I think that's it.</think>