🧠:<think>Okay, so I have this problem about calculating probabilities for a football team's performance over a season. There are two parts. Let me try to tackle them one by one.Starting with the first part: The team plays 38 games. The probabilities for each game are given as winning with 2/3, drawing with 1/6, and losing with 1/6. I need to find the probability that they win exactly 24, draw 8, and lose 6 games. The hint says to use the multinomial theorem. Hmm, okay, I remember that the multinomial distribution generalizes the binomial distribution for more than two outcomes.So, the formula for the multinomial probability is:P = (n!)/(k1! * k2! * ... * km!) * (p1^k1 * p2^k2 * ... * pm^km)Where n is the total number of trials, ki are the number of outcomes for each category, and pi are the probabilities for each category.In this case, n is 38. The categories are wins, draws, and losses. So, k1 is 24 (wins), k2 is 8 (draws), and k3 is 6 (losses). The probabilities are p1 = 2/3, p2 = 1/6, p3 = 1/6.So plugging these into the formula:First, calculate the factorial part: 38! divided by (24! * 8! * 6!). That should give me the multinomial coefficient.Then, multiply that by (2/3)^24 * (1/6)^8 * (1/6)^6.Wait, hold on, (1/6)^8 * (1/6)^6 is the same as (1/6)^(8+6) = (1/6)^14. So, combining the constants, it's (2/3)^24 * (1/6)^14.Let me write that down step by step.First, compute the multinomial coefficient:38! / (24! * 8! * 6!) I might need to calculate this value. But factorials can get really big, so maybe I can compute it using logarithms or approximate it? Wait, maybe I can compute it step by step.Alternatively, I can use the formula for combinations. But since it's a multinomial coefficient, it's the number of ways to arrange 24 wins, 8 draws, and 6 losses in 38 games.Alternatively, I can think of it as:38 choose 24, then 14 choose 8, then 6 choose 6.Which is the same as:C(38,24) * C(14,8) * C(6,6)Which is the same as 38! / (24!14!) * 14! / (8!6!) * 1 / (6!0!) Simplifying, the 14! cancels out, and 6! cancels with the denominator, so we get 38! / (24!8!6!), which is the same as before.So, I can compute this coefficient. Let me see if I can compute it numerically.But factorials of 38, 24, 8, 6 are huge numbers. Maybe I can compute the multinomial coefficient using logarithms or approximate it? Alternatively, maybe I can use a calculator or software, but since I'm doing this manually, perhaps I can find a way to compute it step by step.Alternatively, maybe I can compute the multinomial coefficient as:(38 × 37 × ... × 15) / (24!) for the first part, but that might not help.Wait, maybe I can compute it using the multiplicative formula for multinomial coefficients.The multinomial coefficient can be calculated as:n! / (k1! k2! k3!) = product from i=1 to n of (n - sum_{j=1}^{i-1} kj)! / (ki! )But that might be complicated.Alternatively, maybe I can compute it as:C(38,24) * C(14,8) * C(6,6)Compute each combination separately.C(38,24) = 38! / (24!14!) C(14,8) = 14! / (8!6!) C(6,6) = 1So, let's compute C(38,24):C(38,24) = C(38,14) because C(n,k) = C(n, n -k). So, C(38,14) is easier to compute.C(38,14) = 38! / (14!24!) But again, factorials are too big. Maybe I can compute it using the multiplicative formula:C(n,k) = n*(n-1)*...*(n -k +1)/(k*(k-1)*...*1)So, C(38,14) = (38×37×36×35×34×33×32×31×30×29×28×27×26×25) / (14×13×12×11×10×9×8×7×6×5×4×3×2×1)That's a lot, but maybe I can compute it step by step.Alternatively, maybe I can use logarithms to compute the multinomial coefficient.But perhaps it's better to use the formula for the multinomial probability directly, even if I can't compute the exact value, but maybe I can express it in terms of factorials and exponents.Wait, the question doesn't specify whether to compute the exact numerical value or just express it in terms of factorials and exponents. Hmm, probably, since the numbers are large, it's acceptable to leave it in terms of factorials and exponents, but maybe the problem expects a numerical answer.Alternatively, perhaps I can use Stirling's approximation for factorials, but that might complicate things.Wait, maybe I can compute it using logarithms.Let me try to compute the logarithm of the multinomial coefficient:ln(38! / (24!8!6!)) = ln(38!) - ln(24!) - ln(8!) - ln(6!)Using Stirling's approximation: ln(n!) ≈ n ln n - n + (ln(2πn))/2But that might be too involved.Alternatively, I can use a calculator for factorials, but since I don't have one, maybe I can look up approximate values.Wait, perhaps I can use the fact that 38! is about 3.5568743 × 10^44, 24! is about 6.204484 × 10^23, 8! is 40320, and 6! is 720.So, 38! / (24!8!6!) ≈ (3.5568743 × 10^44) / (6.204484 × 10^23 * 4.032 × 10^4 * 7.2 × 10^2)Let me compute the denominator first:6.204484 × 10^23 * 4.032 × 10^4 = 6.204484 × 4.032 × 10^(23+4) = approx 24.99 × 10^27Then multiply by 7.2 × 10^2: 24.99 × 7.2 × 10^(27+2) = approx 180.7 × 10^29So, denominator is approx 1.807 × 10^31Now, numerator is 3.5568743 × 10^44So, 3.5568743 × 10^44 / 1.807 × 10^31 ≈ (3.5568743 / 1.807) × 10^(44-31) ≈ 1.967 × 10^13So, the multinomial coefficient is approximately 1.967 × 10^13Now, the probability part is (2/3)^24 * (1/6)^14Let me compute that.First, (2/3)^24: 2^24 / 3^242^10 is 1024, so 2^20 is (1024)^2 = 1,048,576, 2^24 = 1,048,576 * 16 = 16,777,2163^24: 3^12 is 531,441, so 3^24 = (531,441)^2 ≈ 281,421,543,681So, (2/3)^24 ≈ 16,777,216 / 281,421,543,681 ≈ 0.0000596Similarly, (1/6)^14: 1 / 6^146^14: 6^10 is 60,466,176, 6^14 = 60,466,176 * 6^4 = 60,466,176 * 1296 ≈ 78,364,164, 60,466,176 * 1000 = 60,466,176,000; 60,466,176 * 200 = 12,093,235,200; 60,466,176 * 96 = approx 5,799, 60,466,176 * 96 = 60,466,176 * 100 - 60,466,176 *4 = 6,046,617,600 - 241,864,704 = 5,804,752,896. So total 60,466,176,000 + 12,093,235,200 + 5,804,752,896 ≈ 78,364,164,096So, 6^14 ≈ 78,364,164,096Thus, (1/6)^14 ≈ 1 / 78,364,164,096 ≈ 1.275 × 10^-11Now, multiplying the two probabilities:(2/3)^24 * (1/6)^14 ≈ 0.0000596 * 1.275 × 10^-11 ≈ 7.58 × 10^-16Now, multiply this by the multinomial coefficient:1.967 × 10^13 * 7.58 × 10^-16 ≈ (1.967 * 7.58) × 10^(13-16) ≈ 14.89 × 10^-3 ≈ 0.01489So, approximately 1.489% chance.Wait, that seems low, but considering the high number of games and the specific number of wins, draws, and losses, it might make sense.Alternatively, maybe I made a mistake in the calculations. Let me check.Wait, when I computed (2/3)^24, I got approximately 0.0000596, which is 5.96 × 10^-5. Then (1/6)^14 is 1.275 × 10^-11. Multiplying them gives 5.96 × 10^-5 * 1.275 × 10^-11 = 7.58 × 10^-16, which is correct.Then, the multinomial coefficient was approx 1.967 × 10^13. Multiplying 1.967e13 * 7.58e-16 = approx 1.967 * 7.58 = approx 14.89, and 10^(13-16) = 10^-3, so 14.89e-3 = 0.01489, or 1.489%.That seems reasonable.So, the probability is approximately 1.489%.Now, moving on to the second part: Comparing this probability with that of their main rivals. There are three other teams, each with probabilities of winning, drawing, and losing each game as 1/2, 1/4, and 1/4 respectively. I need to find the probability that at least one of these teams wins exactly 24, draws 8, and loses 6 games.So, this is a probability of "at least one" out of three teams. The formula for the probability of at least one success in multiple independent trials is 1 - probability that none of them succeed.So, first, I need to find the probability that a single rival team wins exactly 24, draws 8, and loses 6 games. Then, since the teams are independent, the probability that none of them do this is [1 - p]^3, where p is the probability for one team. Then, the probability that at least one does it is 1 - [1 - p]^3.So, first, compute p for one rival team.Using the same multinomial formula as before, but with different probabilities.So, for one rival team, n=38, k1=24, k2=8, k3=6, p1=1/2, p2=1/4, p3=1/4.So, the probability is:P = (38!)/(24!8!6!) * (1/2)^24 * (1/4)^8 * (1/4)^6Simplify the exponents:(1/2)^24 * (1/4)^8 * (1/4)^6 = (1/2)^24 * (1/4)^(8+6) = (1/2)^24 * (1/4)^14But 1/4 is (1/2)^2, so (1/4)^14 = (1/2)^28Thus, the entire probability becomes:(38!)/(24!8!6!) * (1/2)^24 * (1/2)^28 = (38!)/(24!8!6!) * (1/2)^(24+28) = (38!)/(24!8!6!) * (1/2)^52So, P = (38!)/(24!8!6!) * (1/2)^52Wait, but we already computed (38!)/(24!8!6!) earlier as approximately 1.967 × 10^13.So, P ≈ 1.967 × 10^13 * (1/2)^52Compute (1/2)^52: 1 / (2^52). 2^10 is 1024, so 2^20 ≈ 1.048576 × 10^6, 2^30 ≈ 1.073741824 × 10^9, 2^40 ≈ 1.099511627776 × 10^12, 2^50 ≈ 1.125899906842624 × 10^15, so 2^52 = 2^50 * 4 ≈ 4.503599627370496 × 10^15Thus, (1/2)^52 ≈ 2.220446049250313 × 10^-16So, P ≈ 1.967 × 10^13 * 2.220446049250313 × 10^-16 ≈ (1.967 * 2.220446) × 10^(13-16) ≈ 4.367 × 10^-3 ≈ 0.004367, or 0.4367%.So, the probability for one rival team is approximately 0.4367%.Now, the probability that none of the three teams achieve this is [1 - 0.004367]^3.Compute 1 - 0.004367 = 0.995633Then, 0.995633^3 ≈ ?Compute 0.995633^3:First, compute 0.995633 * 0.995633:≈ (1 - 0.004367)^2 ≈ 1 - 2*0.004367 + (0.004367)^2 ≈ 1 - 0.008734 + 0.000019 ≈ 0.991285Then, multiply by 0.995633:0.991285 * 0.995633 ≈ ?Compute 0.991285 * 0.995633:≈ (1 - 0.008715) * (1 - 0.004367) ≈ 1 - 0.008715 - 0.004367 + 0.000038 ≈ 1 - 0.013082 + 0.000038 ≈ 0.986956So, approximately 0.986956Thus, the probability that none of the three teams achieve the exact 24-8-6 record is approximately 0.986956.Therefore, the probability that at least one team does achieve it is 1 - 0.986956 ≈ 0.013044, or 1.3044%.Wait, but let me check the calculations again because approximating can lead to errors.Alternatively, compute [1 - p]^3 where p = 0.004367.So, 1 - p = 0.995633Compute 0.995633^3:First, compute 0.995633 * 0.995633:Let me compute it more accurately.0.995633 * 0.995633:= (1 - 0.004367)^2= 1 - 2*0.004367 + (0.004367)^2= 1 - 0.008734 + 0.00001907≈ 0.99128507Now, multiply this by 0.995633:0.99128507 * 0.995633Let me compute this as:= (0.99128507) * (0.995633)≈ 0.99128507 * (1 - 0.004367)= 0.99128507 - 0.99128507 * 0.004367Compute 0.99128507 * 0.004367 ≈ 0.004325So, ≈ 0.99128507 - 0.004325 ≈ 0.986960Thus, [1 - p]^3 ≈ 0.986960Therefore, the probability of at least one team achieving the record is 1 - 0.986960 ≈ 0.01304, or 1.304%.So, approximately 1.304%.Comparing this with the original team's probability of about 1.489%, the rival teams have a slightly lower chance of at least one of them achieving the exact 24-8-6 record.Wait, but the original team's probability was 1.489%, and the rivals' combined probability is 1.304%, so the original team is slightly more likely.But the question was to compare the probabilities, not necessarily to find which is higher, but just to compute the rival's probability.So, summarizing:1. The probability for the original team is approximately 1.489%.2. The probability for at least one of the three rival teams is approximately 1.304%.Therefore, the original team has a slightly higher chance of achieving exactly 24 wins, 8 draws, and 6 losses compared to their rivals.But wait, in the second part, the question is to compute the probability for the rival teams, not to compare. So, the answer is approximately 1.304%.But let me check if I did everything correctly.Wait, in the first part, I approximated the multinomial coefficient as 1.967 × 10^13, and then multiplied by (2/3)^24 * (1/6)^14 ≈ 7.58 × 10^-16, giving 0.01489 or 1.489%.In the second part, the same multinomial coefficient is used, but with different probabilities: (1/2)^24 * (1/4)^14 = (1/2)^52 ≈ 2.22 × 10^-16, so 1.967 × 10^13 * 2.22 × 10^-16 ≈ 4.367 × 10^-3 or 0.4367% per team.Then, for three teams, the probability that none achieve it is (1 - 0.004367)^3 ≈ 0.98696, so the probability that at least one does is 1 - 0.98696 ≈ 0.01304 or 1.304%.Yes, that seems correct.So, the final answers are approximately 1.489% for the original team and 1.304% for the rivals.But maybe I should express them more precisely, perhaps using exact fractions or more accurate decimal places.Alternatively, perhaps I can compute the exact value using logarithms or more precise approximations.But given the time constraints, I think these approximations are sufficient.So, to recap:1. The probability for the original team is approximately 1.489%.2. The probability for at least one rival team is approximately 1.304%.Therefore, the original team has a slightly higher chance of achieving the exact 24-8-6 record compared to their rivals.But wait, the question only asks for the probabilities, not a comparison, so I just need to provide the numerical values.Alternatively, maybe I can express the probabilities in terms of exact expressions without approximating.For the first part:P = (38!)/(24!8!6!) * (2/3)^24 * (1/6)^14For the second part:P = 1 - [1 - (38!/(24!8!6!)*(1/2)^24*(1/4)^14)]^3But perhaps the problem expects numerical answers.Alternatively, maybe I can compute the exact values using more precise calculations.But given the complexity, I think the approximate values I found are acceptable.So, final answers:1. Approximately 1.489%2. Approximately 1.304%</think>

🧠:<think>Okay, so I have these two problems about a college football coach's defensive strategy. Let me try to figure them out step by step.Starting with the first problem: The probability of the opponent scoring in a single play is p, which is 0.3 initially. The coach's strategy reduces this probability by a factor of k every quarter, and k is 2. I need to find the probability of the opponent scoring in the fourth quarter.Hmm, so every quarter, the probability is reduced by a factor of k. That means each quarter, the probability becomes p divided by k. So, starting with p = 0.3 in the first quarter, then in the second quarter it's 0.3 / 2, third quarter it's (0.3 / 2) / 2, and so on.Wait, so each quarter, the probability is multiplied by 1/k, right? Because reducing by a factor of k is the same as multiplying by 1/k. So, for each subsequent quarter, the probability is p * (1/k)^(quarter number - 1). So, in the first quarter, it's p, second quarter p*(1/k), third p*(1/k)^2, and fourth p*(1/k)^3.Let me write that down:First quarter: p = 0.3Second quarter: p * (1/2) = 0.3 * 0.5 = 0.15Third quarter: 0.3 * (1/2)^2 = 0.3 * 0.25 = 0.075Fourth quarter: 0.3 * (1/2)^3 = 0.3 * 0.125 = 0.0375So, the probability in the fourth quarter is 0.0375. That seems right. Let me double-check: starting at 0.3, each quarter it halves. So, 0.3, 0.15, 0.075, 0.0375. Yep, that's four quarters with each subsequent quarter's probability halved.Okay, so that should be the answer for the first part.Moving on to the second problem: The coach wants to calculate the expected number of points prevented over a game of 4 quarters. Each quarter has 15 plays, and the opponent scores an average of 3 points per successful play. I need to derive a general formula and then compute the numerical result using p = 0.3 and k = 2.Alright, so let's break this down. First, for each quarter, we can calculate the probability of scoring, then find the expected number of successful plays, multiply by points per play, and then sum over all quarters. The points prevented would be the total points the opponent would have scored without the coach's strategy minus the points they actually score with the strategy.Wait, actually, the coach's strategy reduces the probability, so the expected points prevented would be the difference between the expected points without the strategy and with the strategy.But let me think step by step.First, without any strategy, the probability of scoring on each play is p. So, for each play, the expected points are 3 * p. Since each quarter has 15 plays, the expected points per quarter without any strategy would be 15 * 3 * p = 45p. Over 4 quarters, that would be 4 * 45p = 180p.But with the coach's strategy, each quarter the probability is reduced by a factor of k. So, in the first quarter, it's p, second p/k, third p/k², fourth p/k³.Therefore, for each quarter, the expected points would be 15 * 3 * (p / k^(quarter - 1)). So, per quarter, it's 45 * (p / k^(quarter - 1)).Therefore, the total expected points with the strategy would be the sum over quarters 1 to 4 of 45 * (p / k^(quarter - 1)).So, the total expected points with strategy:Total_with = 45p + 45p/k + 45p/k² + 45p/k³Which can be factored as 45p(1 + 1/k + 1/k² + 1/k³)Similarly, the total without strategy is 180p.Therefore, the expected points prevented would be Total_without - Total_with = 180p - 45p(1 + 1/k + 1/k² + 1/k³)Simplify that:180p - 45p(1 + 1/k + 1/k² + 1/k³) = 45p(4 - (1 + 1/k + 1/k² + 1/k³))Alternatively, factor 45p:45p[4 - (1 + 1/k + 1/k² + 1/k³)]Alternatively, we can write it as 45p times the sum from n=0 to 3 of (1/k)^n subtracted from 4.But perhaps it's better to write it as 45p times (4 - sum_{n=0}^{3} (1/k)^n )But let me see, maybe we can write it as 45p times (1 - (1 + 1/k + 1/k² + 1/k³)/4 )Wait, no, the original expression is 180p - 45p(1 + 1/k + 1/k² + 1/k³). Since 180p is 45p*4, so factoring 45p, we get 45p*(4 - (1 + 1/k + 1/k² + 1/k³)).So, that's the general formula.Now, plugging in p = 0.3 and k = 2.First, compute the sum inside the parentheses:1 + 1/2 + 1/4 + 1/8Compute each term:1 = 11/2 = 0.51/4 = 0.251/8 = 0.125Adding them up: 1 + 0.5 = 1.5; 1.5 + 0.25 = 1.75; 1.75 + 0.125 = 1.875So, the sum is 1.875Therefore, 4 - 1.875 = 2.125So, the expression becomes 45p * 2.125Compute 45 * 0.3 first: 45 * 0.3 = 13.5Then, 13.5 * 2.125Compute 13.5 * 2 = 2713.5 * 0.125 = 1.6875So, total is 27 + 1.6875 = 28.6875So, the expected number of points prevented is 28.6875.Wait, let me verify that again.Wait, 45p is 45 * 0.3 = 13.5Then, 4 - sum = 4 - 1.875 = 2.125So, 13.5 * 2.125Compute 13.5 * 2 = 2713.5 * 0.125 = 1.6875Adding them together: 27 + 1.6875 = 28.6875Yes, that seems correct.Alternatively, 2.125 is equal to 17/8, so 13.5 * 17/813.5 is 27/2, so 27/2 * 17/8 = (27*17)/(16) = 459/16 = 28.6875Yes, same result.So, the numerical result is 28.6875 points prevented.Alternatively, as a fraction, 459/16 is 28 and 11/16, which is 28.6875.So, I think that's the answer.But let me think again if I interpreted the problem correctly.The coach measures the effectiveness by calculating the expected number of points prevented over the entire game.Each quarter has 15 plays, opponent scores 3 points per successful play.So, for each quarter, without strategy, expected points would be 15 * 3 * p = 45pWith strategy, each quarter has probability p / k^(quarter -1), so expected points per quarter is 45 * (p / k^(quarter -1))Therefore, total with strategy is sum_{n=0}^{3} 45p / k^n = 45p * sum_{n=0}^{3} (1/k)^nTotal without strategy is 4 * 45p = 180pTherefore, points prevented is 180p - 45p * sum_{n=0}^{3} (1/k)^nWhich is 45p*(4 - sum_{n=0}^{3} (1/k)^n )Yes, that's correct.So, plugging in p=0.3, k=2, we get 45*0.3*(4 - (1 + 0.5 + 0.25 + 0.125)) = 13.5*(4 - 1.875) = 13.5*2.125 = 28.6875So, 28.6875 points prevented.I think that's solid.Final Answer1. The probability of the opponent scoring in the fourth quarter is boxed{0.0375}.2. The expected number of points prevented over the entire game is boxed{28.6875}.</think>

🧠:<think>Okay, so I have this problem about a hotel manager analyzing booking patterns to optimize room allocation and pricing. The hotel has 120 rooms divided into three categories: Standard, Deluxe, and Suite. The probabilities for booking each category are given as 0.5, 0.3, and 0.2 respectively. There are two sub-problems here. Let me tackle them one by one.Starting with Sub-problem 1: If the probability of a room being booked is independent of other rooms being booked, what is the expected number of rooms booked in each category when the hotel is at full capacity?Hmm, so the hotel has 120 rooms in total, and each room can be booked independently with the given probabilities. Since the booking of one room doesn't affect another, this sounds like a binomial distribution problem for each category. But wait, actually, since each room is categorized into one of the three types, maybe it's more like a multinomial distribution?But wait, the problem says the probability of a room being booked is independent of others. So each room has a certain probability of being booked, regardless of the category. Hmm, no, actually, the probabilities given are for each category. So perhaps each room is assigned to a category with those probabilities, and then each room has a certain probability of being booked.Wait, I'm a bit confused. Let me read the problem again."The probability of a room being booked is independent of other rooms being booked." So, each room has a certain probability of being booked, and whether one room is booked doesn't affect another. But the room categories have different probabilities. So, perhaps each room is categorized as Standard, Deluxe, or Suite with probabilities 0.5, 0.3, and 0.2 respectively, and then each room has a certain probability of being booked, independent of others.Wait, but the problem doesn't specify the probability of a room being booked, only the probability of booking each category. Hmm, maybe I need to interpret it differently.Wait, perhaps each room is in a category, and the probability that a room in that category is booked is given. So, for example, each Standard room has a 0.5 probability of being booked, each Deluxe room has a 0.3 probability, and each Suite has a 0.2 probability. And these are independent across rooms.But the hotel has 120 rooms in total, divided into the three categories. So how many rooms are in each category? The problem doesn't specify, so maybe we need to assume that the number of rooms in each category is proportional to their booking probabilities? Or is it that the hotel has 120 rooms, each of which is categorized as Standard, Deluxe, or Suite with probabilities 0.5, 0.3, and 0.2, and then each room has an independent probability of being booked.Wait, that might be it. So, each room is assigned a category with probability 0.5 for Standard, 0.3 for Deluxe, and 0.2 for Suite. Then, each room has a probability of being booked, but the problem doesn't specify that probability. Hmm, that's confusing.Wait, maybe the probabilities given (0.5, 0.3, 0.2) are the probabilities that a room is booked in each category. So, for example, the probability that a Standard room is booked is 0.5, Deluxe is 0.3, and Suite is 0.2. But then, how many rooms are in each category? The problem says the hotel has 120 rooms divided into three categories, but doesn't specify how many in each. Hmm.Wait, maybe the hotel has 120 rooms, and each room is equally likely to be in any category? No, that doesn't make sense because the probabilities are different. Alternatively, perhaps the number of rooms in each category is determined by the probabilities. So, for example, 0.5 of 120 rooms are Standard, 0.3 are Deluxe, and 0.2 are Suite. Let me check: 0.5*120=60, 0.3*120=36, 0.2*120=24. So, 60 Standard, 36 Deluxe, and 24 Suite rooms.If that's the case, then each room in the Standard category has a probability of being booked, but the problem doesn't specify that. Wait, the problem says "the probability of a room being booked is independent of other rooms being booked." So, perhaps each room in each category has a certain probability of being booked, but the problem only gives the probabilities for each category, not per room.Wait, maybe the 0.5, 0.3, and 0.2 are the probabilities that a room in that category is booked. So, for example, each Standard room has a 0.5 chance of being booked, each Deluxe room has a 0.3 chance, and each Suite has a 0.2 chance. Then, since the hotel is at full capacity, meaning all rooms are booked? Wait, no, full capacity would mean all rooms are booked, but the problem says "when the hotel is at full capacity," but the probabilities are given for booking each category. Hmm, maybe I'm overcomplicating.Wait, perhaps the hotel is at full capacity, meaning all 120 rooms are booked, and we need to find the expected number of rooms in each category. But that doesn't make sense because if all rooms are booked, then the number in each category would just be the number of rooms in that category. But the problem doesn't specify how many rooms are in each category, only the probabilities.Wait, maybe the hotel has 120 rooms, and each room is categorized as Standard, Deluxe, or Suite with probabilities 0.5, 0.3, and 0.2 respectively. Then, each room has a probability of being booked, but the problem doesn't specify that. Hmm, this is confusing.Wait, perhaps the 0.5, 0.3, and 0.2 are the probabilities that a room is booked in each category, regardless of the number of rooms in each category. So, for example, the probability that a Standard room is booked is 0.5, but the number of Standard rooms is not given. Hmm, but the hotel has 120 rooms in total.Wait, maybe the hotel has 120 rooms, and each room is assigned to a category with probabilities 0.5, 0.3, and 0.2. So, the expected number of rooms in each category would be 60, 36, and 24 respectively. Then, each room in each category has a probability of being booked, but the problem doesn't specify that. Hmm, but the problem says "the probability of a room being booked is independent of other rooms being booked." So, maybe each room has a certain probability of being booked, but the categories have different probabilities.Wait, I think I need to clarify. The problem states: "the probability of a room being booked is independent of other rooms being booked." So, each room has a certain probability of being booked, and this is independent of other rooms. The probabilities given are for each room category: Standard, Deluxe, and Suite. So, perhaps each room in the Standard category has a 0.5 probability of being booked, each Deluxe room has a 0.3 probability, and each Suite has a 0.2 probability.But then, how many rooms are in each category? The problem says the hotel has 120 rooms divided into three categories, but doesn't specify how many in each. So, maybe we need to assume that the number of rooms in each category is proportional to their booking probabilities? Or is it that the hotel has 120 rooms, each of which is categorized as Standard, Deluxe, or Suite with probabilities 0.5, 0.3, and 0.2, and then each room has an independent probability of being booked.Wait, perhaps the number of rooms in each category is determined by the probabilities. So, 0.5 of 120 is 60 Standard, 0.3 is 36 Deluxe, and 0.2 is 24 Suite. So, the hotel has 60 Standard rooms, 36 Deluxe, and 24 Suite rooms.Then, each room in each category has a probability of being booked. But the problem doesn't specify the booking probability for each room. Wait, the problem says "the probability of a room being booked is independent of other rooms being booked," but it doesn't give the probability. Hmm, maybe I'm missing something.Wait, perhaps the probabilities given (0.5, 0.3, 0.2) are the probabilities that a room is booked in each category. So, for example, each Standard room has a 0.5 chance of being booked, each Deluxe has 0.3, and each Suite has 0.2. Then, since the hotel is at full capacity, meaning all rooms are booked? Wait, no, because if each room has a probability of being booked, then the expected number of booked rooms would be less than the total.Wait, the problem says "when the hotel is at full capacity." So, does that mean all 120 rooms are booked? If so, then the number of rooms in each category would just be the number of rooms in that category, which we don't know. But the problem gives probabilities for booking each category, not the number of rooms in each.Wait, maybe the hotel has 120 rooms, and each room is categorized as Standard, Deluxe, or Suite with probabilities 0.5, 0.3, and 0.2. So, the expected number of rooms in each category is 60, 36, and 24. Then, each room in each category has a probability of being booked, but the problem doesn't specify that. Hmm, this is getting too tangled.Wait, perhaps the problem is simpler. The hotel has 120 rooms, and each room is booked with a probability depending on its category. The categories have probabilities 0.5, 0.3, and 0.2. So, for each room, the probability that it's a Standard room and booked is 0.5, Deluxe is 0.3, and Suite is 0.2. But that doesn't make sense because each room can only be one category.Wait, maybe the probability that a room is booked is 0.5 for Standard, 0.3 for Deluxe, and 0.2 for Suite. So, for each room, depending on its category, it has a certain probability of being booked. Then, the expected number of booked rooms in each category would be the number of rooms in that category multiplied by the booking probability.But we don't know the number of rooms in each category. The problem says the hotel has 120 rooms divided into three categories, but doesn't specify how many in each. So, maybe we need to assume that the number of rooms in each category is proportional to their booking probabilities? So, 0.5:0.3:0.2 ratio, which simplifies to 5:3:2. So, total parts = 10, so 120 rooms divided as 60, 36, 24.Yes, that makes sense. So, the hotel has 60 Standard rooms, 36 Deluxe, and 24 Suite rooms. Then, each room in each category has a probability of being booked equal to their category's probability. So, each Standard room has a 0.5 chance, each Deluxe has 0.3, and each Suite has 0.2.Therefore, the expected number of booked rooms in each category would be:Standard: 60 * 0.5 = 30Deluxe: 36 * 0.3 = 10.8Suite: 24 * 0.2 = 4.8So, the expected number of rooms booked in each category would be 30, 10.8, and 4.8 respectively.Wait, but the problem says "when the hotel is at full capacity." Does that mean all rooms are booked? If so, then the number of rooms in each category would just be 60, 36, and 24, but that contradicts the idea of expected number because if it's at full capacity, all rooms are booked. So, maybe I'm misinterpreting "full capacity."Wait, perhaps "full capacity" refers to the hotel's maximum number of rooms, which is 120, but the booking probabilities are given for each category. So, the expected number of booked rooms in each category would be the number of rooms in that category multiplied by the booking probability.But since we don't know the number of rooms in each category, we can't calculate the expected number. Unless, as I thought earlier, the number of rooms in each category is proportional to their booking probabilities. So, 60, 36, 24.Therefore, the expected number of booked rooms would be 30, 10.8, and 4.8.But let me double-check. The problem says "the probability of a room being booked is independent of other rooms being booked." So, each room has a certain probability of being booked, independent of others. The probabilities for each category are given as 0.5, 0.3, 0.2. So, perhaps each room is categorized as Standard, Deluxe, or Suite with probabilities 0.5, 0.3, 0.2, and then each room has a probability of being booked, but the problem doesn't specify that. Hmm, this is confusing.Wait, maybe the 0.5, 0.3, 0.2 are the probabilities that a room is booked, given its category. So, for example, if a room is Standard, it has a 0.5 chance of being booked, and similarly for others. Then, the number of rooms in each category is 60, 36, 24 as before. So, the expected number of booked rooms would be 30, 10.8, 4.8.Yes, that seems to make sense. So, for Sub-problem 1, the expected number of rooms booked in each category would be 30 Standard, 10.8 Deluxe, and 4.8 Suite.Now, moving on to Sub-problem 2: To maximize revenue, the manager decides to dynamically adjust the pricing. The revenue function for each category is given by:- Standard: ( R_S = 100 + 2x )- Deluxe: ( R_D = 200 + 3x )- Suite: ( R_SU = 300 + 5x )where ( x ) represents the number of rooms booked in that category. Assuming the expected number of bookings from Sub-problem 1 holds true, calculate the total expected revenue for the hotel.So, we have the expected number of bookings for each category: 30, 10.8, and 4.8. We need to plug these into the revenue functions.But wait, the revenue functions are given as ( R_S = 100 + 2x ), etc. So, for each category, the revenue is a linear function of the number of rooms booked. So, for Standard, it's 100 plus 2 times the number of rooms booked. Similarly for the others.So, total revenue would be the sum of revenues from each category.Therefore, total expected revenue ( R ) would be:( R = R_S + R_D + R_SU )Substituting the expected number of bookings:( R_S = 100 + 2*30 = 100 + 60 = 160 )( R_D = 200 + 3*10.8 = 200 + 32.4 = 232.4 )( R_SU = 300 + 5*4.8 = 300 + 24 = 324 )So, total revenue:( R = 160 + 232.4 + 324 = 160 + 232.4 = 392.4; 392.4 + 324 = 716.4 )So, total expected revenue is 716.4.Wait, but let me check the calculations again.For Standard: 100 + 2*30 = 100 + 60 = 160. Correct.Deluxe: 200 + 3*10.8. Let's calculate 3*10.8: 10.8*3 = 32.4. So, 200 + 32.4 = 232.4. Correct.Suite: 300 + 5*4.8. 5*4.8 = 24. So, 300 + 24 = 324. Correct.Adding them up: 160 + 232.4 = 392.4; 392.4 + 324 = 716.4.So, total expected revenue is 716.4.But wait, the revenue functions are given per category, and x is the number of rooms booked in that category. So, if the expected number of rooms booked is 30, 10.8, and 4.8, then plugging these into the revenue functions gives the expected revenue.Yes, that seems correct.So, to summarize:Sub-problem 1: Expected number of rooms booked in each category is 30 Standard, 10.8 Deluxe, and 4.8 Suite.Sub-problem 2: Total expected revenue is 716.4.But let me make sure about Sub-problem 1 again. The hotel has 120 rooms, divided into three categories with probabilities 0.5, 0.3, 0.2. So, number of rooms in each category is 60, 36, 24. Then, each room in each category has a probability of being booked equal to the category's probability. So, expected booked rooms are 60*0.5=30, 36*0.3=10.8, 24*0.2=4.8. Yes, that seems correct.Alternatively, if the probabilities 0.5, 0.3, 0.2 are the probabilities that a room is booked, regardless of category, then the expected number of booked rooms would be 120*(0.5+0.3+0.2)/3, but that doesn't make sense because the probabilities are per category.No, the correct approach is to assume that each room is in a category with probabilities 0.5, 0.3, 0.2, and then each room in a category has a probability of being booked equal to the category's probability. So, the expected number of booked rooms in each category is number of rooms in category * booking probability.Therefore, the answers are as above.</think>

🧠:<think>Alright, so I've got two problems here related to animation techniques for a movie. The first one is about calculating the arc length of a parametric curve, and the second is about analyzing the stability of a crowd simulation model. Let me tackle them one by one.Starting with the first problem: the director wants the arc length of the path from t=0 to t=1. The parametric equations are given as r(t) = (f(t), g(t), h(t)), where f(t) = t³ - 3t, g(t) = 2 sin(t), and h(t) = ln(t + 2). I remember that the formula for the arc length of a parametric curve from t=a to t=b is the integral from a to b of the square root of [ (dx/dt)² + (dy/dt)² + (dz/dt)² ] dt. So, I need to compute the derivatives of f(t), g(t), and h(t) with respect to t, square each, add them up, take the square root, and then integrate from 0 to 1.Let me compute each derivative:First, f(t) = t³ - 3t. The derivative f’(t) is 3t² - 3.Next, g(t) = 2 sin(t). The derivative g’(t) is 2 cos(t).Lastly, h(t) = ln(t + 2). The derivative h’(t) is 1/(t + 2).So, the integrand becomes sqrt[ (3t² - 3)² + (2 cos t)² + (1/(t + 2))² ].Now, I need to compute the integral from 0 to 1 of this expression. Hmm, this looks a bit complicated. I wonder if it's possible to simplify or if I need to use numerical methods.Let me check if the integrand can be simplified. Let's expand (3t² - 3)²:(3t² - 3)² = 9t⁴ - 18t² + 9.(2 cos t)² = 4 cos² t.(1/(t + 2))² = 1/(t + 2)².So, the integrand is sqrt[9t⁴ - 18t² + 9 + 4 cos² t + 1/(t + 2)²].This doesn't seem to simplify nicely. Maybe I can factor out some terms or see if it's a perfect square, but I don't think so. So, perhaps I need to compute this integral numerically.But wait, the problem says "calculate the arc length," so maybe it expects an exact expression or a numerical approximation? Since it's a definite integral from 0 to 1, and the integrand is complicated, I think a numerical approximation is the way to go.I can use methods like Simpson's rule or the trapezoidal rule to approximate the integral. Alternatively, since I might not have a calculator here, maybe I can set up the integral and leave it in terms of an integral expression? But the problem says "calculate," so probably a numerical value is expected.Alternatively, maybe I can use substitution or some other technique. Let me see.Looking at the integrand: sqrt[9t⁴ - 18t² + 9 + 4 cos² t + 1/(t + 2)²]. Hmm, that's quite messy. I don't see an obvious substitution that would simplify this. So, yeah, I think numerical integration is the way to go.But since I don't have a calculator, maybe I can approximate it using a few terms or use a series expansion? Hmm, that might be too time-consuming and not very accurate.Alternatively, perhaps I can note that the integral is complicated and state that it requires numerical methods, but maybe the problem expects me to set up the integral correctly. Let me check the problem statement again.It says, "Calculate the arc length of the curve over this interval." So, maybe just setting up the integral is sufficient, but I think they want the numerical value. Hmm.Wait, maybe I can use a substitution for part of the integrand. Let me see:Looking at 9t⁴ - 18t² + 9, that's 9(t⁴ - 2t² + 1) = 9(t² - 1)². Oh, that's a nice simplification!So, 9t⁴ - 18t² + 9 = 9(t² - 1)².So, the integrand becomes sqrt[9(t² - 1)² + 4 cos² t + 1/(t + 2)²].That's still complicated, but maybe we can write it as sqrt[9(t² - 1)² + 4 cos² t + 1/(t + 2)²].Hmm, perhaps not much better, but at least it's a bit simpler.So, the integral becomes ∫₀¹ sqrt[9(t² - 1)² + 4 cos² t + 1/(t + 2)²] dt.I think this is as simplified as it gets. So, to compute this, I would need to use numerical integration. Since I don't have a calculator, maybe I can use a few points to approximate it.Alternatively, perhaps I can use a substitution for t. Let me see:Let me consider the substitution u = t² - 1. Then du = 2t dt. Hmm, but that doesn't seem to help because of the other terms.Alternatively, maybe a substitution for the entire expression under the square root, but I don't see an obvious one.Alternatively, perhaps I can split the integral into parts or use a series expansion for each term.Wait, maybe I can approximate each term separately and then combine them. Let's see:First, 9(t² - 1)²: At t=0, this is 9(0 - 1)² = 9. At t=1, this is 9(1 - 1)² = 0.4 cos² t: At t=0, this is 4(1) = 4. At t=1, this is 4 cos²(1) ≈ 4*(0.5403)² ≈ 4*0.2919 ≈ 1.1676.1/(t + 2)²: At t=0, this is 1/4 = 0.25. At t=1, this is 1/9 ≈ 0.1111.So, at t=0, the integrand is sqrt(9 + 4 + 0.25) = sqrt(13.25) ≈ 3.6401.At t=1, the integrand is sqrt(0 + 1.1676 + 0.1111) = sqrt(1.2787) ≈ 1.1308.So, the function starts at about 3.64 and decreases to about 1.13 over the interval [0,1]. It's a decreasing function, but how does it behave in between?Maybe I can approximate it using the trapezoidal rule with a few intervals. Let's try with two intervals: t=0, t=0.5, t=1.First, compute the integrand at t=0.5:9(t² - 1)² = 9((0.25) - 1)² = 9(-0.75)² = 9*0.5625 = 5.0625.4 cos²(0.5) ≈ 4*(0.8776)^2 ≈ 4*0.7699 ≈ 3.0796.1/(0.5 + 2)^2 = 1/(2.5)^2 = 1/6.25 = 0.16.So, the integrand at t=0.5 is sqrt(5.0625 + 3.0796 + 0.16) = sqrt(8.3021) ≈ 2.8813.So, now we have three points:t=0: 3.6401t=0.5: 2.8813t=1: 1.1308Using the trapezoidal rule with two intervals (each of width 0.5):Integral ≈ (0.5)/2 * [f(0) + 2f(0.5) + f(1)] = 0.25 * [3.6401 + 2*2.8813 + 1.1308] = 0.25 * [3.6401 + 5.7626 + 1.1308] = 0.25 * 10.5335 ≈ 2.6334.But wait, the trapezoidal rule with two intervals is actually (h/2)[f(a) + 2f(a+h) + f(b)], where h is the step size. So, h=0.5, so it's (0.5)/2 * [f(0) + 2f(0.5) + f(1)] = 0.25 * [3.6401 + 5.7626 + 1.1308] ≈ 0.25 * 10.5335 ≈ 2.6334.But this is just an approximation. Maybe I can do better with more intervals. Let's try with four intervals, so h=0.25.Compute f(t) at t=0, 0.25, 0.5, 0.75, 1.We already have t=0, 0.5, 1.Compute at t=0.25:9(t² -1)^2 = 9((0.0625) -1)^2 = 9*(-0.9375)^2 = 9*(0.8789) ≈ 7.9101.4 cos²(0.25) ≈ 4*(0.9689)^2 ≈ 4*0.9388 ≈ 3.7552.1/(0.25 + 2)^2 = 1/(2.25)^2 ≈ 1/5.0625 ≈ 0.1975.So, integrand ≈ sqrt(7.9101 + 3.7552 + 0.1975) ≈ sqrt(11.8628) ≈ 3.445.At t=0.75:9(t² -1)^2 = 9((0.5625) -1)^2 = 9*(-0.4375)^2 = 9*(0.1914) ≈ 1.7226.4 cos²(0.75) ≈ 4*(0.7317)^2 ≈ 4*0.5353 ≈ 2.1412.1/(0.75 + 2)^2 = 1/(2.75)^2 ≈ 1/7.5625 ≈ 0.1322.So, integrand ≈ sqrt(1.7226 + 2.1412 + 0.1322) ≈ sqrt(3.996) ≈ 1.999.So, now we have:t=0: 3.6401t=0.25: 3.445t=0.5: 2.8813t=0.75: 1.999t=1: 1.1308Using the trapezoidal rule with four intervals (h=0.25):Integral ≈ (0.25)/2 * [f(0) + 2f(0.25) + 2f(0.5) + 2f(0.75) + f(1)] = 0.125 * [3.6401 + 2*3.445 + 2*2.8813 + 2*1.999 + 1.1308]Compute the sum inside:3.6401 + 2*3.445 = 3.6401 + 6.89 = 10.530110.5301 + 2*2.8813 = 10.5301 + 5.7626 = 16.292716.2927 + 2*1.999 = 16.2927 + 3.998 = 20.290720.2907 + 1.1308 = 21.4215Multiply by 0.125: 21.4215 * 0.125 ≈ 2.6777.So, with four intervals, the approximation is about 2.6777.Comparing with the two-interval approximation of 2.6334, it's slightly higher. The function is decreasing, so the trapezoidal rule tends to overestimate when the function is concave down. Maybe the actual value is somewhere around 2.65 to 2.7.Alternatively, maybe using Simpson's rule would give a better approximation. Simpson's rule for n=4 intervals (which is even, so n=4 is okay):Simpson's rule formula: (h/3)[f(a) + 4f(a+h) + 2f(a+2h) + 4f(a+3h) + f(b)]So, h=0.25, so:Integral ≈ (0.25)/3 * [f(0) + 4f(0.25) + 2f(0.5) + 4f(0.75) + f(1)]Compute the sum:f(0) = 3.64014f(0.25) = 4*3.445 ≈ 13.782f(0.5) = 2*2.8813 ≈ 5.76264f(0.75) = 4*1.999 ≈ 7.996f(1) = 1.1308Total sum: 3.6401 + 13.78 + 5.7626 + 7.996 + 1.1308 ≈3.6401 + 13.78 = 17.420117.4201 + 5.7626 = 23.182723.1827 + 7.996 = 31.178731.1787 + 1.1308 ≈ 32.3095Multiply by (0.25)/3 ≈ 0.083333:32.3095 * 0.083333 ≈ 2.6925.So, Simpson's rule gives about 2.6925.Comparing with trapezoidal: 2.6777 vs 2.6925. The difference is about 0.015, so maybe the actual value is around 2.68 or so.Alternatively, maybe I can use more intervals for better accuracy, but since I'm doing this manually, it's time-consuming. Alternatively, perhaps I can accept that the integral is approximately 2.68 units.But wait, let me check if I did the calculations correctly.At t=0.25, f(t) ≈ 3.445At t=0.5, f(t) ≈ 2.8813At t=0.75, f(t) ≈ 1.999At t=1, f(t) ≈ 1.1308So, Simpson's rule with four intervals gives ≈2.6925.Alternatively, maybe I can use the average of trapezoidal and Simpson's for better approximation, but I'm not sure.Alternatively, perhaps I can use a calculator for a better approximation, but since I don't have one, I'll proceed with Simpson's result of approximately 2.69.But wait, let me check if I made any calculation errors.Wait, when I computed f(0.25):9(t² -1)^2 = 9*(0.0625 -1)^2 = 9*(-0.9375)^2 = 9*(0.8789) ≈ 7.9101.4 cos²(0.25): cos(0.25) ≈ 0.9689, so squared is ≈0.9388, multiplied by 4 is ≈3.7552.1/(0.25 + 2)^2 = 1/(2.25)^2 ≈ 0.1975.So, total under sqrt: 7.9101 + 3.7552 + 0.1975 ≈ 11.8628, sqrt ≈ 3.445. Correct.At t=0.75:9(t² -1)^2 = 9*(0.5625 -1)^2 = 9*(-0.4375)^2 = 9*(0.1914) ≈1.7226.4 cos²(0.75): cos(0.75) ≈0.7317, squared ≈0.5353, multiplied by 4 ≈2.1412.1/(0.75 + 2)^2 = 1/(2.75)^2 ≈0.1322.Total under sqrt: 1.7226 + 2.1412 + 0.1322 ≈3.996, sqrt≈1.999. Correct.So, the calculations seem correct.Therefore, the arc length is approximately 2.69 units.But wait, let me check if I can get a better approximation by using more intervals. Let's try with n=8 intervals, h=0.125.But this would involve computing f(t) at t=0, 0.125, 0.25, 0.375, 0.5, 0.625, 0.75, 0.875, 1. That's a lot of calculations, but let me try a few more points to see if the trend continues.Compute f(t) at t=0.125:9(t² -1)^2 = 9*(0.015625 -1)^2 = 9*(-0.984375)^2 ≈9*(0.969) ≈8.721.4 cos²(0.125): cos(0.125) ≈0.9922, squared ≈0.9845, multiplied by 4 ≈3.938.1/(0.125 + 2)^2 = 1/(2.125)^2 ≈1/4.5156 ≈0.2214.Total under sqrt: 8.721 + 3.938 + 0.2214 ≈12.8804, sqrt≈3.589.At t=0.375:9(t² -1)^2 = 9*(0.140625 -1)^2 = 9*(-0.859375)^2 ≈9*(0.7383) ≈6.6447.4 cos²(0.375): cos(0.375) ≈0.9306, squared ≈0.866, multiplied by 4 ≈3.464.1/(0.375 + 2)^2 = 1/(2.375)^2 ≈1/5.6406 ≈0.1773.Total under sqrt: 6.6447 + 3.464 + 0.1773 ≈10.286, sqrt≈3.207.At t=0.625:9(t² -1)^2 = 9*(0.390625 -1)^2 = 9*(-0.609375)^2 ≈9*(0.3712) ≈3.3408.4 cos²(0.625): cos(0.625) ≈0.8109, squared ≈0.6576, multiplied by 4 ≈2.6304.1/(0.625 + 2)^2 = 1/(2.625)^2 ≈1/6.8906 ≈0.1451.Total under sqrt: 3.3408 + 2.6304 + 0.1451 ≈6.1163, sqrt≈2.473.At t=0.875:9(t² -1)^2 = 9*(0.765625 -1)^2 = 9*(-0.234375)^2 ≈9*(0.0549) ≈0.4941.4 cos²(0.875): cos(0.875) ≈0.6414, squared ≈0.4114, multiplied by 4 ≈1.6456.1/(0.875 + 2)^2 = 1/(2.875)^2 ≈1/8.2656 ≈0.121.Total under sqrt: 0.4941 + 1.6456 + 0.121 ≈2.2607, sqrt≈1.5036.So, now we have:t=0: 3.6401t=0.125: 3.589t=0.25: 3.445t=0.375: 3.207t=0.5: 2.8813t=0.625: 2.473t=0.75: 1.999t=0.875: 1.5036t=1: 1.1308Using Simpson's rule with n=8 intervals (h=0.125):Simpson's rule formula: (h/3)[f(a) + 4f(a+h) + 2f(a+2h) + 4f(a+3h) + 2f(a+4h) + 4f(a+5h) + 2f(a+6h) + 4f(a+7h) + f(b)]So, plugging in the values:= (0.125/3)[3.6401 + 4*3.589 + 2*3.445 + 4*3.207 + 2*2.8813 + 4*2.473 + 2*1.999 + 4*1.5036 + 1.1308]Compute each term:4*3.589 ≈14.3562*3.445 ≈6.894*3.207 ≈12.8282*2.8813 ≈5.76264*2.473 ≈9.8922*1.999 ≈3.9984*1.5036 ≈6.0144Now, sum all these up with f(a) and f(b):3.6401 + 14.356 + 6.89 + 12.828 + 5.7626 + 9.892 + 3.998 + 6.0144 + 1.1308Let me add them step by step:Start with 3.6401.+14.356 = 17.9961+6.89 = 24.8861+12.828 = 37.7141+5.7626 = 43.4767+9.892 = 53.3687+3.998 = 57.3667+6.0144 = 63.3811+1.1308 = 64.5119Multiply by (0.125)/3 ≈0.0416667:64.5119 * 0.0416667 ≈2.68799.So, Simpson's rule with n=8 gives approximately 2.688.Comparing with n=4: 2.6925, n=8: 2.688. The values are converging around 2.69.Given that, I think the arc length is approximately 2.69 units.But to be more precise, maybe I can average the two Simpson's results: (2.6925 + 2.688)/2 ≈2.69025.Alternatively, maybe the exact value is around 2.69.But wait, let me check if I can use a calculator for a better approximation. Alternatively, perhaps I can use a series expansion for the integrand.Alternatively, maybe I can use a substitution to make the integral more manageable. Let me think.Wait, the integrand is sqrt[9(t² -1)^2 + 4 cos² t + 1/(t + 2)^2]. Hmm, not sure.Alternatively, perhaps I can approximate each term separately and then combine them.But I think the numerical approximation is the way to go here. So, based on Simpson's rule with n=8, the arc length is approximately 2.69.So, I'll go with that.Now, moving on to the second problem: analyzing the stability of the Cucker-Smale model as n approaches infinity, assuming all agents start with the same initial velocity v0.The Cucker-Smale model is a system of differential equations where each agent's velocity is influenced by the velocities of other agents, with a weighting function that depends on the distance between them.The given system is:dv_i/dt = (1/n) sum_{j=1}^n [ψ(||x_j - x_i||) (v_j - v_i)] / (1 + ||x_j - x_i||²)where ψ(r) = 1/(1 + r²).All agents start with the same initial velocity v0, so v_i(0) = v0 for all i.We need to analyze the stability as n approaches infinity.First, let me recall that in the Cucker-Smale model, agents tend to align their velocities over time, leading to flocking behavior. The stability would depend on whether the system converges to a state where all velocities are equal (i.e., the system reaches consensus).Given that all agents start with the same velocity, v_i(0) = v0, we can check if the velocities remain equal over time.Let me consider the case where all v_i(t) = v(t) for all i. Then, the equation becomes:dv/dt = (1/n) sum_{j=1}^n [ψ(||x_j - x_i||) (v(t) - v(t))] / (1 + ||x_j - x_i||²) = 0.So, if all velocities are equal, the derivative is zero, meaning the system is in equilibrium.But we need to check if this equilibrium is stable. That is, if small perturbations from the equilibrium state decay over time, leading the system back to the equilibrium.Alternatively, if the system is such that the velocities tend to align, then the equilibrium is stable.Given that the interaction strength is modulated by ψ(r) = 1/(1 + r²), which decreases as r increases, meaning that agents farther apart have less influence on each other.As n approaches infinity, the system becomes a mean-field model, where the influence is averaged over all agents.In the mean-field limit, the system can be described by a kinetic equation, but perhaps for stability analysis, we can linearize the system around the equilibrium.Assuming that all velocities are equal, let's consider a small perturbation: v_i(t) = v(t) + ε_i(t), where ε_i(t) is small.Then, the equation becomes:d(v + ε_i)/dt = (1/n) sum_{j=1}^n [ψ(||x_j - x_i||) (v + ε_j - v - ε_i)] / (1 + ||x_j - x_i||²)Simplify:dv/dt + dε_i/dt = (1/n) sum_{j=1}^n [ψ(||x_j - x_i||) (ε_j - ε_i)] / (1 + ||x_j - x_i||²)But since v(t) is the same for all i, and if we assume that v(t) is constant (since dv/dt = 0 in equilibrium), then dv/dt = 0, so:dε_i/dt = (1/n) sum_{j=1}^n [ψ(||x_j - x_i||) (ε_j - ε_i)] / (1 + ||x_j - x_i||²)This is a linear system for the perturbations ε_i.To analyze stability, we can look at the eigenvalues of the operator defining this system. If all eigenvalues have negative real parts, the perturbations decay, and the equilibrium is stable.However, in the mean-field limit as n→∞, the system can be approximated by an integral over the distribution of agents. Let me denote the distribution function as f(x, t), representing the density of agents at position x at time t.Then, the velocity perturbation equation becomes:dε_i/dt = ∫ [ψ(||x' - x||) (ε(x') - ε(x))] / (1 + ||x' - x||²) f(x', t) dx'This is an integral equation, and its stability can be analyzed by considering Fourier modes.Assume that ε(x) is a Fourier mode: ε(x) = e^{ik·x}. Then, the integral becomes:dε/dt = ∫ [ψ(|x'|) (e^{ik·x'} - e^{ik·x})] / (1 + |x'|²) f(x', t) dx'But this might be too abstract. Alternatively, perhaps we can consider the linearized system around the equilibrium and check if the perturbations decay.In the Cucker-Smale model, it's known that under certain conditions, the system converges to a state where all velocities are aligned. The key factor is the interaction kernel ψ(r). In our case, ψ(r) = 1/(1 + r²), which is integrable and decays at infinity.In the mean-field limit, the system is expected to converge to a state where all velocities are equal, provided that the interaction kernel is such that the system is asymptotically flocking.Given that ψ(r) is integrable and positive, it's likely that the system is stable, and the perturbations decay, leading to consensus.Therefore, as n approaches infinity, the system is stable, and all agents' velocities converge to the same value, maintaining the initial velocity v0 in this case.Wait, but in our case, all agents start with the same velocity, so the perturbations are zero. Therefore, the system remains in equilibrium.But if we consider small perturbations, the system should return to the equilibrium if it's stable.Given that the interaction kernel ψ(r) is such that the influence between agents decreases with distance, but is still integrable, the system should exhibit alignment behavior, leading to stability.Therefore, the system is stable as n approaches infinity.So, summarizing:1. The arc length is approximately 2.69 units.2. The system is stable as n approaches infinity, with all agents' velocities converging to the same value.But wait, in the second problem, the director wants to model the interaction dynamics using the Cucker-Smale model. The question is to analyze the stability as n→∞ under the assumption that all agents start with the same initial velocity.Given that, if all agents start with the same velocity, then the system is already in equilibrium, and any perturbations would decay, leading to stability.Therefore, the system is stable.So, the final answers are:1. The arc length is approximately 2.69.2. The system is stable as n approaches infinity.</think>