🧠:<think>Okay, so I have this problem about a corporate CEO who wants to improve the gender diversity on their executive board. The current board has 12 members: 4 women and 8 men. The CEO wants to add new members so that women make up at least 40% of the board. There are two sub-problems to solve here.Starting with Sub-problem 1: Determine the minimum number of new female and male members needed, assuming an equal number of each are added. Hmm, okay. So right now, there are 4 women and 8 men. Let me denote the number of new female members as x and the number of new male members as y. But the problem says we have to add an equal number of male and female members, so x = y. Let's just use x for both.So, after adding x women and x men, the total number of women will be 4 + x, and the total number of men will be 8 + x. The total number of board members will then be 12 + 2x.We need the proportion of women to be at least 40%, so:(4 + x) / (12 + 2x) ≥ 0.4Let me solve this inequality step by step.First, multiply both sides by (12 + 2x) to get rid of the denominator:4 + x ≥ 0.4*(12 + 2x)Calculate the right side:0.4*12 = 4.80.4*2x = 0.8xSo, 4 + x ≥ 4.8 + 0.8xNow, subtract 0.8x from both sides:4 + 0.2x ≥ 4.8Subtract 4 from both sides:0.2x ≥ 0.8Divide both sides by 0.2:x ≥ 4So, x must be at least 4. Since x has to be an integer (you can't add a fraction of a person), the minimum number of new female and male members to add is 4 each.Wait, let me check that. If we add 4 women and 4 men, the total women become 8, total men become 12, total board is 20. 8/20 is 0.4, which is exactly 40%. So, that's the minimum. If we add fewer than 4, say 3, then total women would be 7, total board 18. 7/18 is approximately 38.89%, which is less than 40%. So, yes, 4 is the minimum.Moving on to Sub-problem 2: The CEO also wants to ensure that the average experience of the board doesn't drop below 13 years. Currently, the board has a combined average experience of 15 years. So, the total experience is 12 members * 15 years = 180 years.The new members, both male and female, have an average experience of 10 years. Let's denote the number of new members as n. Since in Sub-problem 1, we added 4 women and 4 men, but here, the number of new members isn't specified yet. Wait, actually, in Sub-problem 2, it's a separate consideration. So, the CEO is looking at both gender diversity and experience diversity. But the problem says, "the CEO also wants to consider the diversity in terms of experience." So, it's a separate problem, not necessarily linked to Sub-problem 1. Hmm, let me read again."Sub-problem 2: The CEO also wants to consider the diversity in terms of experience. Suppose the current board members have a combined average experience of 15 years. If the new members (both male and female) have an average experience of 10 years, calculate the minimum number of new members needed such that the average experience of the entire board does not drop below 13 years."Okay, so it's a separate problem. So, regardless of gender, the CEO wants to add some number of new members (could be any gender, but the average experience is 10 years for all new members). The goal is to find the minimum number of new members such that the overall average experience is at least 13 years.So, let me denote the number of new members as n. Each new member has an average experience of 10 years, so total experience added is 10n.The current total experience is 12*15 = 180 years.After adding n members, total experience becomes 180 + 10n, and total number of members becomes 12 + n.We need the average experience to be at least 13 years:(180 + 10n) / (12 + n) ≥ 13Let me solve this inequality.Multiply both sides by (12 + n):180 + 10n ≥ 13*(12 + n)Calculate the right side:13*12 = 15613*n = 13nSo, 180 + 10n ≥ 156 + 13nSubtract 10n from both sides:180 ≥ 156 + 3nSubtract 156 from both sides:24 ≥ 3nDivide both sides by 3:8 ≥ nSo, n ≤ 8. But wait, we need the average to be at least 13, so n must be such that (180 + 10n)/(12 + n) ≥ 13. So, solving for n, we get n ≤ 8. But wait, that seems counterintuitive. Adding more members with lower average experience would lower the overall average, right? So, if we add n members, the more we add, the lower the average becomes. So, to keep the average above 13, we can't add too many. So, the maximum number of new members we can add without dropping below 13 is 8. But the question is asking for the minimum number of new members needed. Wait, that doesn't make sense. If we add fewer members, the average experience will be higher. So, the minimum number of new members needed to not drop below 13 is actually zero. But that can't be right because the problem says "the minimum number of new members needed such that the average experience does not drop below 13 years." So, perhaps the problem is that adding new members with lower experience will lower the average, so we need to find the maximum number of new members that can be added without dropping below 13. But the wording says "minimum number of new members needed", which is confusing because adding more members would require a higher number, but the average would decrease. Maybe it's a translation issue.Wait, let's read again: "calculate the minimum number of new members needed such that the average experience of the entire board does not drop below 13 years." Hmm, so perhaps the CEO wants to add some new members, but doesn't want the average to drop below 13. So, the minimum number of new members that can be added without causing the average to drop below 13. But actually, adding any number of new members with lower average experience will lower the overall average. So, the more you add, the lower it goes. So, to find the maximum number of new members that can be added without the average dropping below 13. But the question says "minimum number of new members needed". Maybe it's a misstatement, perhaps it's the maximum number. Alternatively, maybe the CEO wants to add a certain number of new members, but wants to ensure that even after adding them, the average doesn't drop below 13. So, given that, the minimum number of new members that can be added without violating the average. But that still doesn't make much sense because adding any number would lower the average. Wait, perhaps the problem is that the CEO wants to add new members, but the average should not drop below 13. So, the question is, what's the minimum number of new members that need to be added so that even if you add that number, the average doesn't drop below 13. But that would mean that adding fewer than that number would cause the average to drop below 13, which is not the case. Wait, no, actually, adding more members would lower the average more. So, the more you add, the lower the average. So, to find the maximum number of new members that can be added without the average dropping below 13. So, perhaps the question is misworded, and it should be "maximum number of new members". Alternatively, maybe the CEO wants to add a certain number of new members, say n, and wants to ensure that the average doesn't drop below 13. So, given that, what is the minimum n such that even if you add n members, the average remains above 13. But that would mean that n can be as low as 0, but that's trivial. Alternatively, perhaps the problem is that the CEO is planning to add some number of new members, and wants to ensure that the average doesn't drop below 13, so we need to find the minimum number of new members that must be added to achieve a certain effect, but it's unclear.Wait, let's go back to the problem statement:"Sub-problem 2: The CEO also wants to consider the diversity in terms of experience. Suppose the current board members have a combined average experience of 15 years. If the new members (both male and female) have an average experience of 10 years, calculate the minimum number of new members needed such that the average experience of the entire board does not drop below 13 years."So, the CEO wants to add new members (with average experience 10) such that the overall average doesn't drop below 13. So, the question is, what's the minimum number of new members that can be added without causing the average to drop below 13. But as I thought earlier, adding more members will lower the average. So, the more you add, the lower the average. Therefore, to find the maximum number of new members that can be added without the average dropping below 13. So, perhaps the question is misworded, and it should be "maximum number of new members". Alternatively, maybe the CEO wants to add a certain number of new members, and wants to ensure that even after adding them, the average is at least 13. So, given that, what's the minimum number of new members that must be added? Wait, that doesn't make sense because adding more members would lower the average, so the minimum number would be 0, which is trivial.Wait, perhaps I misread the problem. Maybe the CEO is planning to add new members, and wants to ensure that the average experience doesn't drop below 13. So, regardless of how many new members are added, the average should stay above 13. But that's impossible because as you add more members with lower average, the overall average will decrease. So, perhaps the problem is to find the maximum number of new members that can be added without the average dropping below 13. So, let's proceed with that interpretation.So, let's denote n as the number of new members. Each new member has an average experience of 10 years. The total experience after adding n members is 180 + 10n, and the total number of members is 12 + n. We need:(180 + 10n)/(12 + n) ≥ 13Multiply both sides by (12 + n):180 + 10n ≥ 13*(12 + n)Calculate the right side:13*12 = 15613n = 13nSo, 180 + 10n ≥ 156 + 13nSubtract 10n from both sides:180 ≥ 156 + 3nSubtract 156 from both sides:24 ≥ 3nDivide both sides by 3:8 ≥ nSo, n ≤ 8. Therefore, the maximum number of new members that can be added without the average experience dropping below 13 is 8. So, if the CEO adds 8 new members, the average experience will be exactly 13. If they add more than 8, the average will drop below 13. Therefore, the minimum number of new members needed to ensure the average doesn't drop below 13 is 0, but that's trivial. However, if the CEO wants to add as many as possible without dropping below 13, the answer is 8. But the problem says "minimum number of new members needed", which is confusing. Maybe it's a translation issue, and they meant "maximum". Alternatively, perhaps the problem is that the CEO wants to add a certain number of new members, and wants to ensure that even after adding them, the average is at least 13. So, given that, what's the minimum number of new members that must be added? But that doesn't make sense because adding more members would lower the average. So, perhaps the problem is misworded, and it should be "maximum number of new members that can be added without the average dropping below 13", which is 8.Alternatively, maybe the problem is that the CEO wants to add new members such that the average experience is at least 13, so the minimum number of new members needed to achieve that. But wait, the current average is 15, which is higher than 13. So, adding new members with lower average will lower the overall average. So, the more you add, the lower it goes. So, to keep the average at least 13, you can't add too many. So, the maximum number of new members you can add is 8. Therefore, the minimum number of new members needed to ensure the average doesn't drop below 13 is 0, but that's trivial. Alternatively, if the CEO wants to add some number of new members, say n, and wants to ensure that the average is at least 13, then n must be ≤8. So, the minimum number of new members that can be added without violating the average is 0, but that's not useful. Alternatively, perhaps the problem is that the CEO wants to add new members, and wants to ensure that the average experience is at least 13, so the minimum number of new members that must be added to achieve that. But that doesn't make sense because adding new members with lower average will lower the overall average. So, the only way to ensure the average doesn't drop below 13 is to not add any new members. But that's trivial.Wait, perhaps I'm overcomplicating. Let's re-express the inequality:(180 + 10n)/(12 + n) ≥ 13We solved it and found n ≤8. So, the maximum number of new members that can be added without the average dropping below 13 is 8. Therefore, if the CEO wants to add new members without dropping the average below 13, they can add up to 8 new members. So, the minimum number of new members needed to ensure the average doesn't drop below 13 is 0, but that's trivial. Alternatively, if the CEO wants to add new members, the maximum number they can add without dropping below 13 is 8. So, perhaps the answer is 8.But the problem says "minimum number of new members needed such that the average experience of the entire board does not drop below 13 years." So, maybe it's asking for the minimum number of new members that must be added to achieve an average of at least 13. But wait, the current average is 15, which is already above 13. So, adding new members with lower average will lower the overall average. Therefore, to keep the average at least 13, the CEO can add up to 8 new members. So, the minimum number of new members needed to ensure the average doesn't drop below 13 is 0, but that's trivial. Alternatively, if the CEO wants to add new members, the maximum number they can add is 8. So, perhaps the answer is 8.Wait, let me think again. If the CEO adds 8 new members, the total experience becomes 180 + 80 = 260, and total members 20. 260/20 = 13, which is exactly 13. So, adding 8 new members will bring the average down to 13. If they add fewer than 8, the average will be higher than 13. So, the minimum number of new members needed to ensure the average doesn't drop below 13 is 0, but that's trivial. Alternatively, if the CEO wants to add new members, the maximum number they can add without dropping below 13 is 8. So, perhaps the answer is 8.But the problem says "minimum number of new members needed such that the average experience does not drop below 13 years." So, perhaps it's asking for the minimum number of new members that must be added to achieve an average of at least 13. But since the current average is 15, which is already above 13, adding any number of new members will lower the average. So, to ensure that the average doesn't drop below 13, the CEO can add up to 8 new members. Therefore, the minimum number of new members needed to ensure the average doesn't drop below 13 is 0, but that's trivial. Alternatively, if the CEO wants to add new members, the maximum number they can add is 8. So, perhaps the answer is 8.Wait, maybe I'm overcomplicating. Let's just solve the inequality again:(180 + 10n)/(12 + n) ≥ 13Multiply both sides by (12 + n):180 + 10n ≥ 156 + 13nSubtract 10n:180 ≥ 156 + 3nSubtract 156:24 ≥ 3nDivide by 3:8 ≥ nSo, n ≤8. Therefore, the maximum number of new members that can be added without the average dropping below 13 is 8. So, the answer is 8.But the problem says "minimum number of new members needed", which is confusing. Maybe it's a translation issue, and they meant "maximum". So, I think the answer is 8.So, to summarize:Sub-problem 1: Add 4 women and 4 men.Sub-problem 2: Add a maximum of 8 new members to keep the average experience at least 13. But since the problem asks for the minimum number, which is 0, but that's trivial, so perhaps the answer is 8.Wait, but in Sub-problem 1, the CEO is adding members to achieve gender diversity, and in Sub-problem 2, it's a separate consideration about experience. So, perhaps the CEO wants to add members for both reasons, but they are separate problems. So, in Sub-problem 2, it's just about experience, regardless of gender. So, the answer is 8.But let me double-check the calculation:If n=8:Total experience = 180 + 8*10 = 260Total members = 12 +8=20260/20=13, which is exactly 13.If n=9:Total experience=180+90=270Total members=21270/21≈12.857, which is below 13.So, yes, n=8 is the maximum number that can be added without dropping below 13. Therefore, the minimum number of new members needed to ensure the average doesn't drop below 13 is 0, but that's trivial. Alternatively, if the CEO wants to add new members, the maximum number is 8. So, perhaps the answer is 8.But the problem says "minimum number of new members needed such that the average experience does not drop below 13 years." So, perhaps it's asking for the minimum number of new members that must be added to achieve an average of at least 13. But since the current average is 15, which is already above 13, adding any number of new members will lower the average. So, to ensure that the average doesn't drop below 13, the CEO can add up to 8 new members. Therefore, the minimum number of new members needed to ensure the average doesn't drop below 13 is 0, but that's trivial. Alternatively, if the CEO wants to add new members, the maximum number they can add is 8. So, perhaps the answer is 8.I think the answer is 8.</think>

🧠:<think>Alright, so I have this problem where a politician is trying to allocate medical resources to different regions, especially focusing on areas with higher populations of individuals with disabilities. The goal is to make sure that everyone has equal access to medical services. There are two parts to this problem: first, formulating an optimization model to minimize the total unmet demand, and second, adding an equity constraint to ensure no region gets too much unless it really needs it.Let me start with the first part. I need to define the problem mathematically. So, we have regions ( R_1 ) to ( R_5 ) with population densities ( P = {100, 200, 150, 120, 180} ). Each region has a certain number of people with disabilities, which is 10% of the population, so ( beta = 0.1 ). The medical resources are of three types, with availabilities ( A = {300, 450, 500} ). First, I need to calculate the demand ( D_i ) for each region. Since the demand is based on population density and the proportion of individuals with disabilities, I can compute ( D_i = P_i times beta ). So, for each region:- ( D_1 = 100 times 0.1 = 10 )- ( D_2 = 200 times 0.1 = 20 )- ( D_3 = 150 times 0.1 = 15 )- ( D_4 = 120 times 0.1 = 12 )- ( D_5 = 180 times 0.1 = 18 )Wait, but these demands seem really low compared to the resource availabilities. Maybe I misunderstood. Perhaps the demand is not just the number of people but the number of medical services needed. Maybe each person with a disability requires multiple resources? Or perhaps the population densities are in thousands? The problem doesn't specify, so I might have to assume that ( P_i ) is the number of people, and ( D_i ) is the number of people with disabilities, each needing one unit of resource. But then, the resources are 300, 450, 500, which are much larger than the total demand.Let me calculate the total demand: 10 + 20 + 15 + 12 + 18 = 75. The total resources are 300 + 450 + 500 = 1250. So, the resources are way more than the demand. That seems odd. Maybe the population densities are in some other units, or perhaps each resource unit serves multiple people. Hmm, the problem says "medical resources" which could be things like hospital beds, doctors, etc., but without more context, I might have to proceed with the given numbers.So, moving on. The first problem is to allocate resources ( M_j ) to regions ( R_i ) to minimize the total unmet demand. Let me define the variables. Let ( x_{ij} ) be the amount of resource ( M_j ) allocated to region ( R_i ). Then, the total allocation to region ( R_i ) is ( sum_{j=1}^m x_{ij} ). The unmet demand for region ( R_i ) would be ( D_i - sum_{j=1}^m x_{ij} ), but since we can't have negative unmet demand, we take the maximum of that and zero. So, the total unmet demand is ( sum_{i=1}^n max(D_i - sum_{j=1}^m x_{ij}, 0) ).But in optimization, we usually avoid using max functions because they are non-linear. Instead, we can introduce a slack variable ( s_i ) which represents the unmet demand for region ( i ). Then, we have the constraint ( sum_{j=1}^m x_{ij} + s_i = D_i ), with ( s_i geq 0 ). The objective function becomes minimizing ( sum_{i=1}^n s_i ).So, the optimization problem is:Minimize ( sum_{i=1}^n s_i )Subject to:For each region ( i ):( sum_{j=1}^m x_{ij} + s_i = D_i )( s_i geq 0 )For each resource ( j ):( sum_{i=1}^n x_{ij} leq A_j )( x_{ij} geq 0 )This is a linear program. Now, considering the given data, let's plug in the numbers.Given ( D = {10, 20, 15, 12, 18} ), and ( A = {300, 450, 500} ). Since the total demand is 75 and the total resources are 1250, the optimal solution would be to allocate all the demand, meaning ( s_i = 0 ) for all ( i ), and the total unmet demand is zero. But perhaps the problem is more complex because each resource might have different types, and each region might require different types of resources. Wait, the problem doesn't specify that. It just says "medical resources" in general. So maybe each resource is interchangeable, and we can allocate any resource to any region.But in that case, the total resources are more than enough, so the optimal allocation is to meet all demands. But maybe I'm missing something. Perhaps each resource has a different cost or something, but the problem doesn't mention that. It just says to minimize the total unmet demand, so the optimal is to allocate as much as possible, which in this case is meeting all demands.But let's proceed formally. The mathematical model is as I described above.Now, moving on to the second part: the equity adjustment problem. The constraint is that no region receives more than 30% of the total resources unless its demand is significantly higher than others, specifically more than 1.5 times the average demand.First, let's compute the average demand. The total demand is 75, so average demand per region is 75/5 = 15. So, a region's demand is significantly higher if it's more than 1.5 * 15 = 22.5. Looking at the demands: 10, 20, 15, 12, 18. None of these exceed 22.5, so none of the regions qualify for the exception. Therefore, the equity constraint applies to all regions: no region can receive more than 30% of the total resources.Total resources are 1250, so 30% of that is 375. Therefore, for each region ( i ), the total allocation ( sum_{j=1}^m x_{ij} leq 375 ).But wait, the total demand is 75, which is much less than 375. So, even if we allocate 75 across all regions, each region's allocation would be way below 375. Therefore, the equity constraint is automatically satisfied because the total demand is much lower than the 30% threshold. So, in this case, the optimal allocation remains the same as in part 1, with all demands met and no unmet demand.But perhaps I'm misunderstanding the equity constraint. Maybe it's not about the total resources allocated to a region, but the proportion of each resource type. Or perhaps it's about the allocation per resource type. Let me re-read the problem.The equity constraint says: "no region receives more than a certain percentage ( alpha ) of the total resources unless its demand is significantly higher than other regions (e.g., more than 1.5 times the average demand)."So, it's about the total resources allocated to a region. If a region's demand is more than 1.5 times the average, it can receive more than ( alpha ) of the total resources. Otherwise, it cannot.In our case, the average demand is 15, so 1.5 times is 22.5. None of the regions have demand above that, so all regions must receive no more than 30% of the total resources.Total resources are 1250, so 30% is 375. Therefore, for each region ( i ), ( sum_{j=1}^m x_{ij} leq 375 ).But since the total demand is 75, and 75 is much less than 5*375=1875, the constraint is not binding. So, the optimal solution remains the same as in part 1.However, if the total demand were higher, say, if the total demand was more than 1875, then the constraint would come into play. But in this case, it doesn't.But perhaps the equity constraint is per resource type. Let me think again. The problem says "no region receives more than a certain percentage ( alpha ) of the total resources". So, it's about the total resources, not per resource type. So, the constraint is on the sum across all resources allocated to a region.Therefore, in our case, since the total demand is 75, and each region's allocation is much less than 375, the constraint doesn't affect the allocation.But to be thorough, let's formalize the constraint. For each region ( i ):If ( D_i > 1.5 times text{average demand} ), then no constraint on the allocation.Else, ( sum_{j=1}^m x_{ij} leq alpha times sum_{j=1}^m A_j ).In our case, average demand is 15, so 1.5*15=22.5. Since all ( D_i leq 20 ), which is less than 22.5, the constraint applies to all regions.Thus, for each region ( i ), ( sum_{j=1}^m x_{ij} leq 0.3 times 1250 = 375 ).But since the total demand is 75, and each region's demand is less than 375, the constraint is not binding. Therefore, the optimal solution remains the same.However, if we had a region with demand higher than 22.5, say, 25, then that region could receive more than 375 resources, but in our case, none do.So, the mathematical model for part 2 is the same as part 1, with the addition of the constraints:For each region ( i ), if ( D_i leq 1.5 times text{average demand} ), then ( sum_{j=1}^m x_{ij} leq alpha times sum_{j=1}^m A_j ).In our case, since all ( D_i leq 22.5 ), we add:For each ( i ), ( sum_{j=1}^m x_{ij} leq 375 ).But since the total demand is 75, and 75 < 5*375, the constraints are not binding.Therefore, the optimal allocation is the same as in part 1, with all demands met and no unmet demand.But perhaps I should consider that each resource type has its own availability, and the equity constraint is per resource type. Let me check the problem statement again.The problem says: "no region receives more than a certain percentage ( alpha ) of the total resources unless its demand is significantly higher than other regions."So, it's about the total resources, not per resource type. So, the constraint is on the sum of all resources allocated to a region.Therefore, the constraints are:For each region ( i ):If ( D_i > 1.5 times text{average demand} ), then no constraint.Else, ( sum_{j=1}^m x_{ij} leq alpha times sum_{j=1}^m A_j ).In our case, since all ( D_i leq 22.5 ), we have:For each ( i ), ( sum_{j=1}^m x_{ij} leq 0.3 times 1250 = 375 ).But since the total demand is 75, and each region's demand is less than 375, the constraints are not binding.Therefore, the optimal allocation is to meet all demands, with ( x_{ij} ) allocated such that each region gets exactly its demand, and the slack variables ( s_i = 0 ).But perhaps I'm missing something. Maybe the resources are different types, and each region has different needs for each resource type. The problem doesn't specify that, so I have to assume that resources are interchangeable.Alternatively, perhaps each resource type has a different cost or benefit, but since the problem only mentions minimizing unmet demand, I think the resources are interchangeable.Therefore, the mathematical models for both parts are as follows.For part 1:Minimize ( sum_{i=1}^5 s_i )Subject to:For each region ( i ):( sum_{j=1}^3 x_{ij} + s_i = D_i )( s_i geq 0 )For each resource ( j ):( sum_{i=1}^5 x_{ij} leq A_j )( x_{ij} geq 0 )For part 2, we add the equity constraints:For each region ( i ):( sum_{j=1}^3 x_{ij} leq 375 )But since the total demand is 75, and 75 < 5*375, this doesn't change the solution.However, if we had a region with demand exceeding 22.5, say, 25, then that region could receive more than 375 resources, but in our case, none do.So, the final models are as described.</think>

🧠:<think>Okay, so I have this problem about optimizing crop yield, and I need to figure out two things. First, calculate the total water distributed across the entire field, and second, determine the optimal coordinates that maximize the crop yield. Let me take it step by step.Starting with the first part: calculating the total water distributed. The water distribution is given by the function ( W(x, y) = A sinleft(frac{pi x}{L_x}right)sinleft(frac{pi y}{L_y}right) ). Since the field is a rectangular grid, I think I need to integrate this function over the entire area to get the total water.So, the total water ( W_{total} ) should be the double integral of ( W(x, y) ) over the field. That is,[W_{total} = int_{0}^{L_x} int_{0}^{L_y} A sinleft(frac{pi x}{L_x}right)sinleft(frac{pi y}{L_y}right) , dy , dx]Hmm, integrating sine functions. I remember that the integral of ( sin(a x) ) is ( -frac{1}{a} cos(a x) ). Let me separate the integrals since the function is a product of functions in x and y.So,[W_{total} = A left( int_{0}^{L_x} sinleft(frac{pi x}{L_x}right) dx right) left( int_{0}^{L_y} sinleft(frac{pi y}{L_y}right) dy right)]Let me compute each integral separately.First, the integral over x:[int_{0}^{L_x} sinleft(frac{pi x}{L_x}right) dx]Let me make a substitution: let ( u = frac{pi x}{L_x} ), so ( du = frac{pi}{L_x} dx ), which means ( dx = frac{L_x}{pi} du ). When x=0, u=0; when x=L_x, u=π.So, substituting,[int_{0}^{pi} sin(u) cdot frac{L_x}{pi} du = frac{L_x}{pi} left[ -cos(u) right]_0^{pi} = frac{L_x}{pi} left( -cos(pi) + cos(0) right)]We know that ( cos(pi) = -1 ) and ( cos(0) = 1 ), so:[frac{L_x}{pi} ( -(-1) + 1 ) = frac{L_x}{pi} (1 + 1) = frac{2 L_x}{pi}]Similarly, the integral over y will be the same, replacing ( L_x ) with ( L_y ):[int_{0}^{L_y} sinleft(frac{pi y}{L_y}right) dy = frac{2 L_y}{pi}]So, putting it all together,[W_{total} = A cdot frac{2 L_x}{pi} cdot frac{2 L_y}{pi} = A cdot frac{4 L_x L_y}{pi^2}]Wait, is that correct? Let me double-check the substitution steps. Yes, I think so. The integral of sine over its period gives a factor of 2, and since we're integrating over the entire length, it's 2 times the length divided by π. So, multiplying both integrals gives 4 times the area divided by π squared. That seems right.So, the total water distributed is ( frac{4 A L_x L_y}{pi^2} ). I think that's the answer for the first part.Moving on to the second part: maximizing the crop yield ( Y(x, y) = k W(x, y)^{alpha} N(x, y)^{beta} ). Given that ( N(x, y) = B e^{-lambda(x^2 + y^2)} ), I need to find the coordinates ( (x_0, y_0) ) that maximize Y.First, let's write out Y in terms of x and y:[Y(x, y) = k left( A sinleft(frac{pi x}{L_x}right)sinleft(frac{pi y}{L_y}right) right)^{alpha} left( B e^{-lambda(x^2 + y^2)} right)^{beta}]Simplify this expression:[Y(x, y) = k A^{alpha} B^{beta} left( sinleft(frac{pi x}{L_x}right)sinleft(frac{pi y}{L_y}right) right)^{alpha} e^{-beta lambda (x^2 + y^2)}]Since k, A, B, α, β, λ, Lx, Ly are constants, the function to maximize is:[f(x, y) = left( sinleft(frac{pi x}{L_x}right)sinleft(frac{pi y}{L_y}right) right)^{alpha} e^{-beta lambda (x^2 + y^2)}]To find the maximum, I need to take the partial derivatives with respect to x and y, set them equal to zero, and solve for x and y.But before jumping into calculus, maybe I can see if the function is separable. Let's see:[f(x, y) = left( sinleft(frac{pi x}{L_x}right) right)^{alpha} left( sinleft(frac{pi y}{L_y}right) right)^{alpha} e^{-beta lambda x^2} e^{-beta lambda y^2}]So, it's a product of functions in x and y. Therefore, the maximum occurs when each component is maximized, but since they are multiplied together, it's not straightforward. However, perhaps the maximum occurs at the same x and y that maximize each term, but considering the trade-off between the sine terms and the exponential decay.Alternatively, maybe we can take the logarithm of f(x,y) to make differentiation easier, since log is a monotonic function.Let me define:[ln f(x, y) = alpha ln left( sinleft(frac{pi x}{L_x}right) right) + alpha ln left( sinleft(frac{pi y}{L_y}right) right) - beta lambda (x^2 + y^2)]Now, to maximize f(x,y), we can maximize ln f(x,y). Let's compute the partial derivatives.First, partial derivative with respect to x:[frac{partial}{partial x} ln f(x, y) = alpha cdot frac{pi}{L_x} cotleft( frac{pi x}{L_x} right) - 2 beta lambda x]Similarly, partial derivative with respect to y:[frac{partial}{partial y} ln f(x, y) = alpha cdot frac{pi}{L_y} cotleft( frac{pi y}{L_y} right) - 2 beta lambda y]Set both partial derivatives equal to zero to find critical points.So, setting the x derivative to zero:[alpha cdot frac{pi}{L_x} cotleft( frac{pi x}{L_x} right) - 2 beta lambda x = 0]Similarly for y:[alpha cdot frac{pi}{L_y} cotleft( frac{pi y}{L_y} right) - 2 beta lambda y = 0]So, we have two equations:1. ( alpha cdot frac{pi}{L_x} cotleft( frac{pi x}{L_x} right) = 2 beta lambda x )2. ( alpha cdot frac{pi}{L_y} cotleft( frac{pi y}{L_y} right) = 2 beta lambda y )These equations are symmetric in x and y, so perhaps the solution will have x and y related in a symmetric way.Let me denote:For x:[cotleft( frac{pi x}{L_x} right) = frac{2 beta lambda L_x}{alpha pi} x]Similarly for y:[cotleft( frac{pi y}{L_y} right) = frac{2 beta lambda L_y}{alpha pi} y]Hmm, these are transcendental equations, meaning they can't be solved algebraically, only numerically. But perhaps we can make some approximations or find a relationship between x and y.Wait, but the problem says "determine the optimal coordinates (x0, y0)". It doesn't specify whether to express them in terms of the given constants or to solve numerically. Since the equations are transcendental, I think we can only express the solution implicitly or suggest that numerical methods are needed.But maybe there's a way to relate x and y. Let me see.Suppose that the optimal point is at the center of the field. That is, x = Lx/2, y = Ly/2. Let's test if that satisfies the equations.Compute cot(π x / Lx) when x = Lx/2: cot(π/2) = 0. So the left side is zero. The right side is 2 β λ x. Unless x=0, which it's not, the right side is non-zero. So, unless β or λ is zero, which they aren't, the center is not the solution.Alternatively, maybe the maximum is at some symmetric point. Let's suppose that x0 = y0. Then, we can write:From the x equation:[cotleft( frac{pi x}{L_x} right) = frac{2 beta lambda L_x}{alpha pi} x]From the y equation:[cotleft( frac{pi x}{L_y} right) = frac{2 beta lambda L_y}{alpha pi} x]But unless Lx = Ly, these are different equations. So, unless Lx = Ly, x0 ≠ y0.Therefore, the optimal coordinates x0 and y0 are solutions to the equations:[cotleft( frac{pi x}{L_x} right) = frac{2 beta lambda L_x}{alpha pi} x][cotleft( frac{pi y}{L_y} right) = frac{2 beta lambda L_y}{alpha pi} y]These equations must be solved numerically for x and y. However, since the problem asks to determine the optimal coordinates, perhaps we can express them in terms of the given constants.Alternatively, maybe we can find a relationship between x and y.Wait, let's think about the behavior of the function. The sine terms are maximum at x = Lx/2, y = Ly/2, but the exponential term is maximum at x=0, y=0. So, the maximum of Y(x,y) is somewhere between the center and the origin.But it's not clear where exactly. So, perhaps the maximum occurs at some point where the trade-off between the increasing sine terms and decreasing exponential terms is balanced.Given that the equations are transcendental, I think the answer is that the optimal coordinates (x0, y0) satisfy the equations:[cotleft( frac{pi x_0}{L_x} right) = frac{2 beta lambda L_x}{alpha pi} x_0][cotleft( frac{pi y_0}{L_y} right) = frac{2 beta lambda L_y}{alpha pi} y_0]Therefore, to find x0 and y0, one would need to solve these equations numerically.Alternatively, if we consider small x and y, perhaps we can approximate cot(z) ≈ 1/z - z/3 for small z. But given that the sine function is maximum at Lx/2 and Ly/2, which are not necessarily small, unless Lx and Ly are large.But without knowing the specific values of the constants, it's hard to make an approximation.Alternatively, maybe we can consider that the maximum occurs at x0 = y0 = 0, but that can't be because the sine terms are zero there, so Y(x,y) would be zero. Similarly, at the edges, the sine terms are zero, so the maximum must be somewhere inside the field.Alternatively, perhaps the maximum is at the point where the derivative of the logarithm is zero, which we already set up.So, in conclusion, the optimal coordinates (x0, y0) are the solutions to the system:[cotleft( frac{pi x_0}{L_x} right) = frac{2 beta lambda L_x}{alpha pi} x_0][cotleft( frac{pi y_0}{L_y} right) = frac{2 beta lambda L_y}{alpha pi} y_0]These equations would need to be solved numerically for x0 and y0 given the specific values of the constants.Alternatively, if we consider that the field is square, i.e., Lx = Ly, then the equations become symmetric, and x0 = y0. Then, we can write:[cotleft( frac{pi x_0}{L} right) = frac{2 beta lambda L}{alpha pi} x_0]But even then, it's still a transcendental equation.So, unless there's a specific substitution or method I'm missing, I think this is as far as we can go analytically. Therefore, the optimal coordinates are the solutions to those two equations.Wait, but maybe I can consider the case where α = β = 1, just to see if it simplifies. Let me try that.If α = β = 1, then the equations become:[cotleft( frac{pi x}{L_x} right) = frac{2 lambda L_x}{pi} x][cotleft( frac{pi y}{L_y} right) = frac{2 lambda L_y}{pi} y]Still transcendental. So, no help.Alternatively, maybe if we set z = π x / Lx, then x = (Lx / π) z. Then, the equation becomes:[cot(z) = frac{2 beta lambda L_x}{alpha pi} cdot frac{Lx}{pi} z = frac{2 beta lambda L_x^2}{alpha pi^2} z]So,[cot(z) = k z]where ( k = frac{2 beta lambda L_x^2}{alpha pi^2} )Similarly for y:[cot(w) = m w]where ( w = frac{pi y}{L_y} ) and ( m = frac{2 beta lambda L_y^2}{alpha pi^2} )So, we have:[cot(z) = k z][cot(w) = m w]These are standard transcendental equations. The solutions can be found graphically or numerically. For example, the solutions for z and w are the points where the cotangent curve intersects the line k z and m w, respectively.Therefore, the optimal x0 and y0 can be found by solving these equations for z and w, then converting back to x and y.But without specific values, we can't find numerical solutions. So, in conclusion, the optimal coordinates are given implicitly by those equations.Therefore, the answer is that the optimal coordinates (x0, y0) satisfy:[cotleft( frac{pi x_0}{L_x} right) = frac{2 beta lambda L_x}{alpha pi} x_0][cotleft( frac{pi y_0}{L_y} right) = frac{2 beta lambda L_y}{alpha pi} y_0]These equations must be solved numerically to find x0 and y0.Wait, but the problem says "determine the optimal coordinates (x0, y0)". So, perhaps the answer is just stating that they satisfy those equations, or maybe expressing x0 and y0 in terms of the given constants.Alternatively, if we consider that the maximum occurs where the derivative of the logarithm is zero, which we have already set up, so perhaps that's the answer.Alternatively, maybe we can express x0 and y0 in terms of the Lambert W function or something, but I don't think so because cotangent is involved, not exponential.Alternatively, perhaps we can make an approximation for small x and y, but as I thought earlier, unless the field is very large, x and y might not be small.Alternatively, maybe we can assume that x0 and y0 are proportional to Lx and Ly, but without more information, it's hard to say.Alternatively, perhaps the maximum occurs at the same point where the water distribution is maximum, but that's at x = Lx/2, y = Ly/2, but as I saw earlier, at that point, the exponential term is e^{-βλ(Lx²/4 + Ly²/4)}, which might be small, so the product might not be maximum there.Alternatively, maybe the maximum is somewhere closer to the origin.Alternatively, perhaps we can set up a ratio between the two terms.Wait, let me think about the ratio of the partial derivatives.Wait, no, since we set each partial derivative to zero separately, they don't directly relate.Alternatively, perhaps we can consider that the optimal x and y are such that the ratio of the partial derivatives is zero, but since both are set to zero, it's not helpful.Alternatively, maybe I can consider the ratio of the two equations.From the x equation:[cotleft( frac{pi x}{L_x} right) = frac{2 beta lambda L_x}{alpha pi} x]From the y equation:[cotleft( frac{pi y}{L_y} right) = frac{2 beta lambda L_y}{alpha pi} y]So, the ratio of the left sides is:[frac{cotleft( frac{pi x}{L_x} right)}{cotleft( frac{pi y}{L_y} right)} = frac{L_x x}{L_y y}]But I don't know if that helps.Alternatively, perhaps if we assume that x0 / Lx = y0 / Ly, meaning the optimal point is scaled similarly in both directions, then we can set x0 = (Lx / Ly) y0, but that might not necessarily be the case.Alternatively, perhaps the optimal x0 and y0 are proportional to Lx and Ly, but again, without more information, it's hard to say.In conclusion, I think the optimal coordinates are given by solving those two transcendental equations, which can't be solved analytically, so the answer is that (x0, y0) satisfy those equations.But maybe the problem expects a different approach. Let me think again.Wait, the crop yield function is Y(x,y) = k W^α N^β. Maybe we can use Lagrange multipliers or something, but since we're maximizing over x and y, it's just taking partial derivatives.Alternatively, maybe we can take the ratio of the partial derivatives and set them equal to the ratio of the variables, but I don't think that helps.Alternatively, maybe we can write the equations as:For x:[cotleft( frac{pi x}{L_x} right) = C_x x]where ( C_x = frac{2 beta lambda L_x}{alpha pi} )Similarly for y:[cotleft( frac{pi y}{L_y} right) = C_y y]where ( C_y = frac{2 beta lambda L_y}{alpha pi} )So, the solutions are x and y such that cot(π x / Lx) is proportional to x, with proportionality constants Cx and Cy.These are standard equations in physics, like the transcendental equations for the modes of a drum or something. The solutions can be found numerically, but not analytically.Therefore, the optimal coordinates are the solutions to these equations, which can be found using numerical methods like Newton-Raphson.So, in conclusion, the optimal coordinates (x0, y0) are given by solving:[cotleft( frac{pi x_0}{L_x} right) = frac{2 beta lambda L_x}{alpha pi} x_0][cotleft( frac{pi y_0}{L_y} right) = frac{2 beta lambda L_y}{alpha pi} y_0]These equations must be solved numerically for x0 and y0.Alternatively, if we consider that the maximum occurs where the derivative of the logarithm is zero, which is what we did, then that's the answer.So, summarizing:1. The total water distributed is ( frac{4 A L_x L_y}{pi^2} ).2. The optimal coordinates (x0, y0) satisfy the transcendental equations above, which must be solved numerically.I think that's as far as I can go without specific numerical values.</think>

🧠:<think>Alright, so I have this problem about evaluating two military strategies using differential equations. Let me try to break it down step by step.Starting with Strategy A. The success rate is modeled by the differential equation:[frac{dS_A}{dt} = alpha S_A(t) (1 - S_A(t)) - beta S_A(t) cdot R(t)]And the resistance function ( R(t) ) is given by:[R(t) = frac{1}{1 + e^{-gamma(t - T_0)}}]I need to find the equilibrium points and analyze their stability. Hmm, okay. So, equilibrium points occur where the derivative is zero, right? So, set ( frac{dS_A}{dt} = 0 ).So, setting the equation to zero:[0 = alpha S_A (1 - S_A) - beta S_A R(t)]Let me factor out ( S_A ):[0 = S_A left[ alpha (1 - S_A) - beta R(t) right]]So, the solutions are either ( S_A = 0 ) or ( alpha (1 - S_A) - beta R(t) = 0 ).Solving the second equation for ( S_A ):[alpha (1 - S_A) = beta R(t)][1 - S_A = frac{beta}{alpha} R(t)][S_A = 1 - frac{beta}{alpha} R(t)]So, the equilibrium points are ( S_A = 0 ) and ( S_A = 1 - frac{beta}{alpha} R(t) ).Wait, but ( R(t) ) is a function of time, so does that mean the equilibrium points are time-dependent? That complicates things a bit because usually, equilibrium points are constant. Maybe I need to think of this as a non-autonomous system, which can have time-dependent equilibria.But perhaps for stability analysis, I need to consider the system as autonomous? Or maybe linearize around the equilibria.Alternatively, maybe I can rewrite the equation in terms of ( R(t) ) and see if I can analyze it as a function of ( t ).But let me first consider the form of ( R(t) ). It's a logistic function, right? It starts at 0 when ( t ) is much less than ( T_0 ), and approaches 1 as ( t ) becomes much larger than ( T_0 ). The parameter ( gamma ) controls the steepness of the transition.So, ( R(t) ) is a smooth transition from 0 to 1 over time. Therefore, the equilibrium ( S_A = 1 - frac{beta}{alpha} R(t) ) will transition from ( 1 - 0 = 1 ) to ( 1 - frac{beta}{alpha} ) as ( t ) increases.Wait, so initially, when ( t ) is small, ( R(t) ) is near 0, so the equilibrium is near 1. As time goes on, ( R(t) ) increases, so the equilibrium decreases towards ( 1 - frac{beta}{alpha} ).But for stability, I need to look at the derivative of ( frac{dS_A}{dt} ) with respect to ( S_A ) at the equilibrium points.So, the function is:[frac{dS_A}{dt} = alpha S_A (1 - S_A) - beta S_A R(t)]Let me compute the derivative with respect to ( S_A ):[frac{d}{dS_A} left( frac{dS_A}{dt} right) = alpha (1 - S_A) - alpha S_A - beta R(t)]Simplify:[= alpha - 2 alpha S_A - beta R(t)]So, at equilibrium points, ( S_A = 0 ) or ( S_A = 1 - frac{beta}{alpha} R(t) ).First, evaluate at ( S_A = 0 ):[frac{d}{dS_A} bigg|_{S_A=0} = alpha - 0 - beta R(t) = alpha - beta R(t)]Since ( R(t) ) is between 0 and 1, and ( alpha ) and ( beta ) are positive constants, the sign of this derivative depends on whether ( alpha > beta R(t) ) or not.If ( alpha > beta R(t) ), the derivative is positive, so the equilibrium ( S_A = 0 ) is unstable. If ( alpha < beta R(t) ), the derivative is negative, so ( S_A = 0 ) is stable.But since ( R(t) ) increases over time, initially ( R(t) ) is small, so ( alpha - beta R(t) ) is positive, making ( S_A = 0 ) unstable. As ( R(t) ) increases, there will be a time when ( alpha = beta R(t) ), beyond which ( S_A = 0 ) becomes stable.Wait, but this is a time-dependent system. So, the stability of the equilibrium points changes over time.Alternatively, maybe I should consider the system as autonomous by treating ( R(t) ) as a parameter that changes with time.But perhaps another approach is to consider the system in different time intervals.When ( t ) is very small, ( R(t) ) is near 0, so the equation becomes:[frac{dS_A}{dt} approx alpha S_A (1 - S_A)]Which is a logistic growth model with equilibrium at ( S_A = 1 ). So, initially, the success rate grows logistically towards 1.As ( t ) increases, ( R(t) ) becomes significant, and the equation becomes:[frac{dS_A}{dt} = alpha S_A (1 - S_A) - beta S_A R(t)]So, the term ( - beta S_A R(t) ) acts as a damping term, reducing the growth rate.At equilibrium, ( S_A = 1 - frac{beta}{alpha} R(t) ). So, as ( R(t) ) approaches 1, the equilibrium approaches ( 1 - frac{beta}{alpha} ).Therefore, the system transitions from a high success rate equilibrium to a lower one as resistance increases.Now, for stability, let's consider the derivative at the non-zero equilibrium.At ( S_A = 1 - frac{beta}{alpha} R(t) ), plug into the derivative:[frac{d}{dS_A} bigg|_{S_A=1 - frac{beta}{alpha} R(t)} = alpha - 2 alpha left(1 - frac{beta}{alpha} R(t)right) - beta R(t)]Simplify:[= alpha - 2 alpha + 2 beta R(t) - beta R(t)][= -alpha + beta R(t)]So, the derivative is ( -alpha + beta R(t) ).Again, since ( R(t) ) is between 0 and 1, the sign depends on ( beta R(t) ) compared to ( alpha ).If ( beta R(t) > alpha ), the derivative is positive, so the equilibrium is unstable. If ( beta R(t) < alpha ), the derivative is negative, so the equilibrium is stable.Wait, but earlier, at ( S_A = 0 ), the derivative was ( alpha - beta R(t) ). So, when ( alpha > beta R(t) ), ( S_A = 0 ) is unstable and ( S_A = 1 - frac{beta}{alpha} R(t) ) is stable because its derivative is ( -alpha + beta R(t) ), which is negative when ( beta R(t) < alpha ).Conversely, when ( beta R(t) > alpha ), the equilibrium ( S_A = 0 ) becomes stable, and the other equilibrium becomes unstable.But since ( R(t) ) increases over time, initially, ( S_A = 1 - frac{beta}{alpha} R(t) ) is stable, and ( S_A = 0 ) is unstable. As ( R(t) ) crosses ( alpha / beta ), the stability flips.So, the system has two equilibrium points, one at 0 and another at ( 1 - frac{beta}{alpha} R(t) ). The stability depends on the value of ( R(t) ).Therefore, the equilibrium points are:1. ( S_A = 0 ), which is stable when ( beta R(t) > alpha ) and unstable otherwise.2. ( S_A = 1 - frac{beta}{alpha} R(t) ), which is stable when ( beta R(t) < alpha ) and unstable otherwise.So, over time, as ( R(t) ) increases, the system transitions from having a stable equilibrium at ( 1 - frac{beta}{alpha} R(t) ) to eventually ( S_A = 0 ) becoming stable.This suggests that initially, the success rate will grow towards ( 1 - frac{beta}{alpha} R(t) ), but as resistance increases beyond ( alpha / beta ), the success rate will start to decline towards 0.Okay, that seems reasonable.Now, moving on to Strategy B. The success rate follows a geometric Brownian motion:[dS_B(t) = mu S_B(t) dt + sigma S_B(t) dW(t)]With ( S_B(0) = S_0 ). I need to find the expected success rate ( E[S_B(t)] ) and the variance of ( S_B(t) ).I remember that geometric Brownian motion is often used to model stock prices, and the solution is a log-normal distribution. The expected value and variance can be derived from the properties of the SDE.The general solution to this SDE is:[S_B(t) = S_0 expleft( left( mu - frac{sigma^2}{2} right) t + sigma W(t) right)]So, to find ( E[S_B(t)] ), we take the expectation of this expression.Since ( W(t) ) is a standard Wiener process, ( W(t) ) has mean 0 and variance ( t ). The exponential of a normal variable is log-normal, and the expectation of a log-normal variable can be computed.Recall that for ( X sim mathcal{N}(mu, sigma^2) ), ( E[e^X] = e^{mu + frac{sigma^2}{2}} ).In our case, the exponent is:[left( mu - frac{sigma^2}{2} right) t + sigma W(t)]Let me denote this exponent as ( Y(t) ):[Y(t) = left( mu - frac{sigma^2}{2} right) t + sigma W(t)]So, ( Y(t) ) is a normal random variable with mean ( left( mu - frac{sigma^2}{2} right) t ) and variance ( sigma^2 t ).Therefore, ( S_B(t) = S_0 e^{Y(t)} ).The expectation ( E[S_B(t)] ) is:[E[S_B(t)] = S_0 E[e^{Y(t)}] = S_0 e^{E[Y(t)] + frac{1}{2} text{Var}(Y(t))}]Because for a normal variable ( Y ), ( E[e^Y] = e^{mu_Y + frac{1}{2} sigma_Y^2} ).So, substituting:[E[Y(t)] = left( mu - frac{sigma^2}{2} right) t][text{Var}(Y(t)) = sigma^2 t]Therefore,[E[S_B(t)] = S_0 e^{left( mu - frac{sigma^2}{2} right) t + frac{1}{2} sigma^2 t} = S_0 e^{mu t}]Because the ( -frac{sigma^2}{2} t ) and ( +frac{sigma^2}{2} t ) cancel out.So, the expected success rate is ( S_0 e^{mu t} ).Now, for the variance of ( S_B(t) ). Since ( S_B(t) ) is log-normal, its variance can be found using the properties of the log-normal distribution.Recall that for ( X sim text{Lognormal}(mu, sigma^2) ), the variance is:[text{Var}(X) = (e^{sigma^2} - 1) e^{2mu}]In our case, ( S_B(t) = S_0 e^{Y(t)} ), where ( Y(t) ) has mean ( mu_Y = left( mu - frac{sigma^2}{2} right) t ) and variance ( sigma_Y^2 = sigma^2 t ).Therefore, ( S_B(t) ) is log-normal with parameters ( mu_Y ) and ( sigma_Y^2 ).So, the variance of ( S_B(t) ) is:[text{Var}(S_B(t)) = (e^{sigma_Y^2} - 1) e^{2 mu_Y}]Substituting ( mu_Y ) and ( sigma_Y^2 ):[text{Var}(S_B(t)) = left( e^{sigma^2 t} - 1 right) e^{2 left( mu - frac{sigma^2}{2} right) t}]Simplify the exponent:[2 left( mu - frac{sigma^2}{2} right) t = 2 mu t - sigma^2 t]So,[text{Var}(S_B(t)) = left( e^{sigma^2 t} - 1 right) e^{2 mu t - sigma^2 t} = left( e^{sigma^2 t} - 1 right) e^{2 mu t} e^{- sigma^2 t}]Simplify further:[= left( 1 - e^{- sigma^2 t} right) e^{2 mu t}]Alternatively, another way to compute variance is:Since ( S_B(t) = S_0 e^{Y(t)} ), then ( text{Var}(S_B(t)) = E[S_B(t)^2] - (E[S_B(t)])^2 ).We already have ( E[S_B(t)] = S_0 e^{mu t} ).Compute ( E[S_B(t)^2] ):[E[S_B(t)^2] = E[ S_0^2 e^{2 Y(t)} ] = S_0^2 E[ e^{2 Y(t)} ]]Again, ( Y(t) ) is normal with mean ( mu_Y = left( mu - frac{sigma^2}{2} right) t ) and variance ( sigma_Y^2 = sigma^2 t ).So, ( E[e^{2 Y(t)}] = e^{2 mu_Y + 2 sigma_Y^2} ).Wait, no, more accurately, for a normal variable ( Y sim mathcal{N}(mu_Y, sigma_Y^2) ), ( E[e^{k Y}] = e^{k mu_Y + frac{1}{2} k^2 sigma_Y^2} ).So, for ( k = 2 ):[E[e^{2 Y(t)}] = e^{2 mu_Y + 2 sigma_Y^2}]Substituting ( mu_Y ) and ( sigma_Y^2 ):[= e^{2 left( mu - frac{sigma^2}{2} right) t + 2 sigma^2 t}]Simplify:[= e^{2 mu t - sigma^2 t + 2 sigma^2 t} = e^{2 mu t + sigma^2 t}]Therefore,[E[S_B(t)^2] = S_0^2 e^{2 mu t + sigma^2 t}]Then, the variance is:[text{Var}(S_B(t)) = E[S_B(t)^2] - (E[S_B(t)])^2 = S_0^2 e^{2 mu t + sigma^2 t} - (S_0 e^{mu t})^2]Simplify:[= S_0^2 e^{2 mu t + sigma^2 t} - S_0^2 e^{2 mu t} = S_0^2 e^{2 mu t} (e^{sigma^2 t} - 1)]Which matches the earlier result.So, putting it all together, the expected success rate for Strategy B is ( S_0 e^{mu t} ), and the variance is ( S_0^2 e^{2 mu t} (e^{sigma^2 t} - 1) ).I think that's it. Let me just recap:For Strategy A, the equilibrium points are ( S_A = 0 ) and ( S_A = 1 - frac{beta}{alpha} R(t) ). The stability depends on whether ( beta R(t) ) is greater than or less than ( alpha ). Initially, ( S_A = 1 - frac{beta}{alpha} R(t) ) is stable, and as resistance increases, it becomes unstable, and ( S_A = 0 ) becomes stable.For Strategy B, the expected success rate grows exponentially as ( S_0 e^{mu t} ), and the variance grows as ( S_0^2 e^{2 mu t} (e^{sigma^2 t} - 1) ), which shows the increasing uncertainty over time due to the stochastic component.I think that covers both parts of the problem.</think>