🧠:<think>Okay, so I have two questions here about statistics and behavioral psychology experiments. Let me try to figure them out step by step.Starting with the first question: It's about deriving a 95% confidence interval for the mean difference in anxiety levels before and after a CBT program. The data is paired, so I think that means each participant's anxiety level is measured both before and after the program. That makes sense because it's the same group of people, so their anxiety levels are related.They gave me the mean anxiety levels: 65 before and 55 after. The standard deviations are 10 before and 8 after. The sample size is 50 participants. Since the data is normally distributed, that should help with the calculations.I remember that for paired data, we calculate the differences for each participant and then find the mean and standard deviation of those differences. Then, we can construct a confidence interval using the t-distribution or z-distribution. Since the sample size is 50, which is pretty large, maybe we can use the z-distribution? But I'm not sure if the standard deviation of the differences is known. Wait, they gave us the standard deviations before and after, but not the standard deviation of the differences. Hmm.So, maybe I need to calculate the standard deviation of the differences first. Let me denote the difference as ( D = A_{text{after}} - A_{text{before}} ). The mean difference ( bar{D} ) would be 55 - 65 = -10. So, on average, anxiety levels decreased by 10 points.Now, to find the standard deviation of D, I think we need the standard deviations of both measurements and the covariance or correlation between them. But wait, they didn't give us the correlation coefficient. Hmm, that complicates things. Without knowing the correlation, I can't compute the exact standard deviation of the differences. Maybe they expect me to assume that the differences have a standard deviation that can be calculated from the given standard deviations?Wait, no, that's not correct. The standard deviation of the differences isn't just the difference of the standard deviations. It's actually calculated using the formula:( sigma_D = sqrt{sigma_{text{before}}^2 + sigma_{text{after}}^2 - 2 rho sigma_{text{before}} sigma_{text{after}}} )Where ( rho ) is the correlation coefficient between the two measurements. But since we don't have ( rho ), maybe we can't compute it? That seems like a problem.Wait, maybe I'm overcomplicating it. Since the data is paired and we have the same participants, perhaps the standard deviation of the differences is given or can be inferred? Wait, no, they didn't provide it. Hmm.Alternatively, maybe they just want me to use the standard deviations of the before and after separately, but that doesn't make sense for the confidence interval of the mean difference. I think I need the standard deviation of the differences.Wait, perhaps the question assumes that the standard deviation of the differences is known? Or maybe they just want me to use the standard deviations provided and treat it as a difference in means with independent samples? But no, the data is paired, so it's not independent.Wait, maybe I can calculate the standard error of the mean difference without knowing the correlation? Let me think. If I don't know the correlation, I can't compute the exact standard deviation of D. But maybe in the absence of information, we can assume that the differences have a standard deviation that's the square root of the sum of variances? That would be:( sigma_D = sqrt{10^2 + 8^2} = sqrt{100 + 64} = sqrt{164} approx 12.8 )But that's only if the two variables are independent, which they are not because they are paired. So that would overestimate the standard deviation. Hmm.Alternatively, if the correlation is positive, which it probably is because higher anxiety before might lead to higher anxiety after, but actually, since the mean decreased, maybe the correlation is negative? I'm not sure.Wait, maybe the question expects me to treat the standard deviation of the differences as the standard error? No, that doesn't make sense.Wait, perhaps they just want me to use the standard deviations of the before and after as if they were independent, but that's not correct for paired data.Wait, maybe I'm missing something. The formula for the standard error of the mean difference in paired data is ( frac{s_D}{sqrt{n}} ), where ( s_D ) is the standard deviation of the differences. But since we don't have ( s_D ), we can't compute it. So maybe the question is missing some data? Or maybe I'm supposed to assume that the standard deviation of the differences is the same as the standard deviation of the before or after? That doesn't seem right.Wait, maybe the standard deviation of the differences can be approximated by the average of the two standard deviations? So (10 + 8)/2 = 9? That seems arbitrary.Alternatively, maybe the question expects me to use the standard deviations of the before and after to compute the standard error for the difference in means, treating them as independent. But that would be:( SE = sqrt{frac{10^2}{50} + frac{8^2}{50}} = sqrt{frac{100 + 64}{50}} = sqrt{frac{164}{50}} approx sqrt{3.28} approx 1.81 )But that's for independent samples. Since the data is paired, this isn't appropriate. So I'm stuck because without the standard deviation of the differences, I can't compute the confidence interval.Wait, maybe the question assumes that the standard deviation of the differences is known? Let me check the question again."Assuming the anxiety levels before and after the program are paired data, derive the 95% confidence interval for the mean difference in anxiety levels before and after the CBT program."They gave the standard deviations before and after, but not the standard deviation of the differences. Hmm. Maybe I need to calculate it using the formula for paired data, but without the correlation, I can't. Maybe they expect me to assume that the differences have a standard deviation that's the square root of the sum of variances? But that's only if they're independent, which they aren't.Wait, perhaps the question is expecting me to use the standard deviations of the before and after as if they were the standard deviation of the differences? That doesn't make sense because the standard deviation of the differences is not the same as the standard deviations of the individual measurements.Wait, maybe I'm overcomplicating it. Let me think about what I know. For paired data, the confidence interval for the mean difference is calculated as:( bar{D} pm t_{alpha/2, n-1} times frac{s_D}{sqrt{n}} )Where ( bar{D} ) is the mean difference, ( t_{alpha/2, n-1} ) is the t-score for the desired confidence level with degrees of freedom ( n-1 ), and ( s_D ) is the standard deviation of the differences.But since we don't have ( s_D ), we can't compute it. So unless they provided it, I can't proceed. Wait, maybe they did provide it indirectly? Let me check the question again.No, they only gave the standard deviations before and after, not the standard deviation of the differences. Hmm. Maybe the question expects me to assume that the standard deviation of the differences is the same as the standard deviation of the before or after? That seems incorrect.Alternatively, maybe they expect me to calculate the standard deviation of the differences using the formula for paired data, but without the correlation, I can't. So perhaps the question is missing some information? Or maybe I'm supposed to make an assumption here.Wait, maybe the standard deviation of the differences can be calculated if we assume that the covariance is zero? But that would mean the variables are uncorrelated, which isn't necessarily true for paired data.Alternatively, maybe the question expects me to use the standard deviations of the before and after to compute the standard error for the difference in means, treating them as independent, even though they are paired. That would be incorrect, but maybe that's what they want.So, if I proceed that way, the standard error would be:( SE = sqrt{frac{10^2}{50} + frac{8^2}{50}} = sqrt{frac{100 + 64}{50}} = sqrt{frac{164}{50}} approx 1.81 )Then, the mean difference is -10. For a 95% confidence interval, the z-score is approximately 1.96 (since n=50 is large, we can use z instead of t). So the confidence interval would be:( -10 pm 1.96 times 1.81 )Calculating that:1.96 * 1.81 ≈ 3.55So the interval would be approximately (-13.55, -6.45)But wait, this is treating the data as independent, which they are not. So this might not be the correct approach. However, since the question didn't provide the standard deviation of the differences, maybe this is the intended method.Alternatively, if we treat it as paired data, we need the standard deviation of the differences. Since we don't have that, perhaps we can't compute it. But maybe the question expects us to use the standard deviations of the before and after as if they were the standard deviation of the differences. That would be incorrect, but let's see.If we take the standard deviation of the differences as, say, 10 (the before standard deviation), then:SE = 10 / sqrt(50) ≈ 1.414Then, the confidence interval would be:-10 ± 1.96 * 1.414 ≈ -10 ± 2.77So (-12.77, -7.23)But that's assuming the standard deviation of the differences is 10, which isn't correct.Alternatively, if we take the average of the two standard deviations: (10 + 8)/2 = 9Then SE = 9 / sqrt(50) ≈ 1.272CI: -10 ± 1.96 * 1.272 ≈ -10 ± 2.49So (-12.49, -7.51)But again, this is an assumption.Wait, maybe the question expects me to use the standard deviation of the differences as the square root of the sum of variances, assuming independence, even though they are paired. So:s_D = sqrt(10^2 + 8^2) = sqrt(164) ≈ 12.8Then SE = 12.8 / sqrt(50) ≈ 1.81CI: -10 ± 1.96 * 1.81 ≈ (-13.55, -6.45)But this is the same as treating them as independent, which is not correct for paired data.Hmm. I'm stuck because without the standard deviation of the differences, I can't compute the exact confidence interval. Maybe the question expects me to use the standard deviations of the before and after as if they were the standard deviation of the differences, but that's not accurate.Wait, maybe the question is referring to the standard deviation of the differences as the standard error, but that's not correct either.Alternatively, maybe the question is expecting me to use the standard deviation of the before and after to calculate the standard error for the difference in means, treating them as independent, even though they are paired. That would be incorrect, but perhaps that's what they want.Given that, I'll proceed with that method, even though it's not the correct approach for paired data.So, mean difference is -10.Standard error is sqrt[(10^2 + 8^2)/50] = sqrt(164/50) ≈ 1.81Using z-score of 1.96 for 95% CI.CI: -10 ± 1.96 * 1.81 ≈ -10 ± 3.55So the 95% CI is approximately (-13.55, -6.45)But I'm not confident about this because it's treating paired data as independent, which isn't correct. However, without the standard deviation of the differences, I can't do it properly.Wait, maybe the question expects me to calculate the standard deviation of the differences using the formula for paired data, but without the correlation, I can't. So perhaps the question is missing some information, or I'm missing something.Alternatively, maybe the standard deviation of the differences is given by the standard deviation of the before minus the standard deviation of the after? That doesn't make sense because standard deviations don't subtract like that.Wait, another thought: maybe the standard deviation of the differences can be approximated by the square root of the average of the variances? So sqrt[(10^2 + 8^2)/2] = sqrt(164/2) = sqrt(82) ≈ 9.06Then SE = 9.06 / sqrt(50) ≈ 1.28CI: -10 ± 1.96 * 1.28 ≈ -10 ± 2.51So (-12.51, -7.49)But again, this is an assumption.I think I'm overcomplicating this. Maybe the question expects me to treat the data as independent and use the standard error as sqrt[(10^2 + 8^2)/50], which is approximately 1.81, leading to a CI of (-13.55, -6.45). But I'm not sure.Alternatively, maybe the standard deviation of the differences is given by the standard deviation of the before, since the after is a reduction. But that's not correct.Wait, perhaps the standard deviation of the differences is the same as the standard deviation of the before or after? That doesn't make sense because the differences could have a different variance.I think I need to make an assumption here. Since the question didn't provide the standard deviation of the differences, maybe they expect me to use the standard deviations of the before and after to compute the standard error for the difference in means, treating them as independent. Even though it's paired data, perhaps that's the intended approach.So, proceeding with that, the 95% CI would be approximately (-13.55, -6.45).But I'm not confident about this. I think the correct approach requires the standard deviation of the differences, which isn't provided. So maybe the question is missing some data, or I'm missing something.Wait, another thought: maybe the standard deviation of the differences can be calculated if we know the correlation between the before and after scores. But since we don't have that, we can't compute it. So perhaps the question is expecting me to assume that the correlation is zero, which would make the standard deviation of the differences sqrt(10^2 + 8^2) ≈ 12.8, leading to SE ≈ 1.81 and CI ≈ (-13.55, -6.45).But that's a big assumption. I don't know if that's what they want.Alternatively, maybe the standard deviation of the differences is the same as the standard deviation of the before or after. If I take it as 10, then SE ≈ 1.414, CI ≈ (-12.77, -7.23). If I take it as 8, SE ≈ 1.131, CI ≈ (-12.29, -7.71).But without knowing the actual standard deviation of the differences, I can't compute it accurately.Wait, maybe the question expects me to use the standard deviation of the differences as the average of the two standard deviations, which is 9, leading to SE ≈ 1.272, CI ≈ (-12.49, -7.51).But again, this is an assumption.I think I need to proceed with the information given, even if it's not perfect. So, I'll calculate the standard error assuming independence, even though it's paired data, because I don't have the standard deviation of the differences.So, SE = sqrt[(10^2 + 8^2)/50] ≈ 1.81CI: -10 ± 1.96 * 1.81 ≈ (-13.55, -6.45)I'll go with that, but I'm aware that it's not the correct approach for paired data.Now, moving on to the second question: The psychologist wants to model the relationship between the reduction in anxiety levels (ΔA) and the number of therapy sessions attended (S) using a linear regression model. The equation is given as ΔA = β0 + β1S + ε.They want to know how to use the least squares method to estimate β0 and β1, and the conditions under which these estimates are unbiased.Okay, least squares method. I remember that in linear regression, the least squares estimates minimize the sum of squared residuals. So, the goal is to find the values of β0 and β1 that make the sum of (ΔA_i - (β0 + β1S_i))^2 as small as possible.Mathematically, the estimates are found by solving the normal equations. The formulas for β1 and β0 are:β1 = Σ[(S_i - S̄)(ΔA_i - ΔĀ)] / Σ[(S_i - S̄)^2]β0 = ΔĀ - β1S̄Where S̄ is the mean of S, and ΔĀ is the mean of ΔA.So, the psychologist would calculate the means of S and ΔA, then compute the covariance between S and ΔA divided by the variance of S to get β1, and then subtract β1 times the mean of S from the mean of ΔA to get β0.As for the conditions for unbiased estimates, I remember that the Gauss-Markov theorem states that the least squares estimators are unbiased and have the minimum variance among all linear unbiased estimators under certain conditions. These conditions are:1. Linearity: The model is linear in parameters.2. Randomness: The error terms have zero mean and are uncorrelated with the regressors.3. No multicollinearity: The regressors are not perfectly correlated with each other (which isn't an issue here since there's only one regressor).4. Homoscedasticity: The error terms have constant variance.5. No autocorrelation: The error terms are uncorrelated with each other.But more specifically, for unbiasedness, the key assumptions are:- The model is correctly specified (no omitted variables or incorrect functional form).- The error terms have zero mean.- The error terms are uncorrelated with the regressors (exogeneity).- No perfect multicollinearity.So, in this case, the psychologist needs to ensure that the relationship between ΔA and S is linear, that the error terms have zero mean, that the number of sessions (S) is not correlated with the error term (i.e., any omitted variables that affect ΔA are not correlated with S), and that there's no perfect multicollinearity (which isn't an issue here since there's only one predictor).If these conditions hold, then the least squares estimates of β0 and β1 will be unbiased.So, summarizing:1. For the confidence interval, I had to make an assumption about the standard deviation of the differences because it wasn't provided. I ended up using the standard error calculated from the standard deviations of the before and after as if they were independent, leading to a CI of approximately (-13.55, -6.45). But I'm aware that this might not be the correct approach for paired data.2. For the regression, the least squares method involves minimizing the sum of squared residuals, leading to formulas for β0 and β1 based on means, covariance, and variance. The estimates are unbiased under the Gauss-Markov assumptions, primarily correct model specification, zero mean errors, exogeneity of the regressor, and no multicollinearity.I think that's about it. I hope I didn't make any mistakes in my reasoning, especially with the first part because I wasn't sure about the standard deviation of the differences.</think>

🧠:<think>Alright, so I have this problem about a filmmaker creating a documentary series. There are two sub-problems here. Let me try to tackle them one by one.Starting with Sub-problem 1. The filmmaker has 10 hours of interviews and 5 hours of archival footage. Each interview can be edited down to a minimum of 15 minutes, and each piece of archival footage can be edited down to a minimum of 10 minutes. She wants to create 6 episodes, each lasting 45 minutes, with each episode containing at least 20% archival footage.First, I need to set up the linear programming problem. Let me define the variables:For each episode ( i ) (where ( i = 1, 2, ldots, 6 )):- ( x_i ) = duration of interviews used in episode ( i ) (in minutes)- ( y_i ) = duration of archival footage used in episode ( i ) (in minutes)Each episode must be 45 minutes long, so the first constraint is:[ x_i + y_i = 45 quad text{for each } i ]Each episode must have at least 20% archival footage. Since 20% of 45 minutes is 9 minutes, we have:[ y_i geq 9 quad text{for each } i ]Also, each interview segment must be at least 15 minutes, so:[ x_i geq 15 quad text{for each } i ]Similarly, each archival footage segment must be at least 10 minutes, so:[ y_i geq 10 quad text{for each } i ]But wait, since each episode is 45 minutes, and ( y_i geq 9 ), but also ( y_i geq 10 ). So the stricter constraint is ( y_i geq 10 ). Similarly, ( x_i geq 15 ).Now, the total interviews used across all episodes can't exceed 10 hours, which is 600 minutes:[ sum_{i=1}^{6} x_i leq 600 ]Similarly, the total archival footage used can't exceed 5 hours, which is 300 minutes:[ sum_{i=1}^{6} y_i leq 300 ]We need to make sure that all variables are non-negative, but since we have minimums, that's already covered.So, summarizing the constraints:1. For each episode ( i ): - ( x_i + y_i = 45 ) - ( x_i geq 15 ) - ( y_i geq 10 )2. Total constraints: - ( sum_{i=1}^{6} x_i leq 600 ) - ( sum_{i=1}^{6} y_i leq 300 )But wait, since each episode is exactly 45 minutes, the total duration across all episodes is ( 6 times 45 = 270 ) minutes. However, the total interviews available are 600 minutes and total archival footage is 300 minutes. So, actually, the total interviews used will be ( sum x_i ) and total archival footage used will be ( sum y_i ). But since each episode is 45 minutes, ( sum x_i + sum y_i = 6 times 45 = 270 ). But the total available is 600 + 300 = 900 minutes, which is way more than 270. So, the constraints on total interviews and archival footage are actually not binding because she only needs 270 minutes total, but she has 900. Wait, that can't be right.Wait, no. Wait, she has 10 hours of interviews and 5 hours of archival footage. So, 10 hours is 600 minutes, 5 hours is 300 minutes. But she's making 6 episodes, each 45 minutes, so total content needed is 6*45=270 minutes. So, she only needs 270 minutes of content, but she has 600+300=900 minutes available. So, actually, the constraints on total interviews and archival footage are not binding because she can use only 270 minutes in total, but she has 900. So, the only constraints are per episode.Wait, but the problem says she wants to optimize the use of this content under certain constraints. So, maybe she wants to use as much as possible? Or is it that she must use all the content? Wait, no, the problem doesn't specify that she has to use all the content. It just says she has access to 10 hours of interviews and 5 hours of archival footage, and she wants to create a series of 6 episodes, each 45 minutes, with each episode containing at least 20% archival footage.So, the total content used will be 6*45=270 minutes, which is less than the total available 900 minutes. So, the constraints on total interviews and archival footage are not binding. Therefore, the only constraints are per episode.But wait, the problem says "optimize the use of this content". So, maybe she wants to use as much as possible? Or perhaps she wants to distribute the content in a way that maximizes some objective, but in Sub-problem 1, it's just about allocation, so maybe it's just to find a feasible allocation.But in the problem statement, it says "formulate and solve the linear programming problem to determine the optimal allocation". So, perhaps the objective is to maximize something, but it's not specified. Wait, in Sub-problem 1, it's just about allocation under constraints, so maybe the objective is to use as much as possible, but since she has more content than needed, perhaps the objective is to minimize the unused content? Or maybe it's just to find a feasible solution.Wait, the problem says "optimize the use of this content under certain constraints to ensure the series is impactful and coherent." So, maybe in Sub-problem 1, the objective is just to find a feasible allocation, but in Sub-problem 2, we have an objective function for emotional impact.Wait, but the problem says "formulate and solve the linear programming problem to determine the optimal allocation". So, perhaps in Sub-problem 1, the objective is to maximize the total content used, but since she can only use 270 minutes, and she has 900, that's not possible. Alternatively, maybe the objective is to maximize the minimum of something, but it's not clear.Wait, maybe I misread. Let me check again."Formulate and solve the linear programming problem to determine the optimal allocation of interviews and archival footage to each episode."So, perhaps the objective is to distribute the content in a way that meets all constraints, but without an explicit objective function, it's just a feasibility problem. But in linear programming, we need an objective function to optimize. So, maybe the objective is to maximize the total content used, but since she can only use 270 minutes, that's fixed. Alternatively, maybe the objective is to maximize the minimum of something, but it's not specified.Wait, perhaps the objective is to use as much as possible of the available content, but since she has more than enough, she can't use all. Alternatively, maybe the objective is to balance the use of interviews and archival footage across episodes.Wait, but the problem doesn't specify an objective function for Sub-problem 1. It just says to determine the optimal allocation. So, perhaps it's a feasibility problem, but in linear programming, we need an objective function. Maybe the objective is to minimize the total unused content, but that would be a bit strange.Alternatively, maybe the objective is to maximize the total duration used, but since that's fixed at 270 minutes, it's not necessary. Hmm.Wait, perhaps the problem is to distribute the content such that each episode meets the constraints, and the total used is 270 minutes, but the allocation is such that the per episode constraints are met. So, maybe the objective is to find any feasible solution, but in linear programming, we need an objective function.Wait, maybe the objective is to maximize the total duration used, but that's fixed. Alternatively, maybe the objective is to maximize the total archival footage used, subject to the constraints. That could be a possible objective.Alternatively, perhaps the objective is to make each episode as balanced as possible, but that's vague.Wait, maybe the problem is just to find a feasible allocation, so the objective function could be arbitrary, like minimizing 0, but that's not useful. Alternatively, maybe the objective is to maximize the total duration, but that's fixed.Wait, perhaps I'm overcomplicating. Maybe the problem is just to set up the constraints and find any feasible solution, but in linear programming, we need an objective function. So, perhaps the objective is to maximize the total duration, but that's fixed, so it's not necessary. Alternatively, maybe the objective is to maximize the total duration of interviews used, subject to the constraints.Wait, let's think. If the objective is to maximize the total interviews used, then we can set that as the objective function. Alternatively, if it's to maximize the total archival footage used, that could be another objective.But the problem doesn't specify, so perhaps the objective is to maximize the total duration used, but that's fixed. Alternatively, maybe the objective is to minimize the total unused content, but that's equivalent to maximizing the total used, which is fixed.Wait, perhaps the problem is just to find a feasible allocation, so the objective function is not important, but in linear programming, we need to specify it. Maybe the objective is to maximize the total duration, but since that's fixed, it's not necessary. Alternatively, maybe the objective is to maximize the total duration of interviews used, subject to the constraints.Alternatively, perhaps the problem is to distribute the content such that each episode meets the constraints, and the total used is 270 minutes, but the allocation is such that the per episode constraints are met. So, maybe the objective is to find any feasible solution, but in linear programming, we need an objective function.Wait, perhaps the problem is to minimize the total unused content, which would be equivalent to maximizing the total used, but since the total used is fixed, that's not necessary.Wait, maybe the problem is just to find a feasible allocation, so the objective function could be to minimize 0, but that's not useful. Alternatively, perhaps the problem is to maximize the total duration of interviews used, subject to the constraints.Wait, let me think again. The problem says "optimize the use of this content under certain constraints to ensure the series is impactful and coherent." So, perhaps in Sub-problem 1, the objective is to find a feasible allocation, and in Sub-problem 2, we have an objective function for emotional impact.But in Sub-problem 1, it says "formulate and solve the linear programming problem to determine the optimal allocation". So, perhaps the objective is to maximize the total duration used, but that's fixed. Alternatively, maybe the objective is to maximize the total duration of interviews used, subject to the constraints.Alternatively, perhaps the objective is to maximize the total duration of archival footage used, subject to the constraints.Wait, but without an explicit objective, it's unclear. Maybe the problem is just to find a feasible allocation, so the objective function is not necessary, but in linear programming, we need to specify it. So, perhaps the objective is to maximize the total duration used, which is fixed, so it's not necessary. Alternatively, maybe the objective is to maximize the total duration of interviews used, subject to the constraints.Wait, perhaps the problem is to maximize the total duration of interviews used, subject to the constraints. That would make sense because interviews are more personal and might be more impactful, so she might want to use as much as possible.Alternatively, maybe the objective is to maximize the total duration of archival footage used, but that's less likely.Wait, but the problem doesn't specify, so perhaps I need to assume that the objective is to find a feasible allocation, so the objective function is not necessary, but in linear programming, we need to specify it. So, perhaps the objective is to minimize the total unused content, which would be equivalent to maximizing the total used, but since the total used is fixed, it's not necessary.Alternatively, perhaps the objective is to balance the use of interviews and archival footage across episodes, but that's vague.Wait, maybe I should proceed with the assumption that the objective is to maximize the total duration of interviews used, subject to the constraints. That would be a reasonable objective because interviews are more personal and might be more impactful.So, let's proceed with that.So, the objective function would be:Maximize ( sum_{i=1}^{6} x_i )Subject to:For each episode ( i ):- ( x_i + y_i = 45 )- ( x_i geq 15 )- ( y_i geq 10 )Total constraints:- ( sum_{i=1}^{6} x_i leq 600 )- ( sum_{i=1}^{6} y_i leq 300 )But wait, since each episode is 45 minutes, the total duration used is 270 minutes, so the total interviews used will be ( sum x_i ), and total archival footage used will be ( sum y_i ). But since ( sum x_i + sum y_i = 270 ), and the total available is 600 + 300 = 900, which is more than enough.So, the constraints on total interviews and archival footage are not binding because she can use only 270 minutes in total, but she has 900. So, the only constraints are per episode.But wait, if we are maximizing the total interviews used, then we need to set ( x_i ) as high as possible, subject to the per episode constraints.So, for each episode, ( x_i ) can be at most 45 - 10 = 35 minutes, because ( y_i geq 10 ). So, the maximum ( x_i ) per episode is 35 minutes.But if we set each ( x_i = 35 ), then ( y_i = 10 ), which meets the per episode constraints. Then, total interviews used would be 6*35=210 minutes, and total archival footage used would be 6*10=60 minutes.But wait, she has 600 minutes of interviews and 300 minutes of archival footage. So, she can use more interviews and more archival footage.Wait, but each episode can have more than 10 minutes of archival footage, so she can have more interviews.Wait, no, because each episode is 45 minutes, so if she uses more archival footage, she has to use less interviews, and vice versa.Wait, but if the objective is to maximize total interviews used, then we need to minimize the total archival footage used, subject to the per episode constraints.So, for each episode, to maximize ( x_i ), we set ( y_i ) to its minimum, which is 10 minutes. Therefore, ( x_i = 45 - 10 = 35 ) minutes.So, for each episode, ( x_i = 35 ), ( y_i = 10 ).Total interviews used: 6*35=210 minutes, which is well within the 600 minutes available.Total archival footage used: 6*10=60 minutes, which is well within the 300 minutes available.So, this allocation meets all constraints and maximizes the total interviews used.Therefore, the optimal allocation is 35 minutes of interviews and 10 minutes of archival footage per episode.Wait, but let me check if this is indeed the optimal.Alternatively, if we set some episodes to have more interviews and some to have less, but since we want to maximize the total interviews, we should set each episode to have as much interviews as possible, which is 35 minutes per episode.Yes, that makes sense.So, the solution is:For each episode ( i ):- ( x_i = 35 ) minutes- ( y_i = 10 ) minutesTotal interviews used: 210 minutesTotal archival footage used: 60 minutesThis meets all constraints and maximizes the total interviews used.Now, moving on to Sub-problem 2.The filmmaker wants to maximize the total emotional impact of the entire series. The emotional impact functions are given as:- Interviews: ( I(x) = 5ln(x + 1) )- Archival footage: ( A(y) = 3sqrt{y} )We need to distribute the interviews and archival footage across the episodes to maximize the total emotional impact, subject to the constraints.First, let's note that the total interviews used in Sub-problem 1 was 210 minutes, and total archival footage was 60 minutes. But in Sub-problem 2, we need to consider the same constraints as in Sub-problem 1, but now with the objective of maximizing the total emotional impact.Wait, but in Sub-problem 1, we found an allocation, but in Sub-problem 2, we need to use the results from Sub-problem 1, which were the allocation of 35 minutes of interviews and 10 minutes of archival footage per episode. But now, we need to determine how to distribute the interviews and archival footage across episodes to maximize the total emotional impact, subject to the constraints.Wait, but the constraints are the same as in Sub-problem 1, right? So, each episode must be 45 minutes, with at least 20% archival footage, and each segment must be at least 15 minutes for interviews and 10 minutes for archival footage.But in Sub-problem 2, the objective is to maximize the total emotional impact, which is the sum over all episodes of ( I(x_i) + A(y_i) ).So, we need to set up a new linear programming problem, but with the objective function being the sum of ( 5ln(x_i + 1) + 3sqrt{y_i} ) for each episode ( i ).But wait, this is a nonlinear objective function because of the logarithm and square root. So, it's not a linear programming problem anymore; it's a nonlinear programming problem.But the problem says "using the results from Sub-problem 1, determine the total emotional impact of the entire series. How should the filmmaker distribute the interviews and archival footage across the episodes to maximize the total emotional impact, subject to the constraints given?"So, perhaps we need to first calculate the total emotional impact using the allocation from Sub-problem 1, and then find a better allocation that maximizes the total emotional impact.So, first, using the allocation from Sub-problem 1:Each episode has ( x_i = 35 ) minutes and ( y_i = 10 ) minutes.So, the emotional impact per episode is:( I(35) + A(10) = 5ln(35 + 1) + 3sqrt{10} )Calculating this:( 5ln(36) + 3sqrt{10} )We can compute this numerically.First, ( ln(36) approx 3.5835 ), so ( 5 * 3.5835 ≈ 17.9175 )Next, ( sqrt{10} ≈ 3.1623 ), so ( 3 * 3.1623 ≈ 9.4869 )Adding them together: 17.9175 + 9.4869 ≈ 27.4044 per episode.Total for 6 episodes: 6 * 27.4044 ≈ 164.4264So, the total emotional impact using the Sub-problem 1 allocation is approximately 164.43.Now, to maximize the total emotional impact, we need to find the optimal distribution of ( x_i ) and ( y_i ) across episodes, subject to the constraints.Given that the emotional impact functions are concave (since the second derivative of ( 5ln(x + 1) ) is negative, and the second derivative of ( 3sqrt{y} ) is also negative), the total emotional impact is a concave function, so the maximum can be found at the boundaries or using some optimization technique.But since it's a nonlinear problem, we might need to use some method like Lagrange multipliers or consider the marginal impact per minute.Alternatively, we can consider that for each episode, the marginal impact of adding a minute to interviews versus archival footage can be compared, and allocate accordingly.Let me think.For each episode, the emotional impact is ( 5ln(x + 1) + 3sqrt{y} ). The derivative with respect to ( x ) is ( 5/(x + 1) ), and with respect to ( y ) is ( 3/(2sqrt{y}) ).To maximize the total emotional impact, we should allocate the minutes in each episode such that the marginal impact per minute is equal across all episodes.Wait, but since each episode is independent, we can optimize each episode separately.Wait, no, because the total interviews and archival footage used across all episodes are constrained by the total available, but in this case, the total available is more than enough, so the constraints are per episode.Wait, no, in Sub-problem 1, the total interviews used were 210 minutes, and total archival footage was 60 minutes, but in Sub-problem 2, we are to find a new allocation, possibly using more or less, but subject to the same constraints as Sub-problem 1, which are:- Each episode is 45 minutes: ( x_i + y_i = 45 )- Each episode has at least 20% archival footage: ( y_i geq 9 ), but also ( y_i geq 10 ) due to the minimum segment length, so ( y_i geq 10 )- Each interview segment is at least 15 minutes: ( x_i geq 15 )- Each archival footage segment is at least 10 minutes: ( y_i geq 10 )But the total interviews and archival footage used are not constrained because she has more than enough.Wait, no, in Sub-problem 1, the total interviews used were 210 minutes, but in Sub-problem 2, we are to find a new allocation, possibly using more or less, but subject to the same constraints as Sub-problem 1, which are per episode.Wait, but the problem says "using the results from Sub-problem 1", which were the allocation of 35 minutes of interviews and 10 minutes of archival footage per episode. But now, in Sub-problem 2, we need to determine how to distribute the interviews and archival footage across episodes to maximize the total emotional impact, subject to the constraints given.Wait, perhaps the total interviews and archival footage used are fixed as in Sub-problem 1, i.e., 210 minutes of interviews and 60 minutes of archival footage, and we need to distribute them across episodes to maximize the total emotional impact.But the problem says "using the results from Sub-problem 1", which were the allocation, but it's not clear if the total used is fixed or not.Alternatively, perhaps in Sub-problem 2, we can use more or less interviews and archival footage, but subject to the same per episode constraints as in Sub-problem 1.Wait, the problem says "subject to the constraints given", which are the same as in Sub-problem 1.So, perhaps in Sub-problem 2, the total interviews and archival footage used are not fixed, and we can use more or less, as long as each episode meets the constraints.But since the total available is 600 minutes of interviews and 300 minutes of archival footage, and the total needed is 270 minutes, we can use any amount up to 270 minutes, but the problem doesn't specify that we have to use all of it.Wait, but in Sub-problem 1, the allocation was 210 minutes of interviews and 60 minutes of archival footage, but in Sub-problem 2, we might need to use more or less, but subject to the same per episode constraints.Wait, perhaps the problem is that in Sub-problem 2, the total interviews and archival footage used are fixed as in Sub-problem 1, i.e., 210 and 60, and we need to distribute them across episodes to maximize the total emotional impact.But the problem says "using the results from Sub-problem 1", which were the allocation, but it's not clear if the total used is fixed or not.Alternatively, perhaps in Sub-problem 2, we can use more or less interviews and archival footage, but subject to the same per episode constraints as in Sub-problem 1.Wait, the problem says "subject to the constraints given", which are the same as in Sub-problem 1.So, perhaps in Sub-problem 2, we can use any amount of interviews and archival footage, as long as each episode meets the constraints, and the total used is up to 270 minutes.But since the total available is 900 minutes, which is more than enough, the only constraints are per episode.Therefore, in Sub-problem 2, we can set up a nonlinear programming problem where we maximize the total emotional impact:Maximize ( sum_{i=1}^{6} [5ln(x_i + 1) + 3sqrt{y_i}] )Subject to:For each episode ( i ):- ( x_i + y_i = 45 )- ( x_i geq 15 )- ( y_i geq 10 )And the total interviews used ( sum x_i leq 600 )And the total archival footage used ( sum y_i leq 300 )But since the total needed is 270 minutes, and the total available is 900, the constraints on total interviews and archival footage are not binding. So, we can ignore them.So, the problem reduces to maximizing the sum of ( 5ln(x_i + 1) + 3sqrt{y_i} ) for each episode, subject to ( x_i + y_i = 45 ), ( x_i geq 15 ), ( y_i geq 10 ).Since each episode is independent, we can optimize each episode separately.So, for each episode, we can find the optimal ( x_i ) and ( y_i ) that maximize ( 5ln(x_i + 1) + 3sqrt{y_i} ), subject to ( x_i + y_i = 45 ), ( x_i geq 15 ), ( y_i geq 10 ).So, let's consider one episode. Let me set up the problem for one episode.Maximize ( 5ln(x + 1) + 3sqrt{y} )Subject to:- ( x + y = 45 )- ( x geq 15 )- ( y geq 10 )We can substitute ( y = 45 - x ), so the problem becomes:Maximize ( 5ln(x + 1) + 3sqrt{45 - x} )Subject to:- ( x geq 15 )- ( 45 - x geq 10 ) => ( x leq 35 )So, ( x in [15, 35] )Now, we can take the derivative of the objective function with respect to ( x ) and set it to zero to find the maximum.Let me compute the derivative:( d/dx [5ln(x + 1) + 3sqrt{45 - x}] = 5/(x + 1) - 3/(2sqrt{45 - x}) )Set this equal to zero:( 5/(x + 1) = 3/(2sqrt{45 - x}) )Multiply both sides by ( 2sqrt{45 - x}(x + 1) ):( 10sqrt{45 - x} = 3(x + 1) )Square both sides to eliminate the square root:( 100(45 - x) = 9(x + 1)^2 )Expand both sides:Left side: ( 4500 - 100x )Right side: ( 9(x^2 + 2x + 1) = 9x^2 + 18x + 9 )Bring all terms to one side:( 4500 - 100x - 9x^2 - 18x - 9 = 0 )Simplify:( -9x^2 - 118x + 4491 = 0 )Multiply both sides by -1:( 9x^2 + 118x - 4491 = 0 )Now, solve for ( x ) using quadratic formula:( x = [-118 pm sqrt{118^2 - 4*9*(-4491)}]/(2*9) )Compute discriminant:( D = 13924 + 4*9*4491 = 13924 + 36*4491 )Calculate 36*4491:4491 * 36:4491 * 30 = 134,7304491 * 6 = 26,946Total: 134,730 + 26,946 = 161,676So, D = 13,924 + 161,676 = 175,600Square root of D: ( sqrt{175600} approx 419 ) (since 419^2 = 175,561, which is close to 175,600)So, ( x = [-118 pm 419]/18 )We discard the negative root because ( x ) must be positive.So, ( x = (-118 + 419)/18 = 299/18 ≈ 16.6111 ) minutesSo, ( x ≈ 16.6111 ) minutes, then ( y = 45 - 16.6111 ≈ 28.3889 ) minutesBut we need to check if this is within the feasible region. ( x ≈ 16.61 ) is greater than 15, and ( y ≈ 28.39 ) is greater than 10, so it's feasible.Now, we need to check if this is indeed a maximum. Let's compute the second derivative.The second derivative of the objective function with respect to ( x ):( d^2/dx^2 [5ln(x + 1) + 3sqrt{45 - x}] = -5/(x + 1)^2 - 3/(4(45 - x)^{3/2}) )Since both terms are negative, the function is concave, so this critical point is indeed a maximum.Therefore, for each episode, the optimal allocation is approximately 16.61 minutes of interviews and 28.39 minutes of archival footage.But since we can't have fractions of a minute in practice, we might need to round, but for the sake of this problem, we can keep it as is.Now, let's compute the total emotional impact for this allocation.First, compute ( I(x) + A(y) ) for one episode:( 5ln(16.61 + 1) + 3sqrt{28.39} )Calculate:( 5ln(17.61) ≈ 5*2.868 ≈ 14.34 )( 3sqrt{28.39} ≈ 3*5.328 ≈ 15.984 )Total per episode: 14.34 + 15.984 ≈ 30.324Total for 6 episodes: 6 * 30.324 ≈ 181.944So, the total emotional impact is approximately 181.94.Comparing this to the allocation from Sub-problem 1, which was approximately 164.43, this is a significant improvement.Therefore, the optimal distribution is approximately 16.61 minutes of interviews and 28.39 minutes of archival footage per episode.But let's check if this allocation uses more interviews and less archival footage than the initial allocation.Wait, in Sub-problem 1, each episode had 35 minutes of interviews and 10 minutes of archival footage, which is more interviews and less archival footage. But in this optimal allocation, we have less interviews and more archival footage, which makes sense because the emotional impact function for archival footage has a higher marginal impact at lower values.Wait, actually, the marginal impact of interviews decreases as ( x ) increases, while the marginal impact of archival footage decreases as ( y ) increases, but the shape of the functions might lead to a higher total impact when we have more archival footage.Wait, let me think again. The emotional impact function for interviews is ( 5ln(x + 1) ), which increases with ( x ), but at a decreasing rate. The emotional impact for archival footage is ( 3sqrt{y} ), which also increases with ( y ), but again at a decreasing rate.So, the optimal allocation balances the marginal impacts of interviews and archival footage.In our calculation, we found that the optimal ( x ) is approximately 16.61 minutes, which is more than the minimum 15 minutes, but less than the 35 minutes in Sub-problem 1. So, by reducing the amount of interviews and increasing the amount of archival footage, we can achieve a higher total emotional impact.Therefore, the filmmaker should distribute the interviews and archival footage across the episodes such that each episode has approximately 16.61 minutes of interviews and 28.39 minutes of archival footage.But let me check if this allocation uses more or less total interviews and archival footage than Sub-problem 1.Total interviews used: 6 * 16.61 ≈ 99.66 minutesTotal archival footage used: 6 * 28.39 ≈ 170.34 minutesComparing to Sub-problem 1:Total interviews: 210 minutesTotal archival footage: 60 minutesSo, in Sub-problem 2, we are using less interviews and more archival footage, which is allowed because the total available is more than enough.Therefore, the optimal allocation is approximately 16.61 minutes of interviews and 28.39 minutes of archival footage per episode, leading to a higher total emotional impact.But let me check if this allocation is indeed feasible.Each episode has ( x_i ≈ 16.61 ) minutes, which is above the minimum of 15 minutes, and ( y_i ≈ 28.39 ) minutes, which is above the minimum of 10 minutes. So, it's feasible.Therefore, the filmmaker should distribute the interviews and archival footage such that each episode has approximately 16.61 minutes of interviews and 28.39 minutes of archival footage to maximize the total emotional impact.But since we can't have fractions of a minute, perhaps we can round to the nearest minute, but for the sake of this problem, we can keep it as is.So, summarizing:Sub-problem 1: Optimal allocation is 35 minutes of interviews and 10 minutes of archival footage per episode, leading to a total emotional impact of approximately 164.43.Sub-problem 2: Optimal allocation is approximately 16.61 minutes of interviews and 28.39 minutes of archival footage per episode, leading to a total emotional impact of approximately 181.94.Therefore, the filmmaker should distribute the interviews and archival footage across the episodes to maximize the total emotional impact by allocating approximately 16.61 minutes of interviews and 28.39 minutes of archival footage per episode.</think>

🧠:<think>Alright, so I have this problem where a CEO is looking at how social movements affect the company's image. They've given me a differential equation that models the change in the corporate image index over time. Let me try to unpack this step by step.First, the problem is divided into two parts. The first part is to derive the explicit form of ( I_t ) given the initial condition ( I_0 ). The second part assumes that the intensity of social movements ( M_t ) follows a sinusoidal pattern, and then we need to find the explicit form of ( I_t ) and analyze its long-term behavior.Starting with part 1. The differential equation given is:[ frac{dI_t}{dt} = -alpha M_t I_t ]This looks like a first-order linear ordinary differential equation (ODE). I remember that such equations can often be solved using an integrating factor. The standard form of a linear ODE is:[ frac{dy}{dt} + P(t)y = Q(t) ]In this case, let me rewrite the given equation to match that form. If I move all terms to one side:[ frac{dI_t}{dt} + alpha M_t I_t = 0 ]So here, ( P(t) = alpha M_t ) and ( Q(t) = 0 ). Since ( Q(t) ) is zero, this is actually a homogeneous equation, which might be simpler to solve.For a homogeneous equation, the solution can be found by separating variables. Let me try that.Rewriting the equation:[ frac{dI_t}{dt} = -alpha M_t I_t ]Separating variables:[ frac{dI_t}{I_t} = -alpha M_t dt ]Now, integrating both sides should give me the solution. The left side integrates with respect to ( I_t ), and the right side integrates with respect to ( t ).Integrating the left side:[ int frac{1}{I_t} dI_t = ln|I_t| + C_1 ]Integrating the right side:[ -alpha int M_t dt + C_2 ]So putting it together:[ ln|I_t| = -alpha int M_t dt + C ]Where ( C = C_2 - C_1 ) is the constant of integration. Exponentiating both sides to solve for ( I_t ):[ I_t = e^{-alpha int M_t dt + C} ]This can be rewritten as:[ I_t = e^{C} cdot e^{-alpha int M_t dt} ]Since ( e^{C} ) is just another constant, let's denote it as ( I_0 ), which is the initial value of ( I_t ) at ( t = 0 ). So,[ I_t = I_0 cdot e^{-alpha int_{0}^{t} M_s ds} ]Wait, hold on. The integral should be from 0 to t because we're considering the change over time starting from the initial condition. So, the solution is:[ I_t = I_0 expleft( -alpha int_{0}^{t} M_s ds right) ]That makes sense. So, the corporate image index decreases exponentially depending on the integral of the social movement intensity over time, scaled by the sensitivity parameter ( alpha ).Okay, so that was part 1. I think that's the explicit form. Let me just double-check. If I take the derivative of ( I_t ) with respect to t, using the chain rule:[ frac{dI_t}{dt} = I_0 expleft( -alpha int_{0}^{t} M_s ds right) cdot left( -alpha M_t right) ]Which simplifies to:[ frac{dI_t}{dt} = -alpha M_t I_t ]Yes, that matches the original differential equation. So, part 1 seems correct.Moving on to part 2. Now, ( M_t ) is given as a sinusoidal function:[ M_t = M_0 sin(omega t) ]So, substituting this into our expression for ( I_t ):[ I_t = I_0 expleft( -alpha int_{0}^{t} M_0 sin(omega s) ds right) ]Let me compute the integral inside the exponent. The integral of ( sin(omega s) ) with respect to s is:[ int sin(omega s) ds = -frac{1}{omega} cos(omega s) + C ]So, evaluating from 0 to t:[ int_{0}^{t} sin(omega s) ds = -frac{1}{omega} cos(omega t) + frac{1}{omega} cos(0) ]Since ( cos(0) = 1 ), this simplifies to:[ int_{0}^{t} sin(omega s) ds = frac{1 - cos(omega t)}{omega} ]Therefore, plugging this back into the expression for ( I_t ):[ I_t = I_0 expleft( -alpha M_0 cdot frac{1 - cos(omega t)}{omega} right) ]Simplify the exponent:[ I_t = I_0 expleft( -frac{alpha M_0}{omega} (1 - cos(omega t)) right) ]Alternatively, this can be written as:[ I_t = I_0 expleft( -frac{alpha M_0}{omega} + frac{alpha M_0}{omega} cos(omega t) right) ]Which can be separated into:[ I_t = I_0 expleft( -frac{alpha M_0}{omega} right) cdot expleft( frac{alpha M_0}{omega} cos(omega t) right) ]So, that's the explicit form of ( I_t ) when ( M_t ) is sinusoidal.Now, analyzing the long-term behavior as ( t ) approaches infinity. Let's consider what happens to ( I_t ) as ( t to infty ).Looking at the expression:[ I_t = I_0 expleft( -frac{alpha M_0}{omega} right) cdot expleft( frac{alpha M_0}{omega} cos(omega t) right) ]The term ( expleft( -frac{alpha M_0}{omega} right) ) is a constant, so it doesn't change with time. The other term is ( expleft( frac{alpha M_0}{omega} cos(omega t) right) ), which oscillates because ( cos(omega t) ) oscillates between -1 and 1.So, the entire expression for ( I_t ) will oscillate over time, scaled by the constant factor ( I_0 expleft( -frac{alpha M_0}{omega} right) ).But wait, let's think about the behavior as ( t to infty ). The cosine function continues to oscillate indefinitely, so the exponential term will oscillate between:[ expleft( -frac{alpha M_0}{omega} right) cdot expleft( -frac{alpha M_0}{omega} right) = expleft( -2frac{alpha M_0}{omega} right) ]and[ expleft( -frac{alpha M_0}{omega} right) cdot expleft( frac{alpha M_0}{omega} right) = exp(0) = 1 ]Wait, no, that's not correct. Let me clarify.The term ( expleft( frac{alpha M_0}{omega} cos(omega t) right) ) oscillates between ( expleft( -frac{alpha M_0}{omega} right) ) and ( expleft( frac{alpha M_0}{omega} right) ). So, the entire ( I_t ) oscillates between:[ I_0 expleft( -frac{alpha M_0}{omega} right) cdot expleft( -frac{alpha M_0}{omega} right) = I_0 expleft( -2frac{alpha M_0}{omega} right) ]and[ I_0 expleft( -frac{alpha M_0}{omega} right) cdot expleft( frac{alpha M_0}{omega} right) = I_0 ]So, ( I_t ) oscillates between ( I_0 expleft( -2frac{alpha M_0}{omega} right) ) and ( I_0 ). Therefore, the corporate image index doesn't settle to a single value but continues to oscillate between these two bounds.However, let's consider the average behavior over time. Since ( cos(omega t) ) has an average value of zero over a full period, the exponent ( frac{alpha M_0}{omega} cos(omega t) ) averages out to zero. Therefore, the average value of ( expleft( frac{alpha M_0}{omega} cos(omega t) right) ) is not zero but actually a positive constant due to the properties of the exponential function.Wait, actually, the average of ( exp(a cos(theta)) ) over ( theta ) from 0 to ( 2pi ) is related to the modified Bessel function of the first kind, ( I_0(a) ). But perhaps that's getting too deep into it.Alternatively, maybe I can consider the logarithm of ( I_t ):[ ln(I_t) = ln(I_0) - frac{alpha M_0}{omega} (1 - cos(omega t)) ]So,[ ln(I_t) = ln(I_0) - frac{alpha M_0}{omega} + frac{alpha M_0}{omega} cos(omega t) ]This shows that ( ln(I_t) ) oscillates sinusoidally around the value ( ln(I_0) - frac{alpha M_0}{omega} ). Therefore, ( I_t ) oscillates around ( I_0 expleft( -frac{alpha M_0}{omega} right) ), with an amplitude that depends on ( frac{alpha M_0}{omega} ).So, in the long term, as ( t ) approaches infinity, ( I_t ) doesn't converge to a single value but continues to oscillate. The oscillations are dampened if ( frac{alpha M_0}{omega} ) is positive, meaning that the average level of ( I_t ) is lower than ( I_0 ), but it still oscillates around that lower average.Wait, but actually, the oscillations aren't dampened in the sense of decreasing amplitude. The amplitude of the oscillations depends on ( frac{alpha M_0}{omega} ). If ( frac{alpha M_0}{omega} ) is a constant, then the amplitude remains constant over time. So, the oscillations are persistent and do not decay.Therefore, the long-term behavior is that ( I_t ) oscillates indefinitely between ( I_0 expleft( -2frac{alpha M_0}{omega} right) ) and ( I_0 ), with the average level being ( I_0 expleft( -frac{alpha M_0}{omega} right) ).But wait, let me think again. The exponent in the expression for ( I_t ) is:[ -frac{alpha M_0}{omega} (1 - cos(omega t)) ]Which simplifies to:[ -frac{alpha M_0}{omega} + frac{alpha M_0}{omega} cos(omega t) ]So, the exponent is oscillating around ( -frac{alpha M_0}{omega} ). Therefore, ( I_t ) is oscillating around ( I_0 expleft( -frac{alpha M_0}{omega} right) ), with the amplitude of oscillation determined by ( frac{alpha M_0}{omega} ).So, in the long term, as ( t ) approaches infinity, ( I_t ) does not approach a limit but continues to oscillate between two values. Therefore, the corporate image index doesn't stabilize but keeps fluctuating due to the periodic nature of the social movements.Alternatively, if we consider the integral over time, the cumulative effect of the social movements might lead to a long-term trend. But in this case, since the social movement intensity is periodic, the integral over each period might contribute a net negative effect, but the oscillations mean that the image index doesn't monotonically decrease but rather fluctuates.Wait, actually, let's compute the integral ( int_{0}^{t} M_s ds ) when ( M_t = M_0 sin(omega t) ). As we saw earlier, it's ( frac{1 - cos(omega t)}{omega} ). So, as ( t ) increases, the integral oscillates between 0 and ( frac{2}{omega} ), right? Because ( 1 - cos(omega t) ) varies between 0 and 2.Therefore, the exponent ( -alpha M_0 cdot frac{1 - cos(omega t)}{omega} ) oscillates between ( 0 ) and ( -frac{2 alpha M_0}{omega} ). So, ( I_t ) oscillates between ( I_0 ) and ( I_0 expleft( -frac{2 alpha M_0}{omega} right) ).Hence, the corporate image index doesn't approach a limit as ( t to infty ) but keeps oscillating between these two bounds. The oscillations are persistent and do not dampen because the social movement intensity is periodic and doesn't decay over time.So, in summary, for part 2, the explicit form of ( I_t ) is:[ I_t = I_0 expleft( -frac{alpha M_0}{omega} (1 - cos(omega t)) right) ]And the long-term behavior is that ( I_t ) oscillates between ( I_0 expleft( -frac{2 alpha M_0}{omega} right) ) and ( I_0 ), without approaching a single value as ( t ) approaches infinity.I think that's the analysis. Let me just recap to make sure I haven't missed anything.1. Solved the ODE by separation of variables, got an exponential decay depending on the integral of ( M_t ).2. Substituted the sinusoidal ( M_t ), computed the integral, and expressed ( I_t ) in terms of exponentials and cosine.3. Analyzed the long-term behavior by looking at the oscillatory nature of the cosine term, concluding that ( I_t ) oscillates between two bounds indefinitely.Yes, that seems thorough. I don't think I made any mistakes in the calculations. The key point is recognizing that the integral of a sine function leads to a cosine term, and then understanding how that affects the exponential in the solution.</think>

🧠:<think>Alright, so I have this problem about Alex constructing a prop with secret compartments. It's a rectangular prism with dimensions a, b, c. Each face has a compartment, so there are 6 compartments in total. Each compartment has dimensions a_i, b_i, c_i, which are each half of the original prism's dimensions. So, for each compartment, a_i = a/2, b_i = b/2, c_i = c/2.First, I need to find the total volume of all compartments. Each compartment's volume is given by V_i = (1/3) * a_i * b_i * c_i. Since each a_i, b_i, c_i is half of a, b, c respectively, plugging those into the formula gives V_i = (1/3)*(a/2)*(b/2)*(c/2). Let me compute that:V_i = (1/3)*(a*b*c)/(2*2*2) = (1/3)*(a*b*c)/8 = (a*b*c)/24.Since there are 6 compartments, the total volume V_total is 6 * V_i = 6*(a*b*c)/24 = (a*b*c)/4.Wait, so the total volume is (a*b*c)/4. Hmm, that seems straightforward. But the question also asks about the conditions under which this total volume is maximized. Since the compartments are each a fixed proportion of the prism's dimensions, the total volume is directly proportional to the volume of the prism itself. So, to maximize the total volume of compartments, Alex should maximize the volume of the prism, which is a*b*c. Therefore, the total volume of compartments is maximized when the prism's volume is maximized, given the constraints on a, b, c.But wait, are there any constraints? The problem doesn't specify any constraints on a, b, c, so technically, the total volume can be made as large as possible by increasing a, b, or c. So, unless there are some practical limitations, like the size of the stage or materials, the maximum volume is unbounded. Maybe I'm missing something here.Alternatively, perhaps the compartments can't overlap, so maybe the way they are placed affects the maximum possible volume? Each compartment is on a face, so each is a smaller prism attached to each face. But if each compartment is half the size of the original prism, then maybe the original prism must be large enough to accommodate them without overlapping. Wait, but the compartments are inside the prism, right? So each compartment is a smaller prism within the original one.Wait, hold on, the problem says "each face of the prism has a hidden compartment that can be accessed by sliding panels." So, each compartment is on a face, meaning each is a smaller prism attached to each face, but within the original prism. So, if each compartment is half the size in each dimension, then the original prism must be at least as large as the compartments.But if each compartment is half the size, then the original prism must be at least twice as large in each dimension as the compartment. But since the compartments are on each face, perhaps they can't extend beyond the original prism. So, each compartment is a smaller prism attached to each face, but within the original.Wait, maybe the compartments are like extensions or protrusions on each face? Or are they internal? The problem says "hidden compartments within the prism," so they must be internal. So, each face has a compartment inside the prism. So, if each compartment is half the size in each dimension, then the original prism must be larger than twice the compartment's dimensions? Wait, no, because the compartments are inside.Wait, perhaps the compartments are each a smaller prism, each attached to a face, but within the original prism. So, if the original prism is a x b x c, then each compartment is (a/2) x (b/2) x (c/2). So, to fit six such compartments, each on a different face, the original prism must be large enough to contain all of them without overlapping.But wait, each compartment is on a different face, so each is on one of the six faces. So, for example, on the front face, there's a compartment extending into the prism, same with the back, left, right, top, and bottom.But if each compartment is (a/2) x (b/2) x (c/2), then the original prism must be at least as large as the compartments in each dimension. But since the compartments are on each face, their placement might interfere with each other.Wait, maybe the original prism is divided into smaller compartments, each attached to a face. So, perhaps the original prism is divided such that each face has a compartment that is half the size in each dimension. So, for example, on the front face, the compartment is a/2 x b/2 x c/2, same for the back, left, right, top, and bottom.But if you have compartments on all six faces, each of size (a/2)x(b/2)x(c/2), how do they fit into the original prism? Because if you have a compartment on the front face, it takes up a/2 x b/2 x c/2, and similarly for the back, but then the total space taken by front and back compartments would be 2*(a/2*b/2*c/2) = a*b*c/4. Similarly, the left and right compartments would take up another a*b*c/4, and the top and bottom another a*b*c/4, totaling 3*(a*b*c)/4. But the original prism is a*b*c, so 3/4 of it is taken by compartments. So, that leaves 1/4 of the original prism's volume as non-compartment space.But in the problem, it's stated that each compartment is attached to each face, so perhaps the original prism is constructed such that each face has a compartment, each of size (a/2)x(b/2)x(c/2). So, the total volume of compartments is 6*(a/2*b/2*c/2)/3, but wait, no, each compartment's volume is (1/3)*(a/2)*(b/2)*(c/2). Wait, why is there a 1/3 factor?Hold on, the volume of each compartment is given by V_i = (1/3)*a_i*b_i*c_i. So, each compartment is a pyramid? Because the formula (1/3)*base area*height is for a pyramid. So, each compartment is a pyramid with base dimensions a_i, b_i, and height c_i, or something like that.Wait, the problem says each compartment is accessed by sliding panels, forming a complex network. So, maybe each compartment is a pyramid-shaped space within the prism, attached to each face. So, each face has a pyramid whose base is a_i x b_i and height c_i.So, if each compartment is a pyramid, then the volume is (1/3)*base area*height. So, for each compartment, base area is (a/2)*(b/2), and height is c/2, so V_i = (1/3)*(a/2*b/2)*(c/2) = (1/3)*(a*b*c)/8 = (a*b*c)/24.Since there are 6 compartments, each on a different face, the total volume is 6*(a*b*c)/24 = (a*b*c)/4.So, that's the total volume of all compartments. Now, the problem asks to express this in terms of a, b, c, which I've done: V_total = (a*b*c)/4.Then, determine the conditions under which this total volume is maximized. Since V_total is proportional to a*b*c, the volume is maximized when a*b*c is maximized. But unless there are constraints on a, b, c, like a fixed surface area or something, the product a*b*c can be increased indefinitely. So, maybe the problem assumes that the original prism has a fixed surface area or something else.Wait, the problem doesn't specify any constraints, so perhaps the total volume is maximized when a, b, c are as large as possible. But that's trivial. Maybe the problem is expecting that the maximum occurs when a = b = c, making the prism a cube? Because often, for a given surface area, the cube maximizes the volume.But since there are no constraints given, I'm not sure. Maybe the problem just wants the expression, and the conditions are that a, b, c are positive real numbers, so the volume is maximized as a, b, c approach infinity. But that seems odd.Alternatively, perhaps the compartments cannot overlap, so the original prism must be large enough to contain all six compartments without overlapping. Each compartment is a pyramid with base (a/2 x b/2) and height c/2. So, for the compartments on opposite faces, like front and back, their heights along the c-axis would be c/2 each, so the total height consumed by front and back compartments would be c/2 + c/2 = c. Similarly, for left and right compartments, their widths would be a/2 each, totaling a. And for top and bottom compartments, their heights along b would be b/2 each, totaling b.Wait, so if each pair of opposite compartments consumes the entire dimension, then the original prism must be at least as large as the sum of the compartments in each dimension. But since each compartment is half the size, the original prism is exactly twice the size in each dimension. So, a = 2*a', b = 2*b', c = 2*c', where a', b', c' are the dimensions of the compartments. But in this case, the compartments are (a/2, b/2, c/2), so the original prism is exactly twice each dimension. So, in that case, the original prism is just twice the size in each dimension, so the compartments fit perfectly without overlapping.But then, if the original prism is exactly twice the size in each dimension, then the total volume of compartments is (a*b*c)/4, and the original prism's volume is a*b*c. So, the compartments take up 1/4 of the original volume, leaving 3/4 for the rest of the prop.But maybe the problem is just asking for the expression, which is (a*b*c)/4, and the conditions for maximum volume would be when a, b, c are as large as possible. But without constraints, it's unbounded.Alternatively, perhaps the compartments must fit within the original prism, so a, b, c must be at least as large as the compartments. But since the compartments are half the size, the original prism must be at least twice as large in each dimension. So, the minimum size of the prism is 2*(a/2) = a, similarly for b and c. So, the original prism must be at least a x b x c, but the compartments are (a/2 x b/2 x c/2). So, actually, the compartments are smaller than the original prism.Wait, I'm getting confused. Let me clarify:Each compartment is on a face of the prism, and each compartment has dimensions a_i = a/2, b_i = b/2, c_i = c/2. So, each compartment is a smaller prism attached to each face. But if the original prism is a x b x c, then each compartment is half the size in each dimension. So, for example, the compartment on the front face would be (a/2) x (b/2) x (c/2). Similarly, the compartment on the back face would be the same. But if you have compartments on both front and back, their combined depth along the c-axis would be c/2 + c/2 = c, which is exactly the depth of the original prism. Similarly, for left and right compartments, their combined width would be a/2 + a/2 = a, and for top and bottom, their combined height would be b/2 + b/2 = b.So, in this case, the original prism is exactly the size needed to contain all six compartments without overlapping. So, the compartments perfectly fit into the original prism, each pair of opposite compartments consuming the entire length, width, or height.Therefore, the total volume of compartments is (a*b*c)/4, as calculated earlier. Since the compartments are fixed proportions of the original prism, the total volume is directly proportional to the original prism's volume. Therefore, to maximize the total volume of compartments, Alex should maximize the volume of the original prism, which is a*b*c. Without any constraints, this can be made arbitrarily large, so the maximum is unbounded. However, if there were constraints, like a fixed surface area or fixed material, then the maximum would be achieved at specific proportions, likely a cube.But since the problem doesn't specify constraints, I think the answer is that the total volume is (a*b*c)/4, and it's maximized when a, b, c are as large as possible. But maybe the problem expects that the compartments are arranged such that the original prism is a cube, so a = b = c, but that's not necessarily stated.Moving on to part 2. Alex wants to connect all compartments through tunnels. Each tunnel has a cross-sectional area A = (1/10)*a*b, and the length is the distance between the centers of adjacent compartments.First, I need to figure out how many tunnels are needed. Since each pair of adjacent compartments shares a tunnel, and there are 6 compartments, each on a face. Each face is adjacent to four other faces, but in terms of compartments, each compartment is adjacent to four others? Wait, no, each face is adjacent to four other faces, but each compartment is on a face, so each compartment is adjacent to four others.But wait, in a rectangular prism, each face is adjacent to four other faces. So, each compartment is adjacent to four others. But in terms of connecting all compartments, we need a network where each compartment is connected to its adjacent ones. So, the number of tunnels would be equal to the number of edges in the graph where each node is a compartment and edges represent adjacency.In a cube, each face is adjacent to four others, but each adjacency is shared between two faces, so the total number of edges is 12. Wait, no, a cube has 6 faces, each face connected to 4 others, but each connection is shared, so total edges are (6*4)/2 = 12. So, 12 tunnels.But wait, in a rectangular prism, the number of adjacencies is the same as a cube, right? Each face is adjacent to four others, so total tunnels would be 12.But let me think again. Each compartment is on a face, and each face is adjacent to four others, so each compartment needs four tunnels. But each tunnel connects two compartments, so total number of tunnels is (6*4)/2 = 12.So, 12 tunnels in total.Each tunnel has a cross-sectional area A = (1/10)*a*b, and a length L = sqrt((a/2)^2 + (b/2)^2). Wait, the problem says the distance between the centers of adjacent compartments on the same face is L = sqrt((a/2)^2 + (b/2)^2). Hmm, wait, adjacent compartments on the same face? Or adjacent compartments on adjacent faces?Wait, the problem says: "the distance between the centers of adjacent compartments it connects." So, each tunnel connects two adjacent compartments, which are on adjacent faces. So, the centers of these compartments are separated by a distance L.But the problem specifies that L is sqrt((a/2)^2 + (b/2)^2). So, that suggests that the centers are offset by a/2 and b/2 in two dimensions. So, perhaps the compartments are on adjacent faces, say front and top, and their centers are offset by a/2 and b/2.Wait, let me visualize the prism. Let's say we have a front face compartment and a top face compartment. The front face compartment is centered at (a/4, b/4, c/2), assuming the prism is from (0,0,0) to (a,b,c). Similarly, the top face compartment is centered at (a/4, 3b/4, c/2). Wait, no, actually, if the compartments are on the faces, their centers would be offset by half their dimensions.Wait, each compartment is a pyramid with base (a/2 x b/2) and height c/2. So, for the front face compartment, its base is on the front face, which is at z=0, and it extends into the prism up to z=c/2. Similarly, the top face compartment is on the top face, which is at y=c, and extends down to y=c - c/2 = c/2.Wait, no, the height is c/2, so if the compartment is on the front face, it goes from z=0 to z=c/2. Similarly, the compartment on the top face goes from y=c to y=c - c/2 = c/2. So, the center of the front face compartment is at (a/4, b/4, c/4), and the center of the top face compartment is at (a/4, 3b/4, 3c/4). Wait, no, the center would be at the midpoint of the compartment.Wait, the compartment on the front face is a pyramid with base (a/2 x b/2) at z=0, and apex at (a/4, b/4, c/2). Similarly, the compartment on the top face is a pyramid with base (a/2 x c/2) at y=c, and apex at (a/4, c/4, c/2). Wait, no, the compartments are on each face, so each has two dimensions along the face and one dimension into the prism.Wait, maybe I'm overcomplicating. The problem says that the distance between the centers of adjacent compartments on the same face is L = sqrt((a/2)^2 + (b/2)^2). Wait, but adjacent compartments on the same face? Each face has only one compartment, right? Each face has one compartment, so there are no adjacent compartments on the same face. So, maybe the problem means adjacent compartments on adjacent faces.Wait, the problem says: "the distance between the centers of the adjacent compartments it connects." So, each tunnel connects two adjacent compartments, which are on adjacent faces. So, the distance between their centers is L = sqrt((a/2)^2 + (b/2)^2). Wait, but why is it sqrt((a/2)^2 + (b/2)^2)?Wait, maybe the centers are offset by a/2 and b/2 in two dimensions. For example, if one compartment is on the front face, its center is at (a/4, b/4, c/4), and the adjacent compartment on the right face is at (3a/4, b/4, c/4). So, the distance between these two centers is sqrt((a/2)^2 + 0 + 0) = a/2. But the problem says L = sqrt((a/2)^2 + (b/2)^2). Hmm, that suggests a diagonal distance.Wait, maybe the centers are offset by a/2 in one dimension and b/2 in another. For example, if one compartment is on the front face, center at (a/4, b/4, c/4), and the adjacent compartment is on the top face, center at (a/4, 3b/4, 3c/4). Then, the distance between these centers is sqrt((0)^2 + (b/2)^2 + (c/2)^2). But the problem says L = sqrt((a/2)^2 + (b/2)^2). So, maybe the distance is along the a and b axes.Wait, perhaps the compartments are on adjacent faces that share an edge, so their centers are offset by a/2 and b/2. For example, front and right face compartments. The front face compartment's center is at (a/4, b/4, c/4), and the right face compartment's center is at (3a/4, b/4, c/4). The distance between them is sqrt((a/2)^2 + 0 + 0) = a/2. But the problem says L = sqrt((a/2)^2 + (b/2)^2). Hmm, maybe I'm miscalculating.Alternatively, perhaps the distance is between centers of compartments on adjacent faces that are not sharing an edge but are on adjacent faces. Wait, no, in a rectangular prism, all adjacent faces share an edge.Wait, maybe the problem is referring to the distance between centers of compartments on adjacent faces, which share an edge, but the distance is calculated in 3D. So, for example, front face compartment center at (a/4, b/4, c/4), and right face compartment center at (3a/4, b/4, c/4). The distance between them is sqrt((a/2)^2 + 0 + 0) = a/2. But the problem says sqrt((a/2)^2 + (b/2)^2). So, maybe the distance is between centers of compartments on adjacent faces that are not aligned along the same axis.Wait, maybe the compartments are on front and top faces, which are adjacent along the y-axis. The front face compartment center is at (a/4, b/4, c/4), and the top face compartment center is at (a/4, 3b/4, 3c/4). The distance between them is sqrt(0 + (b/2)^2 + (c/2)^2). But the problem says sqrt((a/2)^2 + (b/2)^2). Hmm, unless the distance is along the a and b axes, ignoring c.Wait, maybe the problem is considering the projection onto the a-b plane. So, the distance between centers on the same face is sqrt((a/2)^2 + (b/2)^2). But each face only has one compartment, so maybe the distance is between centers of compartments on adjacent faces, but projected onto the face.Wait, I'm getting confused. Let me read the problem again."Calculate the total volume of the tunnels needed to connect all the compartments, given that the distance between the centers of adjacent compartments on the same face is L = sqrt((a/2)^2 + (b/2)^2)."Wait, the problem says "on the same face." But each face has only one compartment. So, maybe it's a typo, and it should be "on adjacent faces." Alternatively, maybe it's considering the distance between centers of compartments that are on the same face, but that doesn't make sense because each face has only one compartment.Alternatively, perhaps the compartments are on the same face, but that face is split into smaller compartments? No, the problem says each face has one compartment. So, maybe the distance is between centers of compartments on adjacent faces, but the problem says "on the same face." Hmm.Alternatively, maybe the distance is between centers of compartments that are on the same face, but considering that the face is divided into smaller compartments. Wait, no, each face has one compartment.Wait, maybe the problem is referring to the distance between centers of compartments on the same face, but each face has multiple compartments? No, the problem says each face has one compartment.I think there might be a misunderstanding here. Let me try to parse the sentence again:"Calculate the total volume of the tunnels needed to connect all the compartments, given that the distance between the centers of adjacent compartments on the same face is L = sqrt((a/2)^2 + (b/2)^2)."So, "adjacent compartments on the same face." But each face has only one compartment. So, perhaps it's a misstatement, and it should be "adjacent compartments on adjacent faces." Alternatively, maybe the compartments are arranged in a way that on the same face, there are multiple compartments, but the problem says each face has one.Alternatively, maybe the compartments are arranged such that each face has multiple compartments, but the problem says each face has one. Hmm.Wait, perhaps the distance L is the distance between centers of compartments that are on adjacent faces, but the problem mistakenly says "on the same face." Alternatively, maybe the distance is between centers of compartments on the same face, but considering that the face is divided into smaller compartments.Wait, I'm stuck here. Let me try to proceed with the assumption that each tunnel connects two compartments on adjacent faces, and the distance between their centers is L = sqrt((a/2)^2 + (b/2)^2). So, each tunnel has length L = sqrt((a/2)^2 + (b/2)^2) and cross-sectional area A = (1/10)*a*b.So, the volume of each tunnel is A * L = (1/10)*a*b * sqrt((a/2)^2 + (b/2)^2).But how many tunnels are there? Earlier, I thought 12, but let me confirm.In a rectangular prism, each face is adjacent to four others, so each compartment is adjacent to four others. Since there are six compartments, each with four connections, but each connection is shared, so total tunnels are (6*4)/2 = 12.So, 12 tunnels in total.Therefore, the total volume of tunnels is 12 * (1/10)*a*b*sqrt((a/2)^2 + (b/2)^2).Simplify this expression:First, sqrt((a/2)^2 + (b/2)^2) = (1/2)*sqrt(a^2 + b^2).So, the volume becomes 12*(1/10)*a*b*(1/2)*sqrt(a^2 + b^2) = 12*(1/20)*a*b*sqrt(a^2 + b^2) = (12/20)*a*b*sqrt(a^2 + b^2) = (3/5)*a*b*sqrt(a^2 + b^2).So, total tunnel volume is (3/5)*a*b*sqrt(a^2 + b^2).Wait, but let me double-check the number of tunnels. If each of the six compartments is connected to four others, that's 6*4 = 24 connections, but each tunnel connects two compartments, so total tunnels are 24/2 = 12. So, yes, 12 tunnels.But wait, in a rectangular prism, each edge is shared by two faces, so the number of adjacent face pairs is 12, which corresponds to the 12 edges of the prism. So, each edge corresponds to a tunnel connecting two compartments. So, yes, 12 tunnels.Therefore, the total tunnel volume is 12*(1/10)*a*b*sqrt((a/2)^2 + (b/2)^2) = (12/10)*a*b*(sqrt(a^2 + b^2)/2) = (6/5)*a*b*(sqrt(a^2 + b^2)/2) = (3/5)*a*b*sqrt(a^2 + b^2).So, total tunnel volume is (3/5)*a*b*sqrt(a^2 + b^2).But let me write it as (3/5)*a*b*sqrt(a^2 + b^2).Alternatively, factor out (a*b):Total tunnel volume = (3/5)*a*b*sqrt(a^2 + b^2).So, that's the total volume of the tunnels.Wait, but let me make sure about the distance L. The problem says "the distance between the centers of adjacent compartments on the same face is L = sqrt((a/2)^2 + (b/2)^2)." But as each face has only one compartment, this seems confusing. Maybe the distance is between centers of compartments on adjacent faces, but the problem says "on the same face." Maybe it's a misstatement, and it should be "on adjacent faces." Alternatively, maybe the compartments are arranged in a way that on the same face, there are multiple compartments, but the problem says each face has one.Alternatively, perhaps the distance is between centers of compartments on the same face, but considering that the face is divided into smaller compartments. Wait, no, each face has one compartment.I think the problem might have a typo, and it should be "on adjacent faces." So, proceeding with that assumption, the total tunnel volume is (3/5)*a*b*sqrt(a^2 + b^2).Alternatively, if the distance is on the same face, but each face has only one compartment, then maybe the distance is zero, which doesn't make sense. So, I think the intended meaning is that the distance is between centers of compartments on adjacent faces, and the distance is sqrt((a/2)^2 + (b/2)^2). So, I'll stick with the calculation above.So, summarizing:1. Total compartment volume: (a*b*c)/4.2. Total tunnel volume: (3/5)*a*b*sqrt(a^2 + b^2).But wait, let me check the calculation again.Each tunnel has volume A * L, where A = (1/10)*a*b, and L = sqrt((a/2)^2 + (b/2)^2) = (1/2)*sqrt(a^2 + b^2).So, each tunnel's volume is (1/10)*a*b*(1/2)*sqrt(a^2 + b^2) = (1/20)*a*b*sqrt(a^2 + b^2).Total tunnels: 12.Total tunnel volume: 12*(1/20)*a*b*sqrt(a^2 + b^2) = (12/20)*a*b*sqrt(a^2 + b^2) = (3/5)*a*b*sqrt(a^2 + b^2).Yes, that's correct.So, the total volume of the tunnels is (3/5)*a*b*sqrt(a^2 + b^2).Therefore, the answers are:1. Total compartment volume: (a*b*c)/4, maximized when a*b*c is maximized.2. Total tunnel volume: (3/5)*a*b*sqrt(a^2 + b^2).But let me write them in boxed form as requested.</think>