answer:<think>Alright, so I have this problem where a fan from Japan wants to attend as many Brantley Gilbert concerts as possible during his North American tour. There are 12 concerts in total, each in different cities on different days. The fan can only go to up to 8 because of his budget. The cost to attend each concert is given by ( C_i = a_i + b_i d_i ). Here, ( a_i ) is the base cost of the ticket, ( b_i ) is the cost per kilometer traveled, and ( d_i ) is the distance from the previous concert city or from Tokyo if it's the first one.The first part of the problem is to figure out which combination of up to 8 concerts he can attend without exceeding his 15,000 budget, while maximizing the number of concerts. The second part is about minimizing the total travel distance if he wants to attend exactly 8 concerts.Okay, starting with the first problem. So, the goal is to maximize the number of concerts attended, which is up to 8, while keeping the total cost within 15,000. The cost for each concert depends on the base cost and the travel cost, which is based on the distance from the previous city or Tokyo.Hmm, this seems like a variation of the knapsack problem. In the classic knapsack, you maximize value with a weight constraint. Here, we want to maximize the number of concerts (which is like maximizing the count) with a budget constraint. But the twist is that the cost of each concert isn't fixed because it depends on the order in which you attend them. The travel cost ( b_i d_i ) depends on the distance from the previous city.So, it's not just a simple knapsack because the cost of each item (concert) depends on the sequence. That makes it more complicated. Maybe it's similar to the Traveling Salesman Problem (TSP) but with a budget constraint instead of a distance constraint.Wait, but the fan doesn't have to visit all cities, just up to 8. So, it's a combination of selecting a subset of cities (concerts) and then finding the order to visit them such that the total cost is minimized, allowing him to attend as many as possible within the budget.But since the number of concerts is limited to 8, and he wants to maximize that number, perhaps we need to find the 8 concerts that can be attended with the lowest total cost, considering the travel costs between them.But the problem is that the travel cost depends on the order. So, if we choose different orders, the total travel cost will vary. Therefore, to minimize the total cost for attending 8 concerts, we need to find the optimal route that visits 8 cities with the minimal total distance, starting from Tokyo.Wait, but the starting point is Tokyo for the first concert, and then each subsequent concert's cost depends on the distance from the previous one. So, the total cost will be the sum of all the base costs ( a_i ) plus the sum of all the travel costs ( b_i d_i ). Since ( b_i ) is constant for all cities, we can factor that out. So, the total cost is ( sum a_i + b sum d_i ).Therefore, to minimize the total cost, we need to minimize both the sum of ( a_i ) and the sum of ( d_i ). But since the fan wants to attend as many concerts as possible, up to 8, within the budget, it's a trade-off between the number of concerts and the total cost.But wait, the problem says "determine the combination of up to 8 concerts that maximizes the total number of concerts attended while staying within budget." So, he wants to attend as many as possible (so up to 8) without exceeding 15,000. So, it's not necessarily about minimizing the total cost, but ensuring that the total cost doesn't exceed 15,000 while attending as many as possible.But since the number is capped at 8, he wants to attend 8 concerts if possible, but if the cost is too high, he might have to attend fewer. So, the problem is to select a subset of concerts, up to 8, such that the total cost is within 15,000, and the number of concerts is maximized.But again, the cost depends on the order because of the travel distances. So, it's not just selecting 8 concerts with the lowest individual costs, because the travel costs depend on the sequence.This seems like a problem that can be modeled as a dynamic traveling salesman problem with a budget constraint. It's a bit complex because the cost is dependent on the path.Alternatively, maybe we can model it as a shortest path problem where each node represents a city, and the edges represent the cost to travel from one city to another, including the base cost of the concert.But with 12 cities, the state space could be manageable. Let me think.In dynamic programming terms, we can represent the state as (current city, number of concerts attended, total cost). But with 12 cities, 8 concerts, and a budget of 15,000, the state space might be too large.Alternatively, since the starting point is fixed (Tokyo), and each concert is in a different city, we can model this as a permutation problem where we select a subset of cities and find the optimal route through them.But this is getting complicated. Maybe another approach is needed.Wait, the problem says "the fan can only attend a maximum of 8 concerts due to budget constraints." So, the budget is 15,000, and the cost is ( C_i = a_i + b_i d_i ). Since ( b_i ) is constant for all cities, let's denote it as ( b ).So, total cost is ( sum_{i=1}^{n} a_i + b sum_{i=1}^{n} d_i ), where ( n ) is the number of concerts attended, up to 8.But the distances ( d_i ) are the distances from the previous city. So, the total travel distance is the sum of the distances between consecutive concerts, starting from Tokyo.Therefore, the total cost is ( sum a_i + b times text{total travel distance} ).So, to minimize the total cost for attending 8 concerts, we need to minimize both the sum of ( a_i ) and the total travel distance.But since the fan wants to maximize the number of concerts, he wants to attend 8 if possible. So, the problem reduces to selecting 8 concerts such that the sum of their base costs plus ( b ) times the total travel distance is less than or equal to 15,000.But how do we model the travel distance? It depends on the order of the concerts. So, the total travel distance is the sum of the distances between consecutive concerts, starting from Tokyo.This seems like the Traveling Salesman Problem with a subset of cities, where we need to select 8 cities to visit, starting from Tokyo, such that the total cost (base costs plus travel costs) is minimized, and then check if it's within the budget.But since the fan wants to attend as many concerts as possible, we need to see if attending 8 is possible. If the minimal total cost for 8 concerts is less than or equal to 15,000, then he can attend 8. Otherwise, he might have to attend fewer.But the problem is to determine the combination of up to 8 concerts that maximizes the number attended while staying within budget. So, if attending 8 is possible, that's the answer. If not, try 7, etc.But how do we determine the minimal total cost for attending 8 concerts? It's equivalent to solving a TSP for 8 cities, starting from Tokyo, with the total cost being the sum of base costs plus travel costs.This is a complex problem because it's a combination of selecting the subset and finding the optimal route.Alternatively, maybe we can simplify it by assuming that the order of the concerts doesn't affect the total travel distance, but that's not true because the distance depends on the sequence.Wait, but maybe we can precompute the distances between all pairs of cities and then model this as a graph where each node is a city, and edges have weights equal to the travel cost ( b times d_i ). Then, the problem becomes finding a path that starts at Tokyo, visits 8 cities, and has the minimal total cost (sum of base costs plus travel costs).But this is similar to the Traveling Salesman Problem with a fixed starting point and a fixed number of cities to visit.Given that there are 12 cities, and we need to visit 8, the number of possible paths is quite large. It might be computationally intensive, but perhaps with some optimization techniques, it can be solved.Alternatively, if we can find a way to approximate the solution, that might be feasible.But since this is a theoretical problem, maybe we can think of it in terms of dynamic programming.Let me try to outline a dynamic programming approach.Define the state as (current city, number of concerts attended, total cost). The goal is to find the minimal total cost for attending 8 concerts, starting from Tokyo.But with 12 cities, 8 concerts, and a budget of 15,000, the state space is 12 * 8 * 15,000, which is 1,440,000 states. That's manageable with some optimizations.Wait, but each state would need to keep track of the current city, the number of concerts attended, and the total cost. For each state, we can transition to other cities, updating the number of concerts and the total cost.But we need to ensure that we don't revisit cities, as each concert is in a different city.So, the state should also include the set of visited cities. But that would make the state space too large because the number of subsets is 2^12 = 4096, multiplied by 12 (current city) and 8 (number of concerts), which is 12 * 8 * 4096 = 393,216. That's still manageable, but the transitions would be complex.Alternatively, since the number of concerts is up to 8, we can represent the state as (current city, number of concerts attended, total cost), and for each state, keep track of the minimal total cost to reach that state.This way, we don't need to track the exact set of visited cities, but just ensure that we don't revisit the same city multiple times. However, this might not be straightforward because the same city can be reached through different paths with different costs.Wait, but if we don't track the visited cities, we might end up revisiting the same city multiple times, which isn't allowed because each concert is in a different city. So, we need a way to ensure that each city is visited at most once.Therefore, the state needs to include the set of visited cities. But as I thought earlier, that leads to a large state space.Alternatively, maybe we can use memoization and pruning to handle this.But perhaps there's a better way. Let's think about the problem differently.Since the fan starts in Tokyo, and each concert is in a different city, the problem is to find a path that starts in Tokyo, visits 8 cities (concerts), without revisiting any, such that the total cost (sum of base costs plus travel costs) is minimized.This is essentially the Traveling Salesman Problem with a fixed starting point and a fixed number of cities to visit (8 out of 12). The goal is to find the minimal cost path.But solving this exactly might be challenging, especially with 12 cities. However, since we're dealing with a theoretical problem, perhaps we can outline the steps to solve it.First, we need to compute the travel distances between all pairs of cities, including Tokyo. Then, we can model this as a graph where each node is a city, and edges have weights equal to the travel cost ( b times d_i ).Next, we can use dynamic programming to find the minimal cost path that starts at Tokyo, visits 8 cities, and doesn't revisit any city. The DP state can be represented as (current city, mask), where mask is a bitmask representing the set of visited cities. The value stored is the minimal total cost to reach that state.The number of states would be 12 (current city) * 2^12 (mask) = 12 * 4096 = 49,152. That's manageable.For each state, we can transition to other cities that haven't been visited yet, updating the mask and the total cost.Once we've filled the DP table, the minimal total cost for visiting 8 cities would be the minimum value among all states where the mask has exactly 8 bits set, and the current city is any of the 12 cities.But wait, the total cost also includes the base costs ( a_i ) for each concert attended. So, in addition to the travel costs, we need to add the base costs for each city visited.Therefore, the DP transition would be:For each state (current city, mask), and for each next city not in mask, the new state would be (next city, mask | (1 << next_city)), and the new total cost would be the current total cost plus the travel cost from current city to next city plus the base cost ( a_{next_city} ).Wait, but the base cost is only added once per concert attended, right? So, when we visit a city, we add its base cost ( a_i ) to the total cost.So, the DP transition should be:DP[next_city][new_mask] = min(DP[next_city][new_mask], DP[current_city][mask] + travel_cost(current_city, next_city) + a_{next_city})But we need to initialize the DP with the starting point. Since the fan starts in Tokyo, which is not one of the concert cities, the first concert will have a travel cost from Tokyo to the first city, plus the base cost ( a_i ) of that city.Wait, actually, the starting point is Tokyo, but the first concert is in one of the 12 cities. So, the first transition is from Tokyo to the first city, with a travel cost ( b times d_i ), where ( d_i ) is the distance from Tokyo to city i. Then, we add the base cost ( a_i ) for that concert.So, the initial state is (Tokyo, mask=0, total_cost=0). Then, for each city i, we can transition to (i, mask=1<<i, total_cost = b * d_i + a_i).Then, for each subsequent step, we transition from city i to city j, adding the travel cost ( b * d_{i,j} ) and the base cost ( a_j ).Wait, but the travel cost between cities is given as ( d_i ), which is the distance from the previous city. So, for each concert after the first, the travel cost is ( b times d_i ), where ( d_i ) is the distance from the previous city.Therefore, the travel cost between two cities i and j is ( b times d_j ), where ( d_j ) is the distance from city i to city j.Wait, no. The problem says ( d_i ) is the distance from the previous concert city. So, for each concert i, ( d_i ) is the distance from the previous city to city i.Therefore, the travel cost for concert i is ( b times d_i ), and this depends on the previous city.So, in the DP, when moving from city i to city j, the travel cost is ( b times d_j ), where ( d_j ) is the distance from city i to city j.Therefore, the DP needs to keep track of the current city to know the travel cost to the next city.So, the state is (current city, mask), and the value is the minimal total cost to reach that state.The transitions would be:For each state (current_city, mask), and for each next_city not in mask:- Compute the travel cost: ( b times d_{next_city} ), where ( d_{next_city} ) is the distance from current_city to next_city.- Compute the new total cost: current_total_cost + travel_cost + ( a_{next_city} ).- Update the DP[next_city][new_mask] if this new total cost is lower than the existing value.The initial state is (Tokyo, mask=0, total_cost=0). From Tokyo, we can go to any city i, with travel cost ( b times d_i ) (distance from Tokyo to i) plus ( a_i ).Once we've filled the DP table, we look for all states where the mask has exactly 8 bits set, and find the minimal total cost among them. If that minimal total cost is less than or equal to 15,000, then the fan can attend 8 concerts. Otherwise, he might have to attend fewer.But this is a bit abstract. Let me try to outline the steps more concretely.1. Precompute the distance matrix between all pairs of cities, including Tokyo. Let's denote the distance from city A to city B as ( d_{A,B} ).2. Initialize the DP table. The DP table will have entries for each city and each possible mask. The value stored is the minimal total cost to reach that state.3. Start with the initial state: current city = Tokyo, mask = 0 (no concerts attended), total cost = 0.4. For each state (current_city, mask), iterate over all possible next cities that haven't been visited yet (i.e., not in mask).5. For each next_city, compute the travel cost from current_city to next_city: ( b times d_{current_city, next_city} ).6. Compute the new total cost: current_total_cost + travel_cost + ( a_{next_city} ).7. Update the DP[next_city][new_mask] if the new_total_cost is less than the current value stored there.8. After processing all transitions, the minimal total cost for attending 8 concerts is the minimum value among all DP[city][mask] where mask has exactly 8 bits set.9. If this minimal total cost is <= 15,000, then the fan can attend 8 concerts. Otherwise, we need to check for 7 concerts, and so on.But this requires knowing the distances between all cities, which aren't provided in the problem statement. So, perhaps the problem expects a general approach rather than a specific numerical answer.Wait, the problem says "Given the distances ( d_i ) between cities and base costs ( a_i ) for each concert...". So, in a real scenario, we would have specific values for ( d_i ) and ( a_i ), and then we could compute the minimal total cost.But since the problem is presented in a general form, maybe the answer should be an outline of the approach rather than a numerical answer.So, for part 1, the approach is:- Model the problem as a Traveling Salesman Problem with a fixed starting point (Tokyo) and a fixed number of cities to visit (up to 8).- Use dynamic programming to find the minimal total cost for visiting 8 cities, considering both the base costs and the travel costs.- If the minimal total cost for 8 concerts is within the budget, that's the optimal solution. Otherwise, reduce the number of concerts and repeat the process until the total cost is within the budget.For part 2, the problem is to minimize the total travel distance while attending exactly 8 concerts. This is similar to part 1, but instead of minimizing the total cost, we're minimizing the total travel distance. However, the total cost is still a factor because the fan has a budget, but in this case, the goal is to minimize travel distance regardless of cost, as long as it's within the budget.Wait, no. The second part says "formulate the problem as an optimization problem and determine the minimal travel distance." So, it's a separate problem where the fan wants to attend exactly 8 concerts, and among all possible combinations and orders, find the one with the minimal total travel distance.But the total travel distance is part of the total cost, which is ( sum a_i + b sum d_i ). So, minimizing the total travel distance would help in minimizing the total cost, but the problem here is to find the minimal travel distance, not necessarily the minimal total cost.But since the fan is attending exactly 8 concerts, the minimal travel distance would correspond to the optimal route visiting those 8 cities with the least total distance.This is essentially the Traveling Salesman Problem for 8 cities, starting from Tokyo, with the goal of minimizing the total travel distance.So, the approach would be similar to part 1, but instead of considering the base costs, we focus solely on minimizing the travel distance.But wait, the problem says "formulate the problem as an optimization problem and determine the minimal travel distance." So, perhaps it's a pure TSP where the objective is to minimize the total travel distance, starting from Tokyo and visiting exactly 8 cities.In that case, the formulation would be:Minimize ( sum_{i=1}^{8} d_i )Subject to:- Starting from Tokyo.- Visiting exactly 8 distinct cities.- Each city is visited exactly once.This can be modeled as a TSP with a fixed starting point and a fixed number of cities to visit.So, the optimization problem can be formulated as an integer linear program, where variables represent whether a city is visited and the order in which they are visited.But perhaps a more straightforward way is to use dynamic programming, similar to part 1, but without considering the base costs.So, the DP state would be (current city, mask), and the value stored is the minimal total travel distance to reach that state.The transitions would be similar, but instead of adding the base costs, we only add the travel distances.Once the DP table is filled, the minimal total travel distance for visiting 8 cities would be the minimum value among all states where the mask has exactly 8 bits set.But again, without specific data, we can't compute the exact minimal distance, but we can outline the approach.So, to summarize:1. For the first part, the problem is a combination of selecting up to 8 concerts and finding the optimal route to minimize the total cost, which includes both base costs and travel costs. This can be modeled as a dynamic traveling salesman problem with a budget constraint, using dynamic programming to find the minimal total cost for 8 concerts and checking if it's within the budget.2. For the second part, the problem is to minimize the total travel distance while attending exactly 8 concerts, which is a pure TSP problem with a fixed starting point and a fixed number of cities to visit. This can be solved using dynamic programming to find the minimal total travel distance.Given that, the answers would involve outlining these approaches rather than providing numerical results, as the specific distances and base costs aren't provided.But wait, the problem says "determine the combination of up to 8 concerts" and "determine the minimal travel distance." So, perhaps it expects a more concrete answer, but without specific data, it's impossible to provide exact numbers.Alternatively, maybe the problem expects recognizing that it's a variation of the TSP and knapsack problem, and formulating it accordingly.In conclusion, for part 1, the optimal solution involves solving a dynamic TSP with a budget constraint to maximize the number of concerts attended, and for part 2, it's a pure TSP to minimize the travel distance for exactly 8 concerts.But since the problem asks to "determine the combination" and "determine the minimal travel distance," perhaps the answer should be in terms of an algorithm or approach rather than specific numbers.However, since the user asked for a detailed thought process, I think I've covered the necessary steps and reasoning.</think>

answer:<think>Okay, so I have two sub-problems to solve here. Let me start with the first one.Sub-problem 1:Jordan has applied to 50 companies, each with a probability ( p = 0.1 ) of getting an interview call. We need to find the probability that he gets at least 5 interview calls. This sounds like a binomial distribution problem because each application is an independent trial with two outcomes: success (interview) or failure (no interview).The binomial distribution formula is:[P(X = k) = C(n, k) times p^k times (1 - p)^{n - k}]Where:- ( n = 50 ) (number of trials)- ( k ) is the number of successes (interviews)- ( p = 0.1 ) (probability of success)But we need the probability of getting at least 5 interviews, which means ( P(X geq 5) ). Calculating this directly would involve summing the probabilities from ( k = 5 ) to ( k = 50 ), which is a lot. Instead, it's easier to calculate the complement: ( P(X geq 5) = 1 - P(X leq 4) ).So, I need to compute ( P(X leq 4) ) and subtract it from 1.Calculating each term from ( k = 0 ) to ( k = 4 ):1. For ( k = 0 ): [ P(X = 0) = C(50, 0) times (0.1)^0 times (0.9)^{50} = 1 times 1 times (0.9)^{50} ] Let me compute ( (0.9)^{50} ). I know that ( ln(0.9) approx -0.10536 ), so ( ln(0.9^{50}) = 50 times (-0.10536) = -5.268 ). Therefore, ( e^{-5.268} approx 0.00517 ).2. For ( k = 1 ): [ P(X = 1) = C(50, 1) times (0.1)^1 times (0.9)^{49} ] ( C(50, 1) = 50 ), so: [ 50 times 0.1 times (0.9)^{49} ] ( (0.9)^{49} = (0.9)^{50} / 0.9 approx 0.00517 / 0.9 approx 0.00574 ) So, ( 50 times 0.1 times 0.00574 = 5 times 0.00574 = 0.0287 )3. For ( k = 2 ): [ P(X = 2) = C(50, 2) times (0.1)^2 times (0.9)^{48} ] ( C(50, 2) = 1225 ) ( (0.1)^2 = 0.01 ) ( (0.9)^{48} = (0.9)^{50} / (0.9)^2 approx 0.00517 / 0.81 approx 0.00638 ) So, ( 1225 times 0.01 times 0.00638 = 12.25 times 0.00638 approx 0.0782 )4. For ( k = 3 ): [ P(X = 3) = C(50, 3) times (0.1)^3 times (0.9)^{47} ] ( C(50, 3) = 19600 ) ( (0.1)^3 = 0.001 ) ( (0.9)^{47} = (0.9)^{50} / (0.9)^3 approx 0.00517 / 0.729 approx 0.0071 ) So, ( 19600 times 0.001 times 0.0071 = 19.6 times 0.0071 approx 0.139 )5. For ( k = 4 ): [ P(X = 4) = C(50, 4) times (0.1)^4 times (0.9)^{46} ] ( C(50, 4) = 230300 ) ( (0.1)^4 = 0.0001 ) ( (0.9)^{46} = (0.9)^{50} / (0.9)^4 approx 0.00517 / 0.6561 approx 0.00788 ) So, ( 230300 times 0.0001 times 0.00788 = 23.03 times 0.00788 approx 0.1817 )Now, summing up all these probabilities:[P(X leq 4) = P(0) + P(1) + P(2) + P(3) + P(4) approx 0.00517 + 0.0287 + 0.0782 + 0.139 + 0.1817]Calculating step by step:- 0.00517 + 0.0287 = 0.03387- 0.03387 + 0.0782 = 0.11207- 0.11207 + 0.139 = 0.25107- 0.25107 + 0.1817 ≈ 0.43277So, ( P(X leq 4) approx 0.43277 ). Therefore, the probability of getting at least 5 interviews is:[P(X geq 5) = 1 - 0.43277 = 0.56723]So, approximately 56.72%.Wait, that seems a bit high. Let me double-check my calculations, especially the exponents and combinations.Wait, for ( k = 2 ), I had ( C(50, 2) = 1225 ), which is correct. ( (0.9)^{48} ) was approximated as 0.00638, which seems okay. Then 1225 * 0.01 * 0.00638 ≈ 0.0782. That seems correct.Similarly, for ( k = 3 ), 19600 * 0.001 * 0.0071 ≈ 0.139. That also seems okay.For ( k = 4 ), 230300 * 0.0001 * 0.00788 ≈ 0.1817. Hmm, 230300 * 0.0001 is 23.03, multiplied by 0.00788 is approximately 0.1817. That seems correct.Adding them up: 0.00517 + 0.0287 + 0.0782 + 0.139 + 0.1817 ≈ 0.43277. So, 1 - 0.43277 ≈ 0.56723, which is about 56.7%.Alternatively, maybe using a calculator or software would give a more precise result, but given the approximations, 56.7% seems reasonable.Sub-problem 2:Now, Jordan has 50 companies with different probabilities ( p_i ) of getting an interview. He needs to allocate effort ( e_i ) such that ( sum e_i = 1 ) and ( e_i geq 0 ), to maximize the expected number of interviews, which is ( sum p_i e_i ).This is a linear programming problem. The objective is to maximize ( sum p_i e_i ) subject to ( sum e_i = 1 ) and ( e_i geq 0 ).In linear programming, to maximize a linear function subject to linear constraints, the optimal solution occurs at a vertex of the feasible region. Since the feasible region is a simplex (because ( sum e_i = 1 ) and ( e_i geq 0 )), the maximum will be achieved by putting all effort into the company with the highest ( p_i ).Wait, that makes sense. Because if you have different ( p_i ), the expected value is a weighted sum, so to maximize it, you should allocate as much as possible to the highest ( p_i ). Since the total effort is 1, you set ( e_j = 1 ) for the company ( j ) with the maximum ( p_j ), and ( e_i = 0 ) for all others.Therefore, the optimal strategy is to focus all effort on the company with the highest probability ( p_i ).But wait, let me think again. Is there any constraint that I'm missing? The problem says "allocate his effort across these 50 companies" with the effort vector ( e ) where each ( e_i ) is non-negative and sums to 1. The objective is to maximize ( sum p_i e_i ).Yes, so in linear programming terms, the objective function is linear, and the feasible region is convex. The maximum of a linear function over a convex set occurs at an extreme point, which in this case is when all effort is concentrated on a single company. Since we want to maximize the sum, we choose the company with the highest ( p_i ).Therefore, the optimal effort allocation is to set ( e_j = 1 ) where ( p_j ) is the maximum among all ( p_i ), and all other ( e_i = 0 ).Alternatively, if there are multiple companies with the same maximum ( p_i ), Jordan could distribute effort among them, but since the problem doesn't specify that, we can assume there's a single maximum.So, the solution is straightforward: allocate all effort to the company with the highest ( p_i ).Final AnswerSub-problem 1: The probability is boxed{0.5672}.Sub-problem 2: Jordan should allocate all effort to the company with the highest ( p_i ), so the optimal effort vector ( mathbf{e} ) has ( e_j = 1 ) for the company ( j ) with the maximum ( p_j ) and ( e_i = 0 ) otherwise. Therefore, the optimal allocation is boxed{e_j = 1} for the company with the highest probability.</think>

answer:<think>Alright, so I've got this problem about a private investor financing a real estate project. Let me try to break it down step by step. First, the project costs 5 million, and the investor is providing the full amount upfront. In return, the investor gets 25% of the annual net revenue for the first 10 years. The net revenue is expected to grow geometrically, starting at 600,000 in the first year with a common ratio of 1.05. After 10 years, the developer can buy back the investor's stake by paying a lump sum, which is the future value of an annuity based on the 10th year's revenue share, compounded at 6%.Okay, so the first sub-problem is to calculate the total amount the investor will receive from the revenue sharing over 10 years. That sounds like I need to compute the sum of a geometric series because the revenue grows each year by 5%.Let me recall the formula for the sum of a geometric series. The sum S of the first n terms where each term is a * r^(k-1) is S = a * (1 - r^n) / (1 - r), where a is the first term and r is the common ratio.In this case, the first term a is 25% of 600,000, which is 150,000. The common ratio r is 1.05, and n is 10 years. So plugging into the formula:S = 150,000 * (1 - 1.05^10) / (1 - 1.05)Wait, let me compute 1.05^10 first. I remember that 1.05^10 is approximately 1.62889. Let me verify that. Yeah, 1.05^10 is roughly 1.62889. So 1 - 1.62889 is -0.62889. Then, 1 - 1.05 is -0.05. So the denominator is -0.05.So S = 150,000 * (-0.62889) / (-0.05) = 150,000 * (0.62889 / 0.05). Let's compute 0.62889 divided by 0.05. That's 12.5778. So 150,000 * 12.5778 is... let me calculate that.150,000 * 12 = 1,800,000, and 150,000 * 0.5778 is approximately 150,000 * 0.5 = 75,000 and 150,000 * 0.0778 ≈ 11,670. So total is 75,000 + 11,670 = 86,670. So total S ≈ 1,800,000 + 86,670 = 1,886,670.Wait, that seems low. Let me double-check my calculations.Alternatively, maybe I should compute each year's revenue share and sum them up. Let's see:Year 1: 0.25 * 600,000 = 150,000Year 2: 0.25 * 600,000 * 1.05 = 150,000 * 1.05 = 157,500Year 3: 150,000 * 1.05^2 ≈ 150,000 * 1.1025 = 165,375Year 4: 150,000 * 1.05^3 ≈ 150,000 * 1.157625 ≈ 173,643.75Year 5: 150,000 * 1.05^4 ≈ 150,000 * 1.21550625 ≈ 182,325.94Year 6: 150,000 * 1.05^5 ≈ 150,000 * 1.27628156 ≈ 191,442.23Year 7: 150,000 * 1.05^6 ≈ 150,000 * 1.34009564 ≈ 201,014.35Year 8: 150,000 * 1.05^7 ≈ 150,000 * 1.40710042 ≈ 211,065.06Year 9: 150,000 * 1.05^8 ≈ 150,000 * 1.47745544 ≈ 221,618.32Year 10: 150,000 * 1.05^9 ≈ 150,000 * 1.55132821 ≈ 232,699.23Now, let's sum all these up:150,000 + 157,500 = 307,500307,500 + 165,375 = 472,875472,875 + 173,643.75 ≈ 646,518.75646,518.75 + 182,325.94 ≈ 828,844.69828,844.69 + 191,442.23 ≈ 1,020,286.921,020,286.92 + 201,014.35 ≈ 1,221,301.271,221,301.27 + 211,065.06 ≈ 1,432,366.331,432,366.33 + 221,618.32 ≈ 1,653,984.651,653,984.65 + 232,699.23 ≈ 1,886,683.88Hmm, so when I sum them up individually, I get approximately 1,886,683.88, which is very close to my initial calculation of 1,886,670. So that seems consistent. So the total amount the investor receives over 10 years is approximately 1,886,684.Wait, but let me check the formula again. The sum of a geometric series is S = a * (1 - r^n) / (1 - r). Here, a is 150,000, r is 1.05, n is 10.So S = 150,000 * (1 - 1.05^10) / (1 - 1.05) = 150,000 * (1 - 1.62889) / (-0.05) = 150,000 * (-0.62889) / (-0.05) = 150,000 * 12.5778 ≈ 1,886,670.Yes, so that's correct. So the total revenue sharing is approximately 1,886,670.Wait, but when I summed them individually, I got 1,886,684, which is almost the same, just a slight difference due to rounding in each year's calculation. So I think 1,886,670 is accurate enough.So that's the first part.Now, the second sub-problem is to determine the lump sum the developer must pay to buy back the investor's stake after 10 years. The lump sum is the future value of an annuity where the annual payment is the 25% revenue share received in the 10th year, compounded at 6%.Wait, so the annual payment for the annuity is the 10th year's revenue share, which is 232,699.23 as calculated earlier. So the annuity payment is 232,699.23 per year, and we need to find the future value of this annuity at the end of 10 years, with an interest rate of 6%.But wait, is it a future value of an ordinary annuity or an annuity due? The problem says it's the future value of an annuity where the annual payment is the 10th year's revenue share. So I think it's an ordinary annuity because the payments are made at the end of each year.Wait, but the 10th year's revenue share is received at the end of year 10, so if we're calculating the future value at the end of year 10, the annuity would consist of just that one payment? That doesn't make sense. Wait, maybe I misread.Wait, the lump sum is calculated as the future value of an annuity where the annual payment is the 25% revenue share received in the 10th year, compounded annually at 6%. So the annual payment is the 10th year's revenue share, which is 232,699.23, and we need to calculate the future value of an annuity with these payments made each year for 10 years, starting from year 1 to year 10, with each payment being 232,699.23, compounded at 6%.Wait, no, that can't be because the 10th year's revenue share is only received at the end of year 10. So perhaps the annuity is for 10 years, with each payment being the 10th year's revenue share, which is 232,699.23. So it's like an annuity where each year, the same amount is paid, which is the amount received in year 10, but that seems odd because in reality, the payments are increasing each year.Wait, maybe I need to clarify. The problem says: "the lump sum is calculated as the future value of an annuity where the annual payment is the 25% revenue share received by the investor in the 10th year, compounded annually at an interest rate of 6%."So, the annual payment for the annuity is the 10th year's revenue share, which is 232,699.23. So the annuity is a 10-year annuity with each payment being 232,699.23, and we need to find its future value at the end of 10 years, compounded at 6%.Wait, but if the payments are made each year for 10 years, starting from year 1 to year 10, each being 232,699.23, then the future value would be calculated as FV = PMT * [(1 + r)^n - 1] / r, where PMT is the annual payment, r is the interest rate, and n is the number of periods.So PMT = 232,699.23, r = 6% = 0.06, n = 10.So FV = 232,699.23 * [(1.06)^10 - 1] / 0.06First, let's compute (1.06)^10. I remember that 1.06^10 is approximately 1.790847.So (1.790847 - 1) = 0.790847Divide that by 0.06: 0.790847 / 0.06 ≈ 13.180783So FV ≈ 232,699.23 * 13.180783 ≈ Let's compute that.First, 232,699.23 * 10 = 2,326,992.30232,699.23 * 3 = 698,097.69232,699.23 * 0.180783 ≈ Let's approximate 232,699.23 * 0.18 = 41,885.86 and 232,699.23 * 0.000783 ≈ 182. So total ≈ 41,885.86 + 182 ≈ 42,067.86So total FV ≈ 2,326,992.30 + 698,097.69 + 42,067.86 ≈ Let's add them up.2,326,992.30 + 698,097.69 = 3,025,090.00 (approximately)3,025,090.00 + 42,067.86 ≈ 3,067,157.86Wait, but let me do a more accurate calculation.Alternatively, 232,699.23 * 13.180783Let me compute 232,699.23 * 13 = 3,025,090.00232,699.23 * 0.180783 ≈ Let's compute 232,699.23 * 0.1 = 23,269.923232,699.23 * 0.08 = 18,615.9384232,699.23 * 0.000783 ≈ 182.00So total ≈ 23,269.923 + 18,615.9384 + 182.00 ≈ 42,067.86So total FV ≈ 3,025,090.00 + 42,067.86 ≈ 3,067,157.86So approximately 3,067,158.Wait, but let me use a calculator for more precision.Alternatively, perhaps I should use the formula directly.FV = PMT * [(1 + r)^n - 1] / rPMT = 232,699.23r = 0.06n = 10So (1.06)^10 = 1.790847So [(1.790847) - 1] / 0.06 = 0.790847 / 0.06 ≈ 13.180783So 232,699.23 * 13.180783 ≈ Let's compute this more accurately.232,699.23 * 13 = 3,025,090.00232,699.23 * 0.180783 ≈ Let's compute 232,699.23 * 0.1 = 23,269.923232,699.23 * 0.08 = 18,615.9384232,699.23 * 0.000783 ≈ 182.00So total ≈ 23,269.923 + 18,615.9384 + 182.00 ≈ 42,067.86So total FV ≈ 3,025,090.00 + 42,067.86 ≈ 3,067,157.86So approximately 3,067,158.Wait, but let me check if I should use the future value of an ordinary annuity or an annuity due. Since the payments are made at the end of each year, it's an ordinary annuity, so the formula I used is correct.Alternatively, if the payments were made at the beginning of each year, it would be an annuity due, but the problem doesn't specify that, so I think ordinary annuity is correct.So, the lump sum the developer must pay is approximately 3,067,158.Wait, but let me double-check the calculation using another method.Alternatively, I can compute the future value factor for an ordinary annuity of 10 years at 6%, which is [(1.06)^10 - 1]/0.06 ≈ (1.790847 - 1)/0.06 ≈ 0.790847 / 0.06 ≈ 13.180783.So, FV = 232,699.23 * 13.180783 ≈ 232,699.23 * 13.180783.Let me compute 232,699.23 * 13 = 3,025,090.00232,699.23 * 0.180783 ≈ 232,699.23 * 0.18 = 41,885.86232,699.23 * 0.000783 ≈ 182.00So total ≈ 41,885.86 + 182.00 ≈ 42,067.86So total FV ≈ 3,025,090.00 + 42,067.86 ≈ 3,067,157.86Yes, so approximately 3,067,158.Wait, but let me check if the 10th year's revenue share is indeed 232,699.23. Earlier, when I summed the revenue shares, the 10th year was approximately 232,699.23, so that's correct.So, to recap:1. Total revenue sharing over 10 years: approximately 1,886,670.2. Lump sum buyback amount: approximately 3,067,158.Wait, but let me make sure I didn't make a mistake in interpreting the buyback option. The problem says the lump sum is calculated as the future value of an annuity where the annual payment is the 25% revenue share received in the 10th year. So, each year, the payment is the same as the 10th year's payment, which is 232,699.23, and this is done for 10 years, starting from year 1 to year 10, with each payment being 232,699.23, compounded at 6%.Yes, that's correct. So the future value of that annuity is what the developer has to pay at the end of 10 years.Alternatively, if the problem had meant that the buyback is based on the revenue share in the 10th year, but only that single payment, then it would be just 232,699.23, but that doesn't make sense because it's a lump sum calculated as an annuity. So I think my interpretation is correct.So, to sum up:1. The total revenue sharing over 10 years is approximately 1,886,670.2. The lump sum buyback amount is approximately 3,067,158.I think that's it. Let me just make sure I didn't make any calculation errors.For the first part, the sum of the geometric series:a = 150,000r = 1.05n = 10S = 150,000 * (1 - 1.05^10) / (1 - 1.05) ≈ 150,000 * (1 - 1.62889) / (-0.05) ≈ 150,000 * 12.5778 ≈ 1,886,670.Yes, that's correct.For the second part, the future value of the annuity:PMT = 232,699.23r = 0.06n = 10FV = 232,699.23 * [(1.06)^10 - 1] / 0.06 ≈ 232,699.23 * 13.180783 ≈ 3,067,158.Yes, that's correct.So, I think I've got it.</think>

answer:<think>Alright, so I've got these two math problems to solve, right? Both are related to Alex's voter sentiment model. Let me take them one at a time.Starting with problem 1. It's about a system of differential equations modeling voter sentiment in two states, A and B. The equations are:[frac{dS_A}{dt} = aS_A + bS_B + c sin(t)][frac{dS_B}{dt} = dS_B + eS_A + f cos(t)]I need to find the general solution for ( S_A(t) ) and ( S_B(t) ). Hmm, okay. So this is a system of linear nonhomogeneous differential equations. I remember that to solve such systems, we can use methods like eigenvalue analysis or Laplace transforms. Since the system is linear, maybe I can write it in matrix form and then find the solution using eigenvalues and eigenvectors.Let me write the system in matrix form:[begin{pmatrix}frac{dS_A}{dt} frac{dS_B}{dt}end{pmatrix}= begin{pmatrix}a & b e & dend{pmatrix}begin{pmatrix}S_A S_Bend{pmatrix}+begin{pmatrix}c sin(t) f cos(t)end{pmatrix}]So, in the form ( mathbf{X}' = Amathbf{X} + mathbf{G}(t) ), where ( mathbf{X} = begin{pmatrix} S_A S_B end{pmatrix} ), ( A ) is the coefficient matrix, and ( mathbf{G}(t) ) is the nonhomogeneous term.To solve this, I think I can find the homogeneous solution first and then find a particular solution.First, let's solve the homogeneous system:[mathbf{X}' = Amathbf{X}]The general solution to this is ( mathbf{X}_h = e^{At}mathbf{C} ), where ( mathbf{C} ) is a constant vector. To find ( e^{At} ), I need to diagonalize matrix ( A ) if possible, which requires finding its eigenvalues and eigenvectors.So, let's find the eigenvalues of ( A ). The characteristic equation is:[det(A - lambda I) = 0]Calculating the determinant:[detleft( begin{pmatrix}a - lambda & b e & d - lambdaend{pmatrix} right) = (a - lambda)(d - lambda) - be = 0]Expanding this:[(ad - alambda - dlambda + lambda^2) - be = 0][lambda^2 - (a + d)lambda + (ad - be) = 0]So, the eigenvalues ( lambda ) are:[lambda = frac{(a + d) pm sqrt{(a + d)^2 - 4(ad - be)}}{2}][= frac{(a + d) pm sqrt{a^2 + 2ad + d^2 - 4ad + 4be}}{2}][= frac{(a + d) pm sqrt{a^2 - 2ad + d^2 + 4be}}{2}][= frac{(a + d) pm sqrt{(a - d)^2 + 4be}}{2}]So, the eigenvalues depend on the constants ( a, b, d, e ). Depending on the discriminant, we can have real and distinct eigenvalues, repeated eigenvalues, or complex eigenvalues.Assuming the eigenvalues are distinct, we can find eigenvectors for each eigenvalue and express the homogeneous solution as a combination of exponential functions multiplied by eigenvectors.But since I don't have specific values for ( a, b, d, e ), I might need to keep the solution in terms of these constants.Alternatively, maybe I can use the method of undetermined coefficients to find a particular solution for the nonhomogeneous system.The nonhomogeneous term is ( mathbf{G}(t) = begin{pmatrix} c sin(t) f cos(t) end{pmatrix} ). So, it's a combination of sine and cosine functions. I can assume a particular solution of the form:[mathbf{X}_p = begin{pmatrix} A sin(t) + B cos(t) C sin(t) + D cos(t) end{pmatrix}]Then, plug this into the original differential equation:[mathbf{X}_p' = A cos(t) - B sin(t) quad text{for } S_A][mathbf{X}_p' = C cos(t) - D sin(t) quad text{for } S_B]So, substituting into the system:1. ( A cos(t) - B sin(t) = a(A sin(t) + B cos(t)) + b(C sin(t) + D cos(t)) + c sin(t) )2. ( C cos(t) - D sin(t) = e(A sin(t) + B cos(t)) + d(C sin(t) + D cos(t)) + f cos(t) )Now, let's equate coefficients for sine and cosine terms on both sides.For equation 1:Coefficient of ( sin(t) ):( -B = aA + bC + c )Coefficient of ( cos(t) ):( A = aB + bD )For equation 2:Coefficient of ( sin(t) ):( -D = eA + dC )Coefficient of ( cos(t) ):( C = eB + dD + f )So, now we have a system of four equations:1. ( -B = aA + bC + c ) -- (1)2. ( A = aB + bD ) -- (2)3. ( -D = eA + dC ) -- (3)4. ( C = eB + dD + f ) -- (4)This is a linear system in variables A, B, C, D. Let me write it in matrix form or try to solve step by step.From equation (2): ( A = aB + bD ). Let's express A in terms of B and D.From equation (3): ( -D = eA + dC ). Substitute A from equation (2):( -D = e(aB + bD) + dC )( -D = a e B + b e D + d C )Let's rearrange:( -D - a e B - b e D = d C )( C = frac{ -D - a e B - b e D }{ d } ) -- (3a)From equation (4): ( C = eB + dD + f ). Substitute C from (3a):( frac{ -D - a e B - b e D }{ d } = eB + dD + f )Multiply both sides by d:( -D - a e B - b e D = d e B + d^2 D + d f )Bring all terms to left:( -D - a e B - b e D - d e B - d^2 D - d f = 0 )Combine like terms:For B: ( -a e B - d e B = -e(a + d) B )For D: ( -D - b e D - d^2 D = -D(1 + b e + d^2) )Constant term: ( -d f )So:( -e(a + d) B - D(1 + b e + d^2) - d f = 0 )Let me write this as:( -e(a + d) B - (1 + b e + d^2) D = d f ) -- (5)Now, from equation (1): ( -B = aA + bC + c ). Substitute A from equation (2) and C from equation (4):( -B = a(aB + bD) + b(eB + dD + f) + c )( -B = a^2 B + a b D + b e B + b d D + b f + c )Bring all terms to left:( -B - a^2 B - a b D - b e B - b d D - b f - c = 0 )Factor terms:For B: ( -1 - a^2 - b e )For D: ( -a b - b d )Constants: ( -b f - c )So:( (-1 - a^2 - b e) B + (-a b - b d) D - b f - c = 0 )Let me write this as:( (-1 - a^2 - b e) B + (-b(a + d)) D = b f + c ) -- (6)Now, we have two equations: (5) and (6):Equation (5): ( -e(a + d) B - (1 + b e + d^2) D = d f )Equation (6): ( (-1 - a^2 - b e) B + (-b(a + d)) D = b f + c )This is a system of two equations in variables B and D. Let me write it as:[begin{cases}- e(a + d) B - (1 + b e + d^2) D = d f (-1 - a^2 - b e) B - b(a + d) D = b f + cend{cases}]Let me denote:Equation (5): ( M B + N D = P )Equation (6): ( Q B + R D = S )Where:M = -e(a + d)N = -(1 + b e + d^2)P = d fQ = -1 - a^2 - b eR = -b(a + d)S = b f + cWe can solve this system using substitution or elimination. Let's use elimination.Multiply equation (5) by R and equation (6) by N:Equation (5)*R: ( M R B + N R D = P R )Equation (6)*N: ( Q N B + R N D = S N )Subtract the two equations:( (M R - Q N) B = P R - S N )So,( B = frac{P R - S N}{M R - Q N} )Similarly, once B is found, substitute back to find D.Let me compute each term:First, compute M R:M = -e(a + d)R = -b(a + d)So, M R = (-e(a + d))(-b(a + d)) = e b (a + d)^2Similarly, Q N:Q = -1 - a^2 - b eN = -(1 + b e + d^2)So, Q N = (-1 - a^2 - b e)(-1 - b e - d^2)Wait, that's a bit complicated. Let me compute it step by step.First, compute M R:M R = (-e(a + d)) * (-b(a + d)) = e b (a + d)^2Compute Q N:Q = -1 - a^2 - b eN = -(1 + b e + d^2)So, Q N = (-1 - a^2 - b e)(-1 - b e - d^2)Let me factor out the negative signs:= (1 + a^2 + b e)(1 + b e + d^2)Hmm, that's a product of two trinomials. It might not simplify easily.Similarly, compute P R:P = d fR = -b(a + d)So, P R = d f * (-b(a + d)) = -b d f (a + d)Compute S N:S = b f + cN = -(1 + b e + d^2)So, S N = (b f + c)(-1 - b e - d^2) = - (b f + c)(1 + b e + d^2)Therefore, the numerator:P R - S N = -b d f (a + d) - [ - (b f + c)(1 + b e + d^2) ]= -b d f (a + d) + (b f + c)(1 + b e + d^2)Similarly, the denominator:M R - Q N = e b (a + d)^2 - (1 + a^2 + b e)(1 + b e + d^2)This is getting really complicated. Maybe there's a better approach.Alternatively, maybe instead of assuming a particular solution with both sine and cosine, I can use complex exponentials. Let me think.Let me consider the nonhomogeneous term as ( mathbf{G}(t) = begin{pmatrix} c sin(t) f cos(t) end{pmatrix} ). I can write this as the imaginary part of ( begin{pmatrix} -i c e^{i t} f e^{i t} end{pmatrix} ), since ( sin(t) = frac{e^{i t} - e^{-i t}}{2i} ) and ( cos(t) = frac{e^{i t} + e^{-i t}}{2} ).So, let me assume a particular solution of the form:[mathbf{X}_p = begin{pmatrix} alpha beta end{pmatrix} e^{i t}]Then, ( mathbf{X}_p' = i begin{pmatrix} alpha beta end{pmatrix} e^{i t} )Substitute into the differential equation:[i begin{pmatrix} alpha beta end{pmatrix} e^{i t} = A begin{pmatrix} alpha beta end{pmatrix} e^{i t} + begin{pmatrix} -i c f end{pmatrix} e^{i t}]Divide both sides by ( e^{i t} ):[i begin{pmatrix} alpha beta end{pmatrix} = A begin{pmatrix} alpha beta end{pmatrix} + begin{pmatrix} -i c f end{pmatrix}]So, rearranged:[(A - i I) begin{pmatrix} alpha beta end{pmatrix} = begin{pmatrix} -i c f end{pmatrix}]This is a linear system for ( alpha ) and ( beta ). Let me write it out:1. ( (a - i) alpha + b beta = -i c )2. ( e alpha + (d - i) beta = f )Let me solve this system. Let me write it as:Equation (1): ( (a - i) alpha + b beta = -i c )Equation (2): ( e alpha + (d - i) beta = f )Let me solve for ( alpha ) and ( beta ). Let's use substitution or elimination.From equation (1):( (a - i) alpha = -i c - b beta )So,( alpha = frac{ -i c - b beta }{ a - i } )Substitute into equation (2):( e left( frac{ -i c - b beta }{ a - i } right) + (d - i) beta = f )Multiply through:( frac{ -i e c - e b beta }{ a - i } + (d - i) beta = f )Multiply numerator and denominator by the conjugate to simplify ( frac{1}{a - i} ):( frac{1}{a - i} = frac{a + i}{a^2 + 1} )So,( frac{ -i e c - e b beta }{ a - i } = (-i e c - e b beta) cdot frac{a + i}{a^2 + 1} )Similarly, expand:( (-i e c)(a + i) / (a^2 + 1) - (e b beta)(a + i)/(a^2 + 1) )Let me compute each part:First term:( (-i e c)(a + i) = -i e c a - i^2 e c = -i e c a + e c ) (since ( i^2 = -1 ))Second term:( - (e b beta)(a + i) = -e b a beta - i e b beta )So, putting it all together:First term: ( (-i e c a + e c) / (a^2 + 1) )Second term: ( (-e b a beta - i e b beta) / (a^2 + 1) )So, equation (2) becomes:( [ (-i e c a + e c) / (a^2 + 1) ] + [ (-e b a beta - i e b beta) / (a^2 + 1) ] + (d - i) beta = f )Multiply all terms by ( a^2 + 1 ) to eliminate denominators:( (-i e c a + e c) + (-e b a beta - i e b beta) + (d - i)(a^2 + 1) beta = f (a^2 + 1) )Let me collect terms with ( beta ):Terms without ( beta ):( -i e c a + e c )Terms with ( beta ):( -e b a beta - i e b beta + (d - i)(a^2 + 1) beta )So, factor ( beta ):( beta [ -e b a - i e b + (d - i)(a^2 + 1) ] )So, the equation is:( (-i e c a + e c) + beta [ -e b a - i e b + d(a^2 + 1) - i(a^2 + 1) ] = f (a^2 + 1) )Let me write this as:( (-i e c a + e c) + beta [ -e b a - i e b + d a^2 + d - i a^2 - i ] = f (a^2 + 1) )Now, let me group the terms in ( beta ):Real parts:( -e b a + d a^2 + d )Imaginary parts:( -e b - a^2 - 1 )So,( beta [ (-e b a + d a^2 + d) + i (-e b - a^2 - 1) ] )Therefore, the equation becomes:( (-i e c a + e c) + beta [ (-e b a + d a^2 + d) + i (-e b - a^2 - 1) ] = f (a^2 + 1) )Now, let me separate real and imaginary parts on both sides.Left side:Real part: ( e c + beta (-e b a + d a^2 + d) )Imaginary part: ( -e c a + beta (-e b - a^2 - 1) )Right side:Real part: ( f (a^2 + 1) )Imaginary part: 0So, equating real and imaginary parts:Real:( e c + beta (-e b a + d a^2 + d) = f (a^2 + 1) ) -- (A)Imaginary:( -e c a + beta (-e b - a^2 - 1) = 0 ) -- (B)From equation (B):( -e c a + beta (-e b - a^2 - 1) = 0 )Solve for ( beta ):( beta (-e b - a^2 - 1) = e c a )So,( beta = frac{ e c a }{ -e b - a^2 - 1 } = frac{ - e c a }{ e b + a^2 + 1 } )Now, substitute ( beta ) into equation (A):( e c + left( frac{ - e c a }{ e b + a^2 + 1 } right) (-e b a + d a^2 + d ) = f (a^2 + 1) )Simplify the left side:First term: ( e c )Second term:( frac{ - e c a }{ e b + a^2 + 1 } times (-e b a + d a^2 + d ) )= ( frac{ e c a (e b a - d a^2 - d ) }{ e b + a^2 + 1 } )So, the equation becomes:( e c + frac{ e c a (e b a - d a^2 - d ) }{ e b + a^2 + 1 } = f (a^2 + 1) )Let me factor out ( e c ):( e c left[ 1 + frac{ a (e b a - d a^2 - d ) }{ e b + a^2 + 1 } right] = f (a^2 + 1) )Let me compute the term inside the brackets:( 1 + frac{ a (e b a - d a^2 - d ) }{ e b + a^2 + 1 } )= ( frac{ (e b + a^2 + 1) + a (e b a - d a^2 - d ) }{ e b + a^2 + 1 } )Expand the numerator:= ( e b + a^2 + 1 + e b a^2 - d a^3 - d a )So, numerator:= ( -d a^3 + e b a^2 + a^2 + (-d a) + e b + 1 )Group like terms:- ( a^3 ): ( -d a^3 )- ( a^2 ): ( e b a^2 + a^2 = a^2 (e b + 1) )- ( a ): ( -d a )- Constants: ( e b + 1 )So, numerator:= ( -d a^3 + (e b + 1) a^2 - d a + e b + 1 )Therefore, the equation is:( e c cdot frac{ -d a^3 + (e b + 1) a^2 - d a + e b + 1 }{ e b + a^2 + 1 } = f (a^2 + 1) )This seems quite complicated. Maybe I made a mistake somewhere. Alternatively, perhaps this approach isn't the most efficient.Wait, maybe instead of assuming a particular solution with both sine and cosine, I could have used the method of variation of parameters. But that might also be involved.Alternatively, perhaps I should stick with the original approach of finding eigenvalues and eigenvectors for the homogeneous system, then finding a particular solution.But given the complexity, maybe it's better to express the general solution in terms of the matrix exponential.The general solution is:( mathbf{X}(t) = e^{At} mathbf{X}_0 + int_0^t e^{A(t - tau)} mathbf{G}(tau) dtau )Where ( mathbf{X}_0 ) is the initial condition.But without knowing the specific eigenvalues, it's hard to write ( e^{At} ) explicitly. So, perhaps the answer is expected to be in terms of the matrix exponential and the integral.Alternatively, if the system is small (2x2), maybe we can find a particular solution using the method of undetermined coefficients as I started earlier, but it's getting too messy.Wait, maybe I can use the fact that the nonhomogeneous term is a combination of sine and cosine, so the particular solution can be expressed as a combination of sine and cosine multiplied by vectors. So, perhaps I can write:( mathbf{X}_p(t) = mathbf{U} sin(t) + mathbf{V} cos(t) )Where ( mathbf{U} ) and ( mathbf{V} ) are constant vectors to be determined.Then, ( mathbf{X}_p' = mathbf{U} cos(t) - mathbf{V} sin(t) )Substitute into the differential equation:( mathbf{U} cos(t) - mathbf{V} sin(t) = A (mathbf{U} sin(t) + mathbf{V} cos(t)) + begin{pmatrix} c sin(t) f cos(t) end{pmatrix} )Equate coefficients of ( sin(t) ) and ( cos(t) ):For ( sin(t) ):( - mathbf{V} = A mathbf{U} + begin{pmatrix} c 0 end{pmatrix} )For ( cos(t) ):( mathbf{U} = A mathbf{V} + begin{pmatrix} 0 f end{pmatrix} )So, we have two equations:1. ( - mathbf{V} = A mathbf{U} + begin{pmatrix} c 0 end{pmatrix} ) -- (1)2. ( mathbf{U} = A mathbf{V} + begin{pmatrix} 0 f end{pmatrix} ) -- (2)Let me write these out in terms of components.Let ( mathbf{U} = begin{pmatrix} U_1 U_2 end{pmatrix} ), ( mathbf{V} = begin{pmatrix} V_1 V_2 end{pmatrix} )From equation (2):( U_1 = a V_1 + b V_2 )( U_2 = e V_1 + d V_2 + f )From equation (1):( -V_1 = a U_1 + b U_2 + c )( -V_2 = e U_1 + d U_2 )So, substitute ( U_1 ) and ( U_2 ) from equation (2) into equation (1):First equation (1):( -V_1 = a (a V_1 + b V_2) + b (e V_1 + d V_2 + f) + c )Expand:( -V_1 = a^2 V_1 + a b V_2 + b e V_1 + b d V_2 + b f + c )Combine like terms:( -V_1 = (a^2 + b e) V_1 + (a b + b d) V_2 + b f + c )Bring all terms to left:( -V_1 - (a^2 + b e) V_1 - (a b + b d) V_2 - b f - c = 0 )Factor:( - (1 + a^2 + b e) V_1 - b (a + d) V_2 - b f - c = 0 ) -- (A)Second equation (1):( -V_2 = e (a V_1 + b V_2) + d (e V_1 + d V_2 + f) )Expand:( -V_2 = a e V_1 + b e V_2 + d e V_1 + d^2 V_2 + d f )Combine like terms:( -V_2 = (a e + d e) V_1 + (b e + d^2) V_2 + d f )Bring all terms to left:( -V_2 - (a e + d e) V_1 - (b e + d^2) V_2 - d f = 0 )Factor:( - (a e + d e) V_1 - (1 + b e + d^2) V_2 - d f = 0 ) -- (B)So, now we have two equations (A) and (B):(A): ( - (1 + a^2 + b e) V_1 - b (a + d) V_2 = b f + c )(B): ( - e (a + d) V_1 - (1 + b e + d^2) V_2 = d f )This is a system of two equations in ( V_1 ) and ( V_2 ). Let me write it as:[begin{cases}- (1 + a^2 + b e) V_1 - b (a + d) V_2 = b f + c - e (a + d) V_1 - (1 + b e + d^2) V_2 = d fend{cases}]Let me write this in matrix form:[begin{pmatrix}- (1 + a^2 + b e) & - b (a + d) - e (a + d) & - (1 + b e + d^2)end{pmatrix}begin{pmatrix}V_1 V_2end{pmatrix}=begin{pmatrix}b f + c d fend{pmatrix}]Let me denote the coefficient matrix as ( C ):[C = begin{pmatrix}- (1 + a^2 + b e) & - b (a + d) - e (a + d) & - (1 + b e + d^2)end{pmatrix}]To solve for ( V_1 ) and ( V_2 ), I can compute the inverse of ( C ) if it's invertible.First, compute the determinant of ( C ):[det(C) = [ - (1 + a^2 + b e) ][ - (1 + b e + d^2) ] - [ - b (a + d) ][ - e (a + d) ]]Simplify:= ( (1 + a^2 + b e)(1 + b e + d^2) - b e (a + d)^2 )This is similar to what I had earlier. Let me compute it step by step.First term:( (1 + a^2 + b e)(1 + b e + d^2) )Let me expand this:= ( 1*(1) + 1*(b e) + 1*(d^2) + a^2*(1) + a^2*(b e) + a^2*(d^2) + b e*(1) + b e*(b e) + b e*(d^2) )= ( 1 + b e + d^2 + a^2 + a^2 b e + a^2 d^2 + b e + b^2 e^2 + b e d^2 )Combine like terms:= ( 1 + 2 b e + d^2 + a^2 + a^2 b e + a^2 d^2 + b^2 e^2 + b e d^2 )Second term:( - b e (a + d)^2 = - b e (a^2 + 2 a d + d^2) = - a^2 b e - 2 a b e d - b e d^2 )So, determinant:= First term - Second term= ( [1 + 2 b e + d^2 + a^2 + a^2 b e + a^2 d^2 + b^2 e^2 + b e d^2] - [ - a^2 b e - 2 a b e d - b e d^2 ] )= ( 1 + 2 b e + d^2 + a^2 + a^2 b e + a^2 d^2 + b^2 e^2 + b e d^2 + a^2 b e + 2 a b e d + b e d^2 )Combine like terms:- Constants: 1- ( b e ): 2 b e- ( d^2 ): d^2- ( a^2 ): a^2- ( a^2 b e ): a^2 b e + a^2 b e = 2 a^2 b e- ( a^2 d^2 ): a^2 d^2- ( b^2 e^2 ): b^2 e^2- ( b e d^2 ): b e d^2 + b e d^2 = 2 b e d^2- ( 2 a b e d ): 2 a b e dSo, determinant:= ( 1 + 2 b e + d^2 + a^2 + 2 a^2 b e + a^2 d^2 + b^2 e^2 + 2 b e d^2 + 2 a b e d )This is quite a complex expression. Let me factor where possible.Notice that ( 2 a b e d ) is a term, and others have ( b e ) or ( d^2 ). It might not factor nicely.Assuming the determinant is non-zero, we can proceed to find ( V_1 ) and ( V_2 ).Using Cramer's rule:( V_1 = frac{ det(C_1) }{ det(C) } )( V_2 = frac{ det(C_2) }{ det(C) } )Where ( C_1 ) is the matrix formed by replacing the first column with the constants, and ( C_2 ) by replacing the second column.Compute ( det(C_1) ):Replace first column with ( begin{pmatrix} b f + c d f end{pmatrix} ):[C_1 = begin{pmatrix}b f + c & - b (a + d) d f & - (1 + b e + d^2)end{pmatrix}]Determinant:= ( (b f + c)( - (1 + b e + d^2) ) - (- b (a + d))(d f) )= ( - (b f + c)(1 + b e + d^2) + b (a + d) d f )Similarly, compute ( det(C_2) ):Replace second column with ( begin{pmatrix} b f + c d f end{pmatrix} ):[C_2 = begin{pmatrix}- (1 + a^2 + b e) & b f + c - e (a + d) & d fend{pmatrix}]Determinant:= ( - (1 + a^2 + b e) d f - (- e (a + d))(b f + c) )= ( - d f (1 + a^2 + b e) + e (a + d)(b f + c) )So, putting it all together:( V_1 = frac{ - (b f + c)(1 + b e + d^2) + b d f (a + d) }{ det(C) } )( V_2 = frac{ - d f (1 + a^2 + b e) + e (a + d)(b f + c) }{ det(C) } )This is getting extremely complicated. I think I might have made a mistake in approach. Maybe instead of trying to find a particular solution, I should accept that the general solution involves the matrix exponential and the integral, which might not simplify nicely without specific constants.Alternatively, perhaps the problem expects a more abstract answer, recognizing that the solution is the sum of the homogeneous solution and a particular solution, which can be expressed using the Green's function or variation of parameters.Given the time constraints and the complexity, I think the answer is expected to be in terms of the matrix exponential and the integral, acknowledging that without specific constants, we can't simplify further.So, the general solution is:( S_A(t) = e^{At}_{11} S_A(0) + e^{At}_{12} S_B(0) + int_0^t e^{A(t - tau)} begin{pmatrix} c sin(tau) f cos(tau) end{pmatrix} dtau )Similarly for ( S_B(t) ).But perhaps more neatly written as:( begin{pmatrix} S_A(t) S_B(t) end{pmatrix} = e^{At} begin{pmatrix} S_A(0) S_B(0) end{pmatrix} + int_0^t e^{A(t - tau)} begin{pmatrix} c sin(tau) f cos(tau) end{pmatrix} dtau )This is the general solution, expressed in terms of the matrix exponential and the integral involving the nonhomogeneous term.Moving on to problem 2. It's about finding the eigenvalues and eigenvectors of matrix ( M ):[M = begin{pmatrix}2 & -1 & 0 -1 & 2 & -1 0 & -1 & 2end{pmatrix}]This is a symmetric tridiagonal matrix, which often appears in problems involving discretized differential equations, like the Laplacian matrix of a path graph. The eigenvalues and eigenvectors can be found analytically.For a tridiagonal matrix with constant diagonals, the eigenvalues can be found using the formula for a symmetric tridiagonal matrix. Specifically, for a matrix of size ( n times n ) with diagonal entries ( a ) and off-diagonal entries ( b ), the eigenvalues are given by:[lambda_k = a + 2 b cosleft( frac{k pi}{n + 1} right), quad k = 1, 2, dots, n]In our case, ( n = 3 ), ( a = 2 ), and ( b = -1 ). So, the eigenvalues are:[lambda_k = 2 + 2 (-1) cosleft( frac{k pi}{4} right) = 2 - 2 cosleft( frac{k pi}{4} right), quad k = 1, 2, 3]Let's compute each:For ( k = 1 ):( lambda_1 = 2 - 2 cosleft( frac{pi}{4} right) = 2 - 2 cdot frac{sqrt{2}}{2} = 2 - sqrt{2} )For ( k = 2 ):( lambda_2 = 2 - 2 cosleft( frac{2pi}{4} right) = 2 - 2 cdot 0 = 2 )For ( k = 3 ):( lambda_3 = 2 - 2 cosleft( frac{3pi}{4} right) = 2 - 2 cdot left( -frac{sqrt{2}}{2} right) = 2 + sqrt{2} )So, the eigenvalues are ( 2 - sqrt{2} ), ( 2 ), and ( 2 + sqrt{2} ).Now, to find the eigenvectors, we can use the fact that for such tridiagonal matrices, the eigenvectors can be expressed in terms of sine functions.Specifically, the eigenvector corresponding to ( lambda_k ) has components:[v_{k,j} = sinleft( frac{j k pi}{n + 1} right), quad j = 1, 2, dots, n]For our case, ( n = 3 ), so:For ( k = 1 ):( v_{1,j} = sinleft( frac{j pi}{4} right) )So, the eigenvector is:( begin{pmatrix} sinleft( frac{pi}{4} right) sinleft( frac{2pi}{4} right) sinleft( frac{3pi}{4} right) end{pmatrix} = begin{pmatrix} frac{sqrt{2}}{2} 1 frac{sqrt{2}}{2} end{pmatrix} )For ( k = 2 ):( v_{2,j} = sinleft( frac{2j pi}{4} right) = sinleft( frac{j pi}{2} right) )So, the eigenvector is:( begin{pmatrix} sinleft( frac{pi}{2} right) sinleft( pi right) sinleft( frac{3pi}{2} right) end{pmatrix} = begin{pmatrix} 1 0 -1 end{pmatrix} )For ( k = 3 ):( v_{3,j} = sinleft( frac{3j pi}{4} right) )So, the eigenvector is:( begin{pmatrix} sinleft( frac{3pi}{4} right) sinleft( frac{6pi}{4} right) sinleft( frac{9pi}{4} right) end{pmatrix} = begin{pmatrix} frac{sqrt{2}}{2} -1 frac{sqrt{2}}{2} end{pmatrix} )However, we should verify these eigenvectors by plugging them back into ( (M - lambda I) mathbf{v} = 0 ).Let's check for ( k = 1 ):( lambda_1 = 2 - sqrt{2} )Eigenvector ( mathbf{v}_1 = begin{pmatrix} frac{sqrt{2}}{2} 1 frac{sqrt{2}}{2} end{pmatrix} )Compute ( M mathbf{v}_1 ):First component:( 2 cdot frac{sqrt{2}}{2} + (-1) cdot 1 + 0 cdot frac{sqrt{2}}{2} = sqrt{2} - 1 )Second component:( (-1) cdot frac{sqrt{2}}{2} + 2 cdot 1 + (-1) cdot frac{sqrt{2}}{2} = -frac{sqrt{2}}{2} + 2 - frac{sqrt{2}}{2} = 2 - sqrt{2} )Third component:( 0 cdot frac{sqrt{2}}{2} + (-1) cdot 1 + 2 cdot frac{sqrt{2}}{2} = -1 + sqrt{2} )So, ( M mathbf{v}_1 = begin{pmatrix} sqrt{2} - 1 2 - sqrt{2} -1 + sqrt{2} end{pmatrix} )Now, ( lambda_1 mathbf{v}_1 = (2 - sqrt{2}) begin{pmatrix} frac{sqrt{2}}{2} 1 frac{sqrt{2}}{2} end{pmatrix} = begin{pmatrix} (2 - sqrt{2}) frac{sqrt{2}}{2} 2 - sqrt{2} (2 - sqrt{2}) frac{sqrt{2}}{2} end{pmatrix} )Compute each component:First component:( (2 - sqrt{2}) frac{sqrt{2}}{2} = frac{2 sqrt{2} - 2}{2} = sqrt{2} - 1 )Second component:( 2 - sqrt{2} )Third component:Same as first: ( sqrt{2} - 1 )So, ( lambda_1 mathbf{v}_1 = begin{pmatrix} sqrt{2} - 1 2 - sqrt{2} sqrt{2} - 1 end{pmatrix} )Comparing with ( M mathbf{v}_1 ), which is ( begin{pmatrix} sqrt{2} - 1 2 - sqrt{2} -1 + sqrt{2} end{pmatrix} ), we see that the third component is ( sqrt{2} - 1 ) in both, so it's correct.Similarly, for ( k = 2 ):( lambda_2 = 2 )Eigenvector ( mathbf{v}_2 = begin{pmatrix} 1 0 -1 end{pmatrix} )Compute ( M mathbf{v}_2 ):First component:( 2 cdot 1 + (-1) cdot 0 + 0 cdot (-1) = 2 )Second component:( (-1) cdot 1 + 2 cdot 0 + (-1) cdot (-1) = -1 + 0 + 1 = 0 )Third component:( 0 cdot 1 + (-1) cdot 0 + 2 cdot (-1) = -2 )So, ( M mathbf{v}_2 = begin{pmatrix} 2 0 -2 end{pmatrix} = 2 begin{pmatrix} 1 0 -1 end{pmatrix} = lambda_2 mathbf{v}_2 ). Correct.For ( k = 3 ):( lambda_3 = 2 + sqrt{2} )Eigenvector ( mathbf{v}_3 = begin{pmatrix} frac{sqrt{2}}{2} -1 frac{sqrt{2}}{2} end{pmatrix} )Compute ( M mathbf{v}_3 ):First component:( 2 cdot frac{sqrt{2}}{2} + (-1) cdot (-1) + 0 cdot frac{sqrt{2}}{2} = sqrt{2} + 1 )Second component:( (-1) cdot frac{sqrt{2}}{2} + 2 cdot (-1) + (-1) cdot frac{sqrt{2}}{2} = -frac{sqrt{2}}{2} - 2 - frac{sqrt{2}}{2} = -2 - sqrt{2} )Third component:( 0 cdot frac{sqrt{2}}{2} + (-1) cdot (-1) + 2 cdot frac{sqrt{2}}{2} = 1 + sqrt{2} )So, ( M mathbf{v}_3 = begin{pmatrix} sqrt{2} + 1 -2 - sqrt{2} 1 + sqrt{2} end{pmatrix} )Now, ( lambda_3 mathbf{v}_3 = (2 + sqrt{2}) begin{pmatrix} frac{sqrt{2}}{2} -1 frac{sqrt{2}}{2} end{pmatrix} = begin{pmatrix} (2 + sqrt{2}) frac{sqrt{2}}{2} - (2 + sqrt{2}) (2 + sqrt{2}) frac{sqrt{2}}{2} end{pmatrix} )Compute each component:First component:( (2 + sqrt{2}) frac{sqrt{2}}{2} = frac{2 sqrt{2} + 2}{2} = sqrt{2} + 1 )Second component:( - (2 + sqrt{2}) )Third component:Same as first: ( sqrt{2} + 1 )So, ( lambda_3 mathbf{v}_3 = begin{pmatrix} sqrt{2} + 1 -2 - sqrt{2} sqrt{2} + 1 end{pmatrix} ), which matches ( M mathbf{v}_3 ). Correct.Therefore, the eigenvalues are ( 2 - sqrt{2} ), ( 2 ), and ( 2 + sqrt{2} ), with corresponding eigenvectors:For ( lambda_1 = 2 - sqrt{2} ):( mathbf{v}_1 = begin{pmatrix} frac{sqrt{2}}{2} 1 frac{sqrt{2}}{2} end{pmatrix} )For ( lambda_2 = 2 ):( mathbf{v}_2 = begin{pmatrix} 1 0 -1 end{pmatrix} )For ( lambda_3 = 2 + sqrt{2} ):( mathbf{v}_3 = begin{pmatrix} frac{sqrt{2}}{2} -1 frac{sqrt{2}}{2} end{pmatrix} )These eigenvectors can be normalized if needed, but as they are, they are correct up to a scalar multiple.So, summarizing:Eigenvalues:1. ( 2 - sqrt{2} )2. ( 2 )3. ( 2 + sqrt{2} )Eigenvectors:1. ( begin{pmatrix} frac{sqrt{2}}{2} 1 frac{sqrt{2}}{2} end{pmatrix} )2. ( begin{pmatrix} 1 0 -1 end{pmatrix} )3. ( begin{pmatrix} frac{sqrt{2}}{2} -1 frac{sqrt{2}}{2} end{pmatrix} )I think that's all for problem 2.</think>

answer:<think>Alright, so I have this problem about theater critics analyzing the impact of idols transitioning into musical theater. They've collected data over 10 years with 100 productions. The critic scores ( C_i ) and audience scores ( A_i ) are given by these equations:[ C_i = 50 + 10 sinleft(frac{pi i}{5}right) - 5I_i ][ A_i = 70 + 15 cosleft(frac{pi i}{10}right) + 3I_i ]where ( I_i ) is 1 if an idol is in production ( i ), and 0 otherwise. The first part asks for the average critic and audience scores for productions with and without idols, assuming exactly half have idols. The second part is about finding the correlation coefficient between critic and audience scores for productions with idols.Okay, let's start with part 1. Since exactly half of the 100 productions feature idols, that means 50 productions have ( I_i = 1 ) and 50 have ( I_i = 0 ). I need to compute the average ( C_i ) and ( A_i ) for both groups.First, let's consider the general form of the equations. Both ( C_i ) and ( A_i ) have a base score plus a sinusoidal component and a term dependent on ( I_i ). So, for each production, the score is a combination of these components.For the average, since we're dealing with averages over 50 productions, we can think about the average of the sinusoidal functions over their periods. The sine and cosine functions have periodicities, so their averages over a full period should be zero.Let me check that. The sine function in ( C_i ) is ( sinleft(frac{pi i}{5}right) ). The period of this sine function is ( frac{2pi}{pi/5} } = 10 ). So every 10 productions, the sine wave completes a full cycle. Similarly, the cosine function in ( A_i ) is ( cosleft(frac{pi i}{10}right) ), which has a period of ( frac{2pi}{pi/10} } = 20 ). So every 20 productions, the cosine completes a full cycle.But we have 100 productions, which is 10 periods for the sine function and 5 periods for the cosine function. So over 100 productions, both functions complete an integer number of periods. Therefore, the average of the sine and cosine terms over 100 productions would be zero.But wait, we're not averaging over all 100 productions, but separately over the 50 productions with idols and the 50 without. So we need to check if the distribution of the sine and cosine terms is the same in both groups.But since the assignment of ( I_i ) is arbitrary (assuming it's random), the distribution of the sine and cosine terms in both groups should be similar. So the average of the sine and cosine terms over 50 productions should still be approximately zero, especially since 50 is a multiple of the periods.Wait, let's see. For the sine function with period 10, 50 is 5 periods. So 50 is a multiple of 10, so the average over 50 would still be zero. Similarly, the cosine function has period 20, and 50 is 2.5 periods. Hmm, 2.5 isn't an integer, so the average over 50 might not be exactly zero. Wait, but 50 is half of 100, which is 5 periods for cosine (since 100 / 20 = 5). So 50 is 2.5 periods, which is half of 5 periods. So perhaps the average over 50 would still be zero? Let me think.Actually, the average of a cosine function over any integer multiple of its period is zero. But 50 isn't an integer multiple of 20. So the average over 50 might not be zero. Hmm, this complicates things.Wait, maybe I need to compute the exact average. Let's consider the sine term first. For ( sinleft(frac{pi i}{5}right) ), over 100 productions, the average is zero. But if we split the 100 into two groups of 50, each group would have 5 periods of the sine function. So each group's average sine term would also be zero.Similarly, for the cosine term ( cosleft(frac{pi i}{10}right) ), over 100 productions, it's 5 periods. So each group of 50 would have 2.5 periods. The average of cosine over 2.5 periods might not be zero. Wait, let's compute it.The average of ( cosleft(frac{pi i}{10}right) ) over 50 terms. Let's denote ( theta_i = frac{pi i}{10} ). So we're summing ( cos(theta_i) ) from ( i = 1 ) to 50. The average would be ( frac{1}{50} sum_{i=1}^{50} cosleft(frac{pi i}{10}right) ).This is equivalent to the average of a cosine function sampled at 50 points over 2.5 periods. The average of a cosine function over any interval is the integral over that interval divided by the interval length. But since we're dealing with discrete points, it's a bit different.Alternatively, we can use the formula for the sum of cosines in an arithmetic sequence. The sum ( sum_{k=0}^{n-1} cos(a + kd) ) is ( frac{sin(n d / 2)}{sin(d / 2)} cosleft(a + frac{(n - 1)d}{2}right) ).In our case, ( a = frac{pi}{10} ), ( d = frac{pi}{10} ), and ( n = 50 ). So the sum is:[ sum_{i=1}^{50} cosleft(frac{pi i}{10}right) = sum_{k=1}^{50} cosleft(frac{pi k}{10}right) ]Let me shift the index to start from 0:[ sum_{k=0}^{49} cosleft(frac{pi (k + 1)}{10}right) = sum_{k=0}^{49} cosleft(frac{pi}{10} + frac{pi k}{10}right) ]Using the formula:[ sum_{k=0}^{n-1} cos(a + kd) = frac{sin(n d / 2)}{sin(d / 2)} cosleft(a + frac{(n - 1)d}{2}right) ]Here, ( a = frac{pi}{10} ), ( d = frac{pi}{10} ), ( n = 50 ). So:Numerator: ( sin(50 * frac{pi}{10} / 2) = sin(5 * pi / 2) = sin(5pi/2) = 1 )Denominator: ( sin(frac{pi}{10} / 2) = sin(pi/20) approx 0.1564 )The argument of the cosine term is:( a + frac{(n - 1)d}{2} = frac{pi}{10} + frac{49 * pi}{10} / 2 = frac{pi}{10} + frac{49pi}{20} = frac{2pi + 49pi}{20} = frac{51pi}{20} )So the sum is:[ frac{1}{sin(pi/20)} cosleft(frac{51pi}{20}right) ]Simplify ( frac{51pi}{20} ). Since ( 51pi/20 = 2pi + 11pi/20 ), and cosine is periodic with period ( 2pi ), so ( cos(51pi/20) = cos(11pi/20) ).So the sum is:[ frac{cos(11pi/20)}{sin(pi/20)} ]Compute this numerically:( cos(11pi/20) approx cos(99 degrees) approx -0.1564 )( sin(pi/20) approx 0.1564 )So the sum is approximately ( -0.1564 / 0.1564 = -1 )Therefore, the sum of the cosine terms over 50 productions is approximately -1. So the average is ( -1 / 50 = -0.02 ). That's very close to zero. So for practical purposes, we can consider the average cosine term over 50 productions as approximately zero.Similarly, for the sine term, since 50 is a multiple of the period (10), the average over 50 would be zero.Therefore, for both groups (with and without idols), the average of the sine and cosine terms is approximately zero.So, for the average critic score ( bar{C} ) and average audience score ( bar{A} ), we can ignore the sine and cosine terms when calculating the averages.So, for productions with idols (( I_i = 1 )):[ bar{C}_{text{with}} = 50 + 10 * 0 - 5 * 1 = 50 - 5 = 45 ][ bar{A}_{text{with}} = 70 + 15 * 0 + 3 * 1 = 70 + 3 = 73 ]For productions without idols (( I_i = 0 )):[ bar{C}_{text{without}} = 50 + 10 * 0 - 5 * 0 = 50 ][ bar{A}_{text{without}} = 70 + 15 * 0 + 3 * 0 = 70 ]Wait, but earlier I considered that the sine and cosine terms average to zero, so their contribution is zero in the average. Therefore, the average scores are as above.But let me double-check. Since the sine and cosine terms average to zero, their contribution is zero in the average, so the only terms contributing are the constants and the ( I_i ) terms.Yes, that makes sense. So the average critic score for productions with idols is 45, and without is 50. The average audience score for with is 73, and without is 70.Wait, but the audience score for with idols is higher, while the critic score is lower. That's interesting. So idols seem to make the audience like the productions more, but critics like them less.But let me make sure about the sine and cosine averages. I approximated them as zero, but for the cosine term, the average was approximately -0.02, which is negligible. So it's safe to consider them as zero.Therefore, the answers for part 1 are:- With idols: ( bar{C} = 45 ), ( bar{A} = 73 )- Without idols: ( bar{C} = 50 ), ( bar{A} = 70 )Now, moving on to part 2: Calculate the correlation coefficient between the critic scores and the audience scores for the productions that feature idols.So, we need to compute the Pearson correlation coefficient between ( C_i ) and ( A_i ) for the 50 productions where ( I_i = 1 ).The Pearson correlation coefficient ( r ) is given by:[ r = frac{text{Cov}(C, A)}{sigma_C sigma_A} ]where ( text{Cov}(C, A) ) is the covariance between ( C ) and ( A ), and ( sigma_C ) and ( sigma_A ) are the standard deviations of ( C ) and ( A ), respectively.First, let's write the expressions for ( C_i ) and ( A_i ) when ( I_i = 1 ):[ C_i = 50 + 10 sinleft(frac{pi i}{5}right) - 5 ][ C_i = 45 + 10 sinleft(frac{pi i}{5}right) ][ A_i = 70 + 15 cosleft(frac{pi i}{10}right) + 3 ][ A_i = 73 + 15 cosleft(frac{pi i}{10}right) ]So, ( C_i = 45 + 10 sinleft(frac{pi i}{5}right) )and ( A_i = 73 + 15 cosleft(frac{pi i}{10}right) )We need to compute the covariance and the standard deviations.First, let's note that the mean of ( C_i ) is 45, and the mean of ( A_i ) is 73, as we found earlier.So, the covariance is:[ text{Cov}(C, A) = frac{1}{n - 1} sum_{i=1}^{n} (C_i - bar{C})(A_i - bar{A}) ]But since we're dealing with the entire population (50 productions), we can use ( frac{1}{n} ) instead of ( frac{1}{n - 1} ). But since the question doesn't specify, I'll assume it's the sample correlation, so using ( n - 1 ).But actually, since we have the exact expressions, maybe we can compute the covariance without summing over all 50 terms.Let me think. The covariance between ( C_i ) and ( A_i ) is:[ text{Cov}(C, A) = E[(C_i - bar{C})(A_i - bar{A})] ]Given that ( C_i = 45 + 10 sin(pi i /5) ) and ( A_i = 73 + 15 cos(pi i /10) ), then:[ (C_i - bar{C}) = 10 sin(pi i /5) ][ (A_i - bar{A}) = 15 cos(pi i /10) ]So,[ text{Cov}(C, A) = E[10 sin(pi i /5) * 15 cos(pi i /10)] ][ = 150 E[sin(pi i /5) cos(pi i /10)] ]We can use the identity:[ sin a cos b = frac{1}{2} [sin(a + b) + sin(a - b)] ]So,[ sin(pi i /5) cos(pi i /10) = frac{1}{2} [sin(pi i /5 + pi i /10) + sin(pi i /5 - pi i /10)] ][ = frac{1}{2} [sin(3pi i /10) + sin(pi i /10)] ]Therefore,[ text{Cov}(C, A) = 150 * frac{1}{2} E[sin(3pi i /10) + sin(pi i /10)] ][ = 75 [E[sin(3pi i /10)] + E[sin(pi i /10)]] ]Now, we need to compute the expected value (average) of ( sin(3pi i /10) ) and ( sin(pi i /10) ) over the 50 productions where ( I_i = 1 ).Wait, but the index ( i ) here is for the productions with idols. So we need to consider the values of ( i ) where ( I_i = 1 ). However, the problem doesn't specify which specific productions have idols, only that exactly half do. So, to compute the expectation, we need to know how the idol productions are distributed in terms of ( i ).But since the problem doesn't specify, we can assume that the idol productions are randomly distributed over the 100 productions. Therefore, the distribution of ( i ) for idol productions is uniform over the 100 productions. So, the average of ( sin(3pi i /10) ) and ( sin(pi i /10) ) over 50 randomly selected ( i ) from 1 to 100 would be the same as the average over all 100 productions.But wait, the average of ( sin(3pi i /10) ) over 100 productions is:Since the period of ( sin(3pi i /10) ) is ( frac{2pi}{3pi/10} } = frac{20}{3} approx 6.666 ). So over 100 productions, which is approximately 15 full periods, the average would be zero.Similarly, ( sin(pi i /10) ) has period 20, so over 100 productions, it's 5 periods, so average is zero.Therefore, the expected value of both ( sin(3pi i /10) ) and ( sin(pi i /10) ) over the 50 idol productions is zero.Wait, but is that correct? Because if the idol productions are randomly distributed, the average over 50 would still be zero, as the sine function is symmetric.Therefore, ( E[sin(3pi i /10)] = 0 ) and ( E[sin(pi i /10)] = 0 ), so the covariance is zero.But that can't be right because the sine and cosine functions might have some phase relationship that could lead to a non-zero covariance.Wait, no, because we're taking the expectation over all ( i ), and the sine functions average to zero. So the covariance is zero.But wait, let's think again. The covariance is between ( C_i ) and ( A_i ), which are both functions of ( i ). So, even though individually their sine terms average to zero, their product might not.Wait, no, we already expressed the covariance as the expectation of the product of deviations, which we expanded into the expectation of the product of sine and cosine terms, which we transformed into the sum of sines. Since each sine term averages to zero, the covariance is zero.Therefore, the covariance is zero, which would imply that the correlation coefficient is zero.But that seems counterintuitive because both ( C_i ) and ( A_i ) are functions of ( i ), so they might be related.Wait, let's think differently. Maybe the covariance isn't zero because the sine and cosine functions are related through the index ( i ). Let me check the exact relationship.We have:( C_i = 45 + 10 sin(pi i /5) )( A_i = 73 + 15 cos(pi i /10) )Note that ( pi i /5 = 2pi i /10 ), so ( sin(pi i /5) = sin(2pi i /10) ). So, ( C_i ) is a sine function with twice the frequency of the cosine function in ( A_i ).So, ( C_i ) is ( sin(2theta) ) and ( A_i ) is ( cos(theta) ), where ( theta = pi i /10 ).We can use the identity ( sin(2theta) = 2sintheta costheta ). So,( C_i = 45 + 10 * 2 sin(theta) cos(theta) = 45 + 20 sin(theta) cos(theta) )But ( A_i = 73 + 15 cos(theta) )So, ( C_i ) is related to ( sin(theta) cos(theta) ), and ( A_i ) is related to ( cos(theta) ). Therefore, ( C_i ) is proportional to ( sin(theta) ) times ( A_i )'s term.But since we're looking at the covariance between ( C_i ) and ( A_i ), which is ( E[(C_i - bar{C})(A_i - bar{A})] ), and we've expressed ( C_i - bar{C} = 10 sin(2theta) ) and ( A_i - bar{A} = 15 cos(theta) ), so the product is ( 150 sin(2theta) cos(theta) ).Using the identity ( sin(2theta) cos(theta) = frac{1}{2} [sin(3theta) + sin(theta)] ), as before.So, the covariance is ( 150 * frac{1}{2} E[sin(3theta) + sin(theta)] ), which is ( 75 [E[sin(3theta)] + E[sin(theta)]] ).But ( theta = pi i /10 ), and ( i ) ranges over 50 randomly selected values from 1 to 100. So, the average of ( sin(3theta) ) and ( sin(theta) ) over these 50 values.But since the distribution of ( i ) is uniform over 1 to 100, the average of ( sin(3theta) ) and ( sin(theta) ) over 50 randomly selected ( i ) is the same as the average over all 100 ( i ).So, let's compute the average of ( sin(3theta) ) and ( sin(theta) ) over all 100 ( i ).For ( sin(3theta) ), ( theta = pi i /10 ), so ( 3theta = 3pi i /10 ). The average of ( sin(3pi i /10) ) over 100 ( i ) is zero because it's a sine function over 100 points, which is 5 periods (since period is 20). Similarly, the average of ( sin(pi i /10) ) over 100 ( i ) is zero.Therefore, the covariance is zero.Wait, but that seems to suggest that ( C_i ) and ( A_i ) are uncorrelated, which might be the case because one is a sine function and the other is a cosine function with a different frequency.But let me think again. If ( C_i ) is a function of ( sin(2theta) ) and ( A_i ) is a function of ( cos(theta) ), then their covariance depends on the relationship between ( sin(2theta) ) and ( cos(theta) ).But since ( sin(2theta) = 2sintheta costheta ), then ( sin(2theta) ) is proportional to ( sintheta costheta ). So, the covariance between ( sin(2theta) ) and ( costheta ) is proportional to the covariance between ( sintheta costheta ) and ( costheta ), which is ( E[sintheta cos^2theta] ).But the expectation of ( sintheta cos^2theta ) over a full period is zero because ( sintheta ) is an odd function and ( cos^2theta ) is even, so their product is odd, and the integral over a symmetric interval is zero.Therefore, the covariance is indeed zero.So, the correlation coefficient ( r ) is zero divided by the product of standard deviations, which is zero.But wait, let's compute the standard deviations to be thorough.First, for ( C_i ):[ C_i = 45 + 10 sin(pi i /5) ]The variance of ( C_i ) is the variance of ( 10 sin(pi i /5) ). The variance of ( sin(x) ) over a full period is ( frac{1}{2} ), so the variance of ( 10 sin(x) ) is ( 10^2 * frac{1}{2} = 50 ). Therefore, the standard deviation ( sigma_C = sqrt{50} approx 7.071 ).Similarly, for ( A_i ):[ A_i = 73 + 15 cos(pi i /10) ]The variance of ( 15 cos(x) ) is ( 15^2 * frac{1}{2} = 112.5 ). So, ( sigma_A = sqrt{112.5} approx 10.6066 ).Therefore, the correlation coefficient is:[ r = frac{0}{7.071 * 10.6066} = 0 ]So, the correlation coefficient is zero.But wait, is that correct? Because ( C_i ) and ( A_i ) are both functions of ( i ), but their frequencies are different, so they might be orthogonal, leading to zero covariance.Yes, that makes sense. So, the correlation coefficient is zero.But let me double-check by considering specific values. Let's take a few values of ( i ) and see.For example, let's take ( i = 1 ):( C_1 = 45 + 10 sin(pi/5) approx 45 + 10 * 0.5878 approx 50.878 )( A_1 = 73 + 15 cos(pi/10) approx 73 + 15 * 0.9511 approx 87.2665 )( i = 2 ):( C_2 = 45 + 10 sin(2pi/5) approx 45 + 10 * 0.9511 approx 54.511 )( A_2 = 73 + 15 cos(2pi/10) = 73 + 15 cos(pi/5) approx 73 + 15 * 0.8090 approx 84.135 )( i = 3 ):( C_3 = 45 + 10 sin(3pi/5) approx 45 + 10 * 0.9511 approx 54.511 )( A_3 = 73 + 15 cos(3pi/10) approx 73 + 15 * 0.5878 approx 81.817 )( i = 4 ):( C_4 = 45 + 10 sin(4pi/5) approx 45 + 10 * 0.5878 approx 50.878 )( A_4 = 73 + 15 cos(4pi/10) = 73 + 15 cos(2pi/5) approx 73 + 15 * 0.3090 approx 76.635 )( i = 5 ):( C_5 = 45 + 10 sin(5pi/5) = 45 + 10 sin(pi) = 45 )( A_5 = 73 + 15 cos(5pi/10) = 73 + 15 cos(pi/2) = 73 + 0 = 73 )Looking at these, ( C_i ) and ( A_i ) don't seem to have a clear linear relationship. For example, when ( C_i ) is high (around 54.5), ( A_i ) is around 84, but when ( C_i ) decreases, ( A_i ) also decreases, but not in a perfectly linear way. However, since we're dealing with sinusoidal functions of different frequencies, their relationship isn't linear, so the covariance could indeed be zero.Therefore, the correlation coefficient is zero.So, summarizing:1. Average scores: - With idols: Critic = 45, Audience = 73 - Without idols: Critic = 50, Audience = 702. Correlation coefficient between critic and audience scores for idol productions: 0Final Answer1. The average critic score for productions with idols is boxed{45} and the average audience score is boxed{73}. For productions without idols, the average critic score is boxed{50} and the average audience score is boxed{70}.2. The correlation coefficient between critic and audience scores for productions with idols is boxed{0}.</think>

answer:<think>Alright, so I've got these two probability questions to solve related to cricket matches at the Ayoubi Cricket Stadium. Let me take them one at a time and think through each step carefully.Starting with the first question:1. Probability that Team B's run rate exceeds Team A's by more than 0.5 runs per over.Okay, so both teams have their average run rates and variances given. Team A has an average of 3.8 runs per over with a variance of 0.16, and Team B has an average of 4.2 runs per over with a variance of 0.25. We need to find the probability that in a randomly chosen match, Team B's run rate is more than 0.5 runs per over higher than Team A's.Hmm, since both run rates are normally distributed, the difference between them should also be normally distributed. Let me recall that if X and Y are two independent normal variables, then X - Y is also normal with mean μ_X - μ_Y and variance σ_X² + σ_Y².So, let me define D = Y - X, where Y is Team B's run rate and X is Team A's run rate. Then:- Mean of D, μ_D = μ_Y - μ_X = 4.2 - 3.8 = 0.4 runs per over.- Variance of D, σ_D² = σ_Y² + σ_X² = 0.25 + 0.16 = 0.41.- Therefore, standard deviation σ_D = sqrt(0.41) ≈ 0.6403.We need P(D > 0.5). So, we can standardize this to find the Z-score:Z = (0.5 - μ_D) / σ_D = (0.5 - 0.4) / 0.6403 ≈ 0.1 / 0.6403 ≈ 0.1562.Now, looking at the standard normal distribution table, a Z-score of 0.1562 corresponds to a cumulative probability of approximately 0.5616. But since we want P(D > 0.5), which is the area to the right of Z=0.1562, we subtract this from 1:P(D > 0.5) = 1 - 0.5616 = 0.4384.Wait, is that correct? Let me double-check. If the mean difference is 0.4, and we're looking for when it's more than 0.5, which is just a little above the mean. So the probability should be less than 0.5, which matches the 0.4384 result. Hmm, that seems plausible.Moving on to the second question:2. Probability related to the number of wickets lost per match.We have that the average number of wickets lost is 7 with a standard deviation of 2. Assuming a normal distribution, we need two probabilities:a) The probability that the number of wickets lost is between 5 and 10.b) The probability that the number of wickets lost exceeds the average by more than twice the standard deviation.Starting with part a:We need P(5 < W < 10), where W is the number of wickets. Let's standardize both 5 and 10.First, Z1 = (5 - 7)/2 = (-2)/2 = -1.Z2 = (10 - 7)/2 = 3/2 = 1.5.Looking up these Z-scores:- For Z = -1, the cumulative probability is 0.1587.- For Z = 1.5, the cumulative probability is approximately 0.9332.Therefore, P(5 < W < 10) = P(Z < 1.5) - P(Z < -1) = 0.9332 - 0.1587 = 0.7745.So, about 77.45% probability.Now, part b:Probability that the number of wickets exceeds the average by more than twice the standard deviation. The average is 7, standard deviation is 2, so twice the standard deviation is 4. Therefore, exceeding the average by more than 4 would mean W > 7 + 4 = 11.So, we need P(W > 11).Calculating the Z-score:Z = (11 - 7)/2 = 4/2 = 2.Looking up Z = 2, the cumulative probability is 0.9772. Therefore, the probability that W is less than or equal to 11 is 0.9772, so the probability that W > 11 is 1 - 0.9772 = 0.0228, or 2.28%.Wait, but the question says "exceeds the average by more than twice the standard deviation." So, does that mean W > μ + 2σ, which is 11? Yes, that's correct. So, 2.28% is the probability.Let me just recap to make sure I didn't make any mistakes.For the first question, I calculated the difference in run rates as a normal variable, found the Z-score for 0.5, and subtracted from 1 to get the probability. That seems right.For the second question, part a was straightforward, calculating the area between two Z-scores. Part b was about the upper tail beyond μ + 2σ, which is a standard calculation.I think I did everything correctly, but let me just verify the Z-scores and the corresponding probabilities once more.First question:Z = (0.5 - 0.4)/0.6403 ≈ 0.1562. The cumulative probability for 0.1562 is approximately 0.5616, so 1 - 0.5616 ≈ 0.4384. That seems correct.Second question, part a:Z1 = -1, cumulative 0.1587; Z2 = 1.5, cumulative 0.9332. Difference is 0.7745. Correct.Part b:Z = 2, cumulative 0.9772, so 1 - 0.9772 = 0.0228. Correct.I think all steps are solid. Maybe I should check if the run rates are independent? The problem says to assume normal distribution, but it doesn't specify independence. However, in such problems, unless stated otherwise, we usually assume independence. So, that's probably fine.Also, for the wickets, the distribution is normal, but the number of wickets is discrete. However, since the problem says to assume a normal distribution, we can proceed as if it's continuous, which is standard practice in such cases.Alright, I think I'm confident with these answers.Final Answer1. The probability is boxed{0.4384}.2. a) The probability is boxed{0.7745}. b) The probability is boxed{0.0228}.</think>