🧠:<think>Okay, so I have this problem about a writer documenting an Olympic dressage rider's career. There are two parts: first, modeling the rider's performance scores with a sinusoidal function, and second, calculating the correlation coefficient between the rider's scores and the team's scores. Let me tackle them one by one.Starting with the first part. The function given is S(t) = A sin(Bt + C) + D. I need to find A, B, C, and D. The data provided includes the highest score, lowest score, the period between peaks, and the time of the first peak.First, the highest score is 95 and the lowest is 75. I remember that in a sinusoidal function, the amplitude A is half the difference between the maximum and minimum values. So, let me calculate that.A = (Max - Min)/2 = (95 - 75)/2 = 20/2 = 10. So, A is 10.Next, the average performance score D is the midpoint between the highest and lowest scores. So, D = (Max + Min)/2 = (95 + 75)/2 = 170/2 = 85. So, D is 85.Now, the performance peaks every 4 years. The period of a sinusoidal function is given by 2π/B. So, if the period is 4, then 2π/B = 4. Solving for B, we get B = 2π/4 = π/2. So, B is π/2.Now, the tricky part is finding the phase shift C. The first peak occurs at t = 2. In the sine function, the maximum occurs at π/2. So, we can set up the equation:Bt + C = π/2 when t = 2.We already know B is π/2, so plugging in:(π/2)(2) + C = π/2Simplify:π + C = π/2Subtract π from both sides:C = π/2 - π = -π/2.So, C is -π/2.Let me double-check that. If t = 2, then Bt + C = (π/2)(2) - π/2 = π - π/2 = π/2. Yes, that makes sense because sin(π/2) is 1, which is the maximum. So, that seems correct.So, putting it all together, the function is S(t) = 10 sin( (π/2)t - π/2 ) + 85.Wait, let me write it as S(t) = 10 sin( (π/2)t - π/2 ) + 85.Alternatively, since sin(x - π/2) is equal to -cos(x), so maybe we can write it as S(t) = -10 cos( (π/2)t ) + 85. But since the question just asks for the values of A, B, C, D, I think the first form is acceptable.So, A = 10, B = π/2, C = -π/2, D = 85.Moving on to the second part: calculating the correlation coefficient ρ between the rider's scores and the team's scores.Given data:Rider's scores: [85, 90, 87, 93, 88]Team's scores: [340, 360, 355, 370, 345]First, I need to compute the means of both sets of scores.For the rider's scores:Sum = 85 + 90 + 87 + 93 + 88 = let's compute step by step.85 + 90 = 175175 + 87 = 262262 + 93 = 355355 + 88 = 443Mean rider score, r_bar = 443 / 5 = 88.6For the team's scores:Sum = 340 + 360 + 355 + 370 + 345340 + 360 = 700700 + 355 = 10551055 + 370 = 14251425 + 345 = 1770Mean team score, T_bar = 1770 / 5 = 354Now, I need to compute the numerator and denominator of the correlation coefficient formula.Numerator: sum of (r_i - r_bar)(T_i - T_bar) for each i.Denominator: sqrt(sum of (r_i - r_bar)^2) multiplied by sqrt(sum of (T_i - T_bar)^2)Let me create a table for each term.First, list the rider's scores, team's scores, deviations from mean, and products.i | r_i | T_i | r_i - r_bar | T_i - T_bar | (r_i - r_bar)(T_i - T_bar)---|-----|-----|-------------|-------------|----------------------------1 | 85 |340 |85 - 88.6 = -3.6 |340 - 354 = -14 | (-3.6)(-14) = 50.42 |90 |360 |90 - 88.6 = 1.4 |360 - 354 = 6 | (1.4)(6) = 8.43 |87 |355 |87 - 88.6 = -1.6 |355 - 354 = 1 | (-1.6)(1) = -1.64 |93 |370 |93 - 88.6 = 4.4 |370 - 354 = 16 | (4.4)(16) = 70.45 |88 |345 |88 - 88.6 = -0.6 |345 - 354 = -9 | (-0.6)(-9) = 5.4Now, compute the numerator:Sum of products: 50.4 + 8.4 - 1.6 + 70.4 + 5.4Let's add them step by step:50.4 + 8.4 = 58.858.8 - 1.6 = 57.257.2 + 70.4 = 127.6127.6 + 5.4 = 133So, numerator = 133Now, compute the denominator.First, compute sum of (r_i - r_bar)^2:(-3.6)^2 = 12.96(1.4)^2 = 1.96(-1.6)^2 = 2.56(4.4)^2 = 19.36(-0.6)^2 = 0.36Sum: 12.96 + 1.96 + 2.56 + 19.36 + 0.36Compute step by step:12.96 + 1.96 = 14.9214.92 + 2.56 = 17.4817.48 + 19.36 = 36.8436.84 + 0.36 = 37.2So, sum of (r_i - r_bar)^2 = 37.2Similarly, compute sum of (T_i - T_bar)^2:(-14)^2 = 1966^2 = 361^2 = 116^2 = 256(-9)^2 = 81Sum: 196 + 36 + 1 + 256 + 81Compute step by step:196 + 36 = 232232 + 1 = 233233 + 256 = 489489 + 81 = 570So, sum of (T_i - T_bar)^2 = 570Now, denominator is sqrt(37.2) * sqrt(570)Compute sqrt(37.2): approximately 6.1Compute sqrt(570): approximately 23.87Multiply them: 6.1 * 23.87 ≈ 145.6But let me compute more accurately.First, sqrt(37.2):37.2 is between 36 (6^2) and 49 (7^2). 6.1^2 = 37.21, which is very close. So, sqrt(37.2) ≈ 6.1Similarly, sqrt(570):23^2 = 529, 24^2=576. So, sqrt(570) is between 23 and 24. Let's compute 23.87^2:23.87^2 = (23 + 0.87)^2 = 23^2 + 2*23*0.87 + 0.87^2 = 529 + 40.02 + 0.7569 ≈ 570. So, sqrt(570) ≈ 23.87So, denominator ≈ 6.1 * 23.87 ≈ let's compute 6 * 23.87 = 143.22, plus 0.1 * 23.87 = 2.387, total ≈ 145.607So, denominator ≈ 145.607Therefore, ρ = 133 / 145.607 ≈ 0.913Wait, let me compute 133 / 145.607.Divide numerator and denominator by 10: 13.3 / 14.5607 ≈ 0.913So, approximately 0.913.But let me compute it more precisely.145.607 * 0.9 = 131.0463145.607 * 0.91 = 131.0463 + 145.607*0.01 = 131.0463 + 1.45607 ≈ 132.5023145.607 * 0.913 ≈ 132.5023 + 145.607*0.003 ≈ 132.5023 + 0.4368 ≈ 132.9391But our numerator is 133, so 133 / 145.607 ≈ 0.913So, approximately 0.913.But let me use a calculator approach.Compute 133 divided by 145.607.145.607 goes into 133 zero times. So, 0.But wait, 133 is less than 145.607, so it's 0. something.Compute 133 / 145.607:Multiply numerator and denominator by 1000: 133000 / 145607 ≈Compute 133000 ÷ 145607.Since 145607 * 0.9 = 131,046.3Subtract: 133,000 - 131,046.3 = 1,953.7So, 0.9 + (1,953.7 / 145,607) ≈ 0.9 + 0.0134 ≈ 0.9134So, approximately 0.9134.So, rounding to three decimal places, ρ ≈ 0.913.But let me check if I did all calculations correctly.Wait, let me recalculate the numerator:Sum of (r_i - r_bar)(T_i - T_bar):First row: (85 - 88.6)(340 - 354) = (-3.6)(-14) = 50.4Second row: (90 - 88.6)(360 - 354) = (1.4)(6) = 8.4Third row: (87 - 88.6)(355 - 354) = (-1.6)(1) = -1.6Fourth row: (93 - 88.6)(370 - 354) = (4.4)(16) = 70.4Fifth row: (88 - 88.6)(345 - 354) = (-0.6)(-9) = 5.4Adding these: 50.4 + 8.4 = 58.8; 58.8 - 1.6 = 57.2; 57.2 + 70.4 = 127.6; 127.6 + 5.4 = 133. Correct.Sum of (r_i - r_bar)^2:(-3.6)^2 = 12.961.4^2 = 1.96(-1.6)^2 = 2.564.4^2 = 19.36(-0.6)^2 = 0.36Sum: 12.96 + 1.96 = 14.92; 14.92 + 2.56 = 17.48; 17.48 + 19.36 = 36.84; 36.84 + 0.36 = 37.2. Correct.Sum of (T_i - T_bar)^2:(-14)^2 = 1966^2 = 361^2 = 116^2 = 256(-9)^2 = 81Sum: 196 + 36 = 232; 232 + 1 = 233; 233 + 256 = 489; 489 + 81 = 570. Correct.So, numerator is 133, denominator is sqrt(37.2)*sqrt(570) ≈ 6.1*23.87 ≈ 145.607So, ρ ≈ 133 / 145.607 ≈ 0.913.Therefore, the correlation coefficient is approximately 0.913.But let me check if I can compute it more accurately.Alternatively, maybe I can compute it step by step without approximating sqrt(37.2) and sqrt(570).Compute sqrt(37.2):37.2 = 372/10. Let's compute sqrt(372)/sqrt(10).sqrt(372) is between 19^2=361 and 20^2=400. 19.28^2 ≈ 372.19.28^2 = (19 + 0.28)^2 = 361 + 2*19*0.28 + 0.28^2 = 361 + 10.64 + 0.0784 ≈ 371.7184, which is close to 372.So, sqrt(372) ≈ 19.28, so sqrt(37.2) = sqrt(372)/sqrt(10) ≈ 19.28 / 3.1623 ≈ 6.1.Similarly, sqrt(570):570 is between 23^2=529 and 24^2=576.Compute 23.87^2 as before, which is approximately 570.So, sqrt(570) ≈ 23.87.Thus, denominator ≈ 6.1 * 23.87 ≈ 145.607.So, 133 / 145.607 ≈ 0.913.Therefore, the correlation coefficient is approximately 0.913.But let me see if I can compute it more precisely.Alternatively, maybe I can compute it using fractions.But perhaps it's better to use the exact values.Alternatively, use more precise decimal places.But given the context, 0.913 is sufficient.Alternatively, if I compute 133 / 145.607:145.607 * 0.9 = 131.0463Subtract: 133 - 131.0463 = 1.9537Now, 1.9537 / 145.607 ≈ 0.0134So, total is 0.9 + 0.0134 ≈ 0.9134So, approximately 0.9134.Rounding to three decimal places, 0.913.Therefore, the correlation coefficient is approximately 0.913.So, summarizing:For part 1:A = 10B = π/2C = -π/2D = 85For part 2:ρ ≈ 0.913I think that's it.Final Answer1. The values are ( A = boxed{10} ), ( B = boxed{dfrac{pi}{2}} ), ( C = boxed{-dfrac{pi}{2}} ), and ( D = boxed{85} ).2. The correlation coefficient is ( rho = boxed{0.913} ).</think>

🧠:<think>Okay, so I have this problem about a retired police captain organizing monthly meetings. The room's seating capacity follows the Fibonacci sequence, and I need to figure out two things: first, the smallest n where the seating capacity exceeds 10,000, and second, the total number of chairs arranged in a circle by that nth meeting, given that the circumference increases by 3 units each month starting from 15 units.Let me start with the first part. The seating capacity follows the Fibonacci sequence, starting with 1 and 1. So, the sequence is 1, 1, 2, 3, 5, 8, 13, 21, and so on. Each term is the sum of the two preceding ones. I need to find the smallest n such that the nth Fibonacci number is greater than 10,000.Hmm, Fibonacci numbers grow exponentially, so it shouldn't take too many terms to exceed 10,000. Maybe I can list them out until I pass 10,000. Let me try that.Starting with F(1) = 1, F(2) = 1.F(3) = F(2) + F(1) = 1 + 1 = 2F(4) = F(3) + F(2) = 2 + 1 = 3F(5) = 3 + 2 = 5F(6) = 5 + 3 = 8F(7) = 8 + 5 = 13F(8) = 13 + 8 = 21F(9) = 21 + 13 = 34F(10) = 34 + 21 = 55F(11) = 55 + 34 = 89F(12) = 89 + 55 = 144F(13) = 144 + 89 = 233F(14) = 233 + 144 = 377F(15) = 377 + 233 = 610F(16) = 610 + 377 = 987F(17) = 987 + 610 = 1597F(18) = 1597 + 987 = 2584F(19) = 2584 + 1597 = 4181F(20) = 4181 + 2584 = 6765F(21) = 6765 + 4181 = 10946Okay, so F(21) is 10946, which is the first Fibonacci number exceeding 10,000. So, n is 21.Wait, let me double-check that. Maybe I made a mistake in the calculations.F(1) = 1F(2) = 1F(3) = 2F(4) = 3F(5) = 5F(6) = 8F(7) = 13F(8) = 21F(9) = 34F(10) = 55F(11) = 89F(12) = 144F(13) = 233F(14) = 377F(15) = 610F(16) = 987F(17) = 1597F(18) = 2584F(19) = 4181F(20) = 6765F(21) = 10946Yes, that seems correct. So n is 21.Now, moving on to the second part. The captain arranges chairs in circular patterns where each chair is 1 unit apart. The circumference increases linearly by 3 units every month, starting from 15 units for the first meeting. I need to find the total number of chairs that can be arranged in a circle by the 21st meeting.Wait, chairs arranged in a circle with each chair 1 unit apart. So, the number of chairs would be equal to the circumference divided by the distance between chairs, right? Since each chair is 1 unit apart, the number of chairs is equal to the circumference.But the circumference increases by 3 units each month, starting from 15 units. So, the circumference for the nth meeting is 15 + 3*(n-1). Because the first month is 15, the second is 18, the third is 21, etc.So, for the 21st meeting, the circumference would be 15 + 3*(21 - 1) = 15 + 3*20 = 15 + 60 = 75 units.Therefore, the number of chairs is 75, since each chair is 1 unit apart. So, 75 chairs can be arranged.But wait, the problem says "the total number of chairs that can be arranged in a circle by the nth meeting." Hmm, does that mean the cumulative total from the first meeting up to the 21st meeting? Or just the number for the 21st meeting?Looking back at the problem: "determine the total number of chairs that can be arranged in a circle by the nth meeting you identified in the first sub-problem."Hmm, the wording is a bit ambiguous. It could be interpreted as the total number of chairs used in all meetings up to the nth meeting, or just the number for the nth meeting.But considering the context, since each meeting is a separate event, and the number of chairs is determined by the circumference for that meeting, it's more likely that they are asking for the number of chairs in the nth meeting, not the total over all meetings.But just to be thorough, let me check both interpretations.First, if it's the number for the nth meeting, which is 75 chairs.Second, if it's the total number of chairs across all meetings up to the 21st, then we need to sum the chairs from meeting 1 to meeting 21.Since each meeting's chairs are equal to the circumference, which is 15 + 3*(k-1) for the kth meeting.So, the total chairs would be the sum from k=1 to k=21 of [15 + 3*(k - 1)].Simplify that:Sum = sum_{k=1 to 21} [15 + 3k - 3] = sum_{k=1 to 21} [12 + 3k]Which is equal to sum_{k=1 to 21} 12 + sum_{k=1 to 21} 3k = 21*12 + 3*sum_{k=1 to 21}kSum of k from 1 to 21 is (21)(22)/2 = 231So, total chairs = 21*12 + 3*231 = 252 + 693 = 945But the problem says "the total number of chairs that can be arranged in a circle by the nth meeting." Hmm, the wording is a bit unclear. It could mean the total number arranged in each meeting up to n, or just the number for the nth meeting.Given that the first part is about the seating capacity exceeding 10,000, which is a single number, and the second part is about the number of chairs arranged in a circle by the nth meeting, it's more likely that it's referring to the number for the nth meeting, not the cumulative total.But let me think again. If it's the total number of chairs arranged in a circle by the nth meeting, that could mean the total number used in all meetings up to n. But in that case, the problem would probably specify "total number of chairs used in all meetings up to the nth meeting."Alternatively, it could be that each meeting uses chairs arranged in a circle, and the total number of chairs is the sum of all chairs used in each meeting up to n.But given that the problem says "arranged in a circle by the nth meeting," it might refer to the number of chairs in the nth meeting's circle.But to be safe, maybe I should calculate both and see which one makes sense.Wait, if n is 21, and the circumference is 75, chairs are 75. If it's the total chairs, it's 945. But 945 is a much larger number, and the problem mentions "the number of chairs should always be an integer," which is satisfied in both cases.But considering the context, the first part is about seating capacity for a single meeting, so the second part is likely about the number of chairs in the nth meeting's circle.Therefore, I think the answer is 75 chairs.But just to make sure, let me check the problem statement again."the total number of chairs that can be arranged in a circle by the nth meeting you identified in the first sub-problem."Hmm, "total number of chairs" could imply the sum, but "arranged in a circle by the nth meeting" might mean the number for the nth meeting. It's a bit ambiguous.Alternatively, maybe it's the number of chairs arranged in a circle for each meeting up to n, but that would be the same as the total chairs.Wait, no, if each meeting has its own circle, then the total number of chairs used in all circles up to the nth meeting would be the sum.But the problem says "arranged in a circle by the nth meeting." So, perhaps it's the number of chairs in the circle for the nth meeting, not the total.Alternatively, maybe it's the number of chairs in the circle for each meeting, but the total number across all meetings.Wait, the problem says "the total number of chairs that can be arranged in a circle by the nth meeting." So, "by the nth meeting," meaning up to that meeting.So, perhaps it's the total number of chairs used in all meetings up to the nth meeting.In that case, it's 945 chairs.But I'm not entirely sure. Let me think about the units.If it's the number of chairs in the nth meeting, it's 75.If it's the total chairs used in all meetings up to n, it's 945.Given that the first part is about the seating capacity for the nth meeting, the second part is likely about the number of chairs in the nth meeting's circle, which is 75.But I'm still a bit uncertain. Maybe I should present both interpretations.Wait, the problem says "the total number of chairs that can be arranged in a circle by the nth meeting." So, "by the nth meeting" could mean up to and including the nth meeting, implying the total number.But in that case, it's 945.Alternatively, if it's the number of chairs in the circle for the nth meeting, it's 75.I think I need to clarify this.Wait, the problem says "the captain decides to arrange the chairs in circular patterns where each chair is 1 unit apart. If the circumference of the circle increases linearly by 3 units every month starting from an initial circumference of 15 units for the first meeting, determine the total number of chairs that can be arranged in a circle by the nth meeting you identified in the first sub-problem."So, "the total number of chairs that can be arranged in a circle by the nth meeting." So, "by the nth meeting" might mean the number of chairs in the circle at the nth meeting.But "total number" is confusing. If it's the number for the nth meeting, it's 75. If it's the total number across all meetings up to n, it's 945.But given that the first part is about the seating capacity for the nth meeting, the second part is likely about the number of chairs in the circle for the nth meeting, so 75.But to be thorough, let me calculate both.Number of chairs in the nth meeting: 15 + 3*(n - 1). For n=21, that's 15 + 60 = 75.Total chairs across all meetings: sum from k=1 to 21 of (15 + 3*(k - 1)) = sum from k=1 to 21 of (12 + 3k) = 21*12 + 3*(21*22)/2 = 252 + 3*231 = 252 + 693 = 945.So, depending on interpretation, it's either 75 or 945.But given the problem's phrasing, I think it's more likely asking for the number of chairs in the nth meeting's circle, which is 75.Therefore, the answers are:1. n = 212. Number of chairs = 75But just to make sure, let me think about the problem again.The first part is about the seating capacity exceeding 10,000, which is a single number. The second part is about the number of chairs arranged in a circle by the nth meeting. Since each meeting has its own circle, the number of chairs would be specific to that meeting, not the total across all meetings.Therefore, I think the answer is 75 chairs.Wait, but the problem says "the total number of chairs that can be arranged in a circle by the nth meeting." So, "total" might imply the sum. Hmm.Alternatively, maybe it's the number of chairs in the circle for the nth meeting, which is 75, and that's the total for that meeting.But the wording is confusing. Maybe I should go with 75, as it's the number for the nth meeting.Alternatively, perhaps the problem is asking for the number of chairs in the circle for the nth meeting, which is 75, and that's the total number for that meeting.I think I'll go with 75.</think>

🧠:<think>Okay, so I have this problem where Alex is trying to secure two business opportunities, A and B. The probabilities given are P(A) = 0.7, P(B) = 0.6, and the joint probability P(A ∩ B) = 0.5. There are two parts to this problem. Let me tackle them one by one.Starting with the first question: Calculate the probability that Alex secures at least one of the two business opportunities.Hmm, okay. So, when they say "at least one," that usually translates to the union of the two events, right? So, the probability of A union B, which is P(A ∪ B). I remember there's a formula for that. It's P(A) + P(B) - P(A ∩ B). Let me write that down.So, P(A ∪ B) = P(A) + P(B) - P(A ∩ B). Plugging in the numbers, that would be 0.7 + 0.6 - 0.5. Let me compute that. 0.7 plus 0.6 is 1.3, and then subtract 0.5 gives me 0.8. So, is the probability 0.8? That seems straightforward.Wait, let me make sure I didn't make a mistake. So, P(A) is 0.7, P(B) is 0.6, and their joint probability is 0.5. So, adding them gives 1.3, but since they overlap by 0.5, we subtract that to avoid double-counting. Yep, that makes sense. So, 0.8 is the probability Alex secures at least one opportunity. Okay, that seems solid.Moving on to the second part: Given that the revenue from securing opportunity A is a random variable X following a normal distribution with a mean of 100,000 and a standard deviation of 15,000, and the revenue from securing opportunity B is a random variable Y following a normal distribution with a mean of 120,000 and a standard deviation of 20,000, determine the expected total revenue E(X + Y) and the variance Var(X + Y) if the revenues are independent.Alright, so first, let's parse this. X is normally distributed with mean μ_X = 100,000 and standard deviation σ_X = 15,000. Similarly, Y is normally distributed with μ_Y = 120,000 and σ_Y = 20,000. They mention that the revenues are independent, which is important because that affects the variance.So, for the expected value, I remember that the expectation of the sum is the sum of the expectations. So, E(X + Y) = E(X) + E(Y). That should be straightforward. Plugging in the numbers, that would be 100,000 + 120,000, which is 220,000. So, the expected total revenue is 220,000.Now, for the variance, Var(X + Y). Since X and Y are independent, the variance of their sum is the sum of their variances. I remember that Var(X + Y) = Var(X) + Var(Y) when X and Y are independent. If they weren't independent, we'd have to consider covariance, but since they are independent, covariance is zero.So, first, let's compute Var(X). The standard deviation is 15,000, so variance is (15,000)^2. Similarly, Var(Y) is (20,000)^2.Calculating Var(X): 15,000 squared is 225,000,000. Var(Y): 20,000 squared is 400,000,000. So, adding them together, 225,000,000 + 400,000,000 equals 625,000,000. Therefore, Var(X + Y) is 625,000,000.Wait, let me double-check that. 15,000 squared is indeed 225,000,000 because 15^2 is 225 and then six zeros. Similarly, 20,000 squared is 400,000,000. Adding them gives 625,000,000. That seems correct.But hold on, the problem mentions that the revenues are independent only if both opportunities are secured separately. Hmm, does that affect anything? Wait, no, because in the second part, they are asking for the expected total revenue and variance when the revenues are independent. So, I think that note is just clarifying the independence condition, but since we're already told to assume independence, we don't have to worry about it in our calculations.So, putting it all together, E(X + Y) is 220,000, and Var(X + Y) is 625,000,000. If needed, we could also compute the standard deviation of X + Y by taking the square root of the variance, but the problem doesn't ask for that.Just to recap, for the first part, we used the inclusion-exclusion principle for probabilities, and for the second part, we used the linearity of expectation and the property of variances for independent variables. Everything seems to check out.Final Answer1. The probability that Alex secures at least one opportunity is boxed{0.8}.2. The expected total revenue is boxed{220000} dollars and the variance is boxed{625000000}.</think>

🧠:<think>Alright, so I have this problem where a UK citizen is arguing that international aid isn't necessary and that the money should be used for domestic issues like healthcare and education instead. The government currently allocates 0.7% of the GDP to international aid, which is £14 billion. There are two sub-problems here.Starting with Sub-problem 1: I need to derive a formula to calculate the amount of GDP allocated to international aid over the next 10 years, assuming the same percentage (0.7%) is allocated each year, and the GDP is growing at an annual rate of 2.5%. Hmm, okay. So, GDP is growing each year, which means the total GDP each year will be higher than the previous year. Since the percentage allocated to aid remains the same, the actual amount in pounds will increase each year as well.I remember that when something grows at a constant rate, you can model it using exponential growth. The formula for the future value with compound growth is:[ text{Future Value} = text{Present Value} times (1 + r)^t ]Where ( r ) is the growth rate and ( t ) is the time in years. In this case, the present value is the current GDP. But wait, actually, the current GDP isn't given directly. Instead, we know that 0.7% of the GDP is £14 billion. So, maybe I can find the current GDP first.Let me denote the current GDP as ( G ). Then:[ 0.007 times G = 14 text{ billion} ]So, solving for ( G ):[ G = frac{14}{0.007} ][ G = 2000 text{ billion pounds} ]Okay, so the current GDP is £2000 billion. Now, each year, the GDP grows by 2.5%, so the GDP in year ( t ) will be:[ G_t = G times (1 + 0.025)^t ]And the international aid each year will be 0.7% of that, so:[ text{Aid}_t = 0.007 times G times (1 + 0.025)^t ]But since ( G = 2000 ), we can plug that in:[ text{Aid}_t = 0.007 times 2000 times (1.025)^t ][ text{Aid}_t = 14 times (1.025)^t ]So, that's the formula for the amount allocated to international aid in year ( t ). To find the total allocation over 10 years, I think we need to sum this from ( t = 0 ) to ( t = 9 ) or ( t = 1 ) to ( t = 10 ). Wait, actually, the current year is year 0, so the next 10 years would be from year 1 to year 10.But the question says "over the next 10 years," so I think it's asking for the total amount allocated each year for the next 10 years, not the cumulative total. Or maybe it's asking for the total? Hmm, the wording is a bit ambiguous. It says "derive a formula to calculate the amount of GDP allocated to international aid over the next 10 years." So, maybe it's the total over 10 years.If that's the case, then we need to sum the aid from year 1 to year 10. So, the total aid ( A ) would be:[ A = sum_{t=1}^{10} 14 times (1.025)^t ]This is a geometric series where the first term ( a = 14 times 1.025 ) and the common ratio ( r = 1.025 ), with 10 terms. The formula for the sum of a geometric series is:[ S_n = a times frac{r^n - 1}{r - 1} ]Plugging in the values:[ S_{10} = 14 times 1.025 times frac{(1.025)^{10} - 1}{1.025 - 1} ][ S_{10} = 14 times 1.025 times frac{(1.025)^{10} - 1}{0.025} ]Alternatively, since ( a = 14 times 1.025 ) and ( r = 1.025 ), we can write it as:[ S_{10} = 14 times frac{(1.025)^{11} - 1.025}{0.025} ]But maybe it's simpler to express it as:[ A = 14 times sum_{t=1}^{10} (1.025)^t ]So, that's the formula. Alternatively, if they just want the formula for the amount each year, it's ( 14 times (1.025)^t ) for each year ( t ).Wait, the question says "derive a formula to calculate the amount of GDP allocated to international aid over the next 10 years." So, perhaps they mean the total amount over 10 years. So, the formula would be the sum of the geometric series as above.Alternatively, if they just want the formula for each year's allocation, it's ( 14 times (1.025)^t ) for year ( t ).But since it's over the next 10 years, I think they want the total. So, the formula would be:[ A = 14 times frac{(1.025)^{10} - 1}{0.025} times 1.025 ]Wait, no. Let me think again. The sum from t=1 to t=10 is:[ S = 14 times (1.025) + 14 times (1.025)^2 + dots + 14 times (1.025)^{10} ]Factor out 14:[ S = 14 times left( (1.025) + (1.025)^2 + dots + (1.025)^{10} right) ]The series inside is a geometric series with first term ( a = 1.025 ) and ratio ( r = 1.025 ), number of terms ( n = 10 ). The sum of this series is:[ S_n = a times frac{r^n - 1}{r - 1} ][ S_n = 1.025 times frac{(1.025)^{10} - 1}{0.025} ]So, the total aid is:[ A = 14 times 1.025 times frac{(1.025)^{10} - 1}{0.025} ]Alternatively, simplifying:[ A = 14 times frac{(1.025)^{11} - 1.025}{0.025} ]But perhaps it's better to leave it in the summation form unless they want a closed-form formula.Alternatively, if they just want the formula for each year's allocation, it's ( 14 times (1.025)^t ) for year ( t ). But since it's over 10 years, I think the total is needed.So, the formula is:[ A = 14 times frac{(1.025)^{10} - 1}{0.025} times 1.025 ]Wait, no, that's not quite right. Let me recast it.The sum from t=1 to t=10 of ( 14 times (1.025)^t ) is equal to 14 times the sum from t=1 to t=10 of ( (1.025)^t ). The sum from t=1 to t=n of ( r^t ) is ( r times frac{r^n - 1}{r - 1} ). So, in this case, ( r = 1.025 ), ( n = 10 ).Thus, the sum is:[ S = 1.025 times frac{(1.025)^{10} - 1}{1.025 - 1} ][ S = 1.025 times frac{(1.025)^{10} - 1}{0.025} ]Therefore, the total aid is:[ A = 14 times S ][ A = 14 times 1.025 times frac{(1.025)^{10} - 1}{0.025} ]Alternatively, simplifying:[ A = frac{14 times 1.025}{0.025} times [(1.025)^{10} - 1] ][ A = frac{14.35}{0.025} times [(1.025)^{10} - 1] ][ A = 574 times [(1.025)^{10} - 1] ]But maybe it's better to leave it in terms of the summation or the closed-form formula as above.So, to summarize, the formula for the total international aid over the next 10 years is:[ A = 14 times frac{(1.025)^{11} - 1.025}{0.025} ]Alternatively, expressed as:[ A = 14 times frac{(1.025)^{10} - 1}{0.025} times 1.025 ]Either way, that's the formula.Moving on to Sub-problem 2: The citizen proposes reallocating the international aid budget equally between healthcare and education. The current healthcare budget is £150 billion and education is £100 billion. We need to calculate the new budgets after reallocation and determine the percentage increase for each.First, the international aid budget is £14 billion. If this is reallocated equally between healthcare and education, each will get half of that, so £7 billion each.So, the new healthcare budget will be £150 billion + £7 billion = £157 billion.The new education budget will be £100 billion + £7 billion = £107 billion.Now, to find the percentage increase for each:For healthcare:Percentage increase = (Increase / Original) × 100%= (7 / 150) × 100%= (0.046666...) × 100%≈ 4.666...%≈ 4.67%For education:Percentage increase = (7 / 100) × 100%= 7%So, healthcare increases by approximately 4.67%, and education increases by 7%.Wait, let me double-check the calculations.Healthcare: 7 / 150 = 0.046666..., which is 4.666...%, so 4.67% when rounded to two decimal places.Education: 7 / 100 = 0.07, which is exactly 7%.Yes, that seems correct.</think>