🧠:<think>Okay, so I've got this problem about optimizing a network system using graph algorithms and integer linear programming. Let me try to break it down step by step.First, the problem is divided into two parts. The first part is about finding the shortest path using the Floyd-Warshall algorithm, and the second part is about formulating an optimization problem to add up to k new edges to improve network performance.Starting with part 1: I need to find the shortest path from a source node s to a target node t using the Floyd-Warshall algorithm. I remember that Floyd-Warshall is an algorithm for finding the shortest paths between all pairs of nodes in a graph. It's useful when we need to compute all pairs of shortest paths, not just from a single source. But in this case, since we're only interested in the shortest path from s to t, maybe there's a more efficient algorithm like Dijkstra's. However, the question specifically asks for Floyd-Warshall, so I have to go with that.I think the Floyd-Warshall algorithm works by progressively improving an estimate of the shortest path between all pairs of nodes. It does this by considering each node as an intermediate node and checking if going through that node provides a shorter path. The algorithm uses a dynamic programming approach, updating a distance matrix over time.The pseudocode for Floyd-Warshall usually initializes the distance matrix with the adjacency matrix, where the distance from a node to itself is zero and there's no edge where the adjacency matrix has zero. Then, for each intermediate node k, it checks for each pair of nodes i and j whether the path from i to j through k is shorter than the current known distance.So, the pseudocode would look something like this:Initialize dist as a copy of A.For k from 1 to n: For i from 1 to n: For j from 1 to n: If dist[i][j] > dist[i][k] + dist[k][j]: Update dist[i][j] to dist[i][k] + dist[k][j]After this, dist[s][t] will have the shortest path from s to t.But wait, the question mentions that A_{ij} is the weight of the edge from i to j, and zero if there's no edge. So, in the initial distance matrix, we have to set the distance between i and j as A_{ij} if there's an edge, otherwise, it's infinity, except for the diagonal which is zero. Hmm, but in the problem statement, it's written that A_{ij} is zero if there's no edge. So, in the distance matrix, we should set dist[i][j] = A_{ij} if i != j, else zero. But for nodes without a direct edge, the initial distance is zero, which might not be correct because in reality, the distance should be infinity if there's no direct edge. So, maybe I need to adjust the initialization step.Wait, no, in the adjacency matrix, A_{ij} is the weight if there's an edge, otherwise zero. So, in the distance matrix, we need to set dist[i][j] = A_{ij} if there's an edge, else infinity. But since A_{ij} is zero when there's no edge, we can't directly use A as the initial distance matrix. So, perhaps the initialization should be:For all i, j: If i == j: dist[i][j] = 0 Else if A[i][j] > 0: dist[i][j] = A[i][j] Else: dist[i][j] = infinityThat makes sense because if A[i][j] is zero, it means there's no direct edge, so the initial distance is infinity, except for the diagonal which is zero.So, the pseudocode should include this initialization step. Then, proceed with the three nested loops as before.Regarding the time complexity, Floyd-Warshall runs in O(n^3) time because it has three nested loops, each going through all n nodes. This is because for each intermediate node k, it checks all pairs (i, j) and updates their distances if going through k provides a shorter path. So, regardless of the number of edges, it's O(n^3), which is polynomial but can be slow for very large n. However, since it computes all pairs of shortest paths, it's efficient in that sense.Moving on to part 2: After finding the shortest path, we need to optimize the network by adding up to k new edges with given weights. The goal is to improve the network's performance, which I assume means reducing the shortest path from s to t or maybe improving other metrics, but since the first part was about the shortest path, I think the focus is on reducing the distance from s to t.To formulate this as an integer linear programming problem, I need to define decision variables, an objective function, and constraints.First, the decision variables. We need to decide which edges to add. Since we can add up to k edges, each edge addition is a binary variable. Let me denote x_{ij} as a binary variable where x_{ij} = 1 if we add an edge from i to j, and 0 otherwise. But wait, the problem says "up to k new edges," so we might need another variable to count the number of edges added. Alternatively, we can have a constraint that the sum of x_{ij} over all possible edges is less than or equal to k.But before that, I need to think about how adding edges affects the shortest path. The new edges can potentially provide shorter paths from s to t. So, after adding some edges, the new shortest path might be shorter than the original one.But how do we model this in ILP? The challenge is that the shortest path depends on the graph structure, which is now being modified by the addition of edges. So, we need to model the shortest path in the augmented graph.One approach is to model the shortest path problem within the ILP formulation. That is, we can include variables that represent the shortest distances and the paths taken.Alternatively, since we're adding edges, we can consider that the new edges can create new paths, and we want to choose which edges to add such that the resulting shortest path is minimized.But this seems a bit complex because the shortest path is a function of the graph, which is being modified by the decision variables. So, perhaps we need to model the shortest path constraints within the ILP.Let me recall that in ILP, we can model shortest path constraints using variables that represent the distance from the source to each node. Let me denote d_j as the shortest distance from s to node j. Then, for each node j, we have:d_j <= d_i + w_{ij} for all edges (i, j) in the original graph.Additionally, for the new edges we add, if we add an edge from i to j with weight w_{ij}^new, then we have:d_j <= d_i + w_{ij}^newBut since adding an edge is a decision variable, we need to link x_{ij} with the constraints. So, for each potential new edge (i, j), if x_{ij} = 1, then the constraint d_j <= d_i + w_{ij}^new must hold. However, in ILP, we can't have conditional constraints directly, so we need to linearize this.One way to do this is to include the constraint for all possible new edges, but only activate it when x_{ij} = 1. To do this, we can use big-M constraints. For each potential new edge (i, j), we can write:d_j <= d_i + w_{ij}^new + M*(1 - x_{ij})Where M is a large enough constant, larger than any possible distance in the graph. This way, when x_{ij} = 1, the constraint becomes active, and when x_{ij} = 0, the constraint is effectively ignored because M*(1 - x_{ij}) becomes M, which is larger than any possible distance.But wait, we also need to consider that adding an edge from i to j might allow for a shorter path, but we also have to consider the reverse. For example, adding an edge from j to i could also affect the shortest path if it creates a cycle that reduces the overall distance. However, since we're only concerned with the shortest path from s to t, maybe we don't need to consider all possible edges, just those that can potentially reduce the distance from s to t.But to keep it general, perhaps we should consider all possible edges, but in practice, the number of potential edges is n(n-1), which could be large, but since we're limited to adding up to k edges, the decision variables x_{ij} will help select which ones to add.So, putting this together, the ILP formulation would have:Decision variables:- x_{ij} ∈ {0, 1} for all pairs (i, j), indicating whether we add an edge from i to j.- d_j ∈ ℝ for all nodes j, representing the shortest distance from s to j.Objective function:Minimize d_tSubject to:1. For the original graph: For each edge (i, j) in E: d_j <= d_i + A_{ij}2. For the new edges: For each potential new edge (i, j): d_j <= d_i + w_{ij}^new + M*(1 - x_{ij})3. The number of new edges added: Σ_{i,j} x_{ij} <= k4. The source node distance: d_s = 05. The non-negativity constraints: d_j >= 0 for all jWait, but in the original graph, the edges are already present, so their constraints are always active. The new edges' constraints are only active if x_{ij} = 1.But also, we need to ensure that the distances are consistent with the graph. So, in addition to the constraints from the original edges and the new edges (if added), we also need to ensure that the distances satisfy the triangle inequality.But in ILP, it's challenging to model all possible paths, so we rely on the constraints for each edge to enforce the shortest path.However, I think the way I wrote it, the constraints for the original edges are sufficient because they enforce that the distance d_j is at most the distance through each original edge. Similarly, for the new edges, if added, they provide additional constraints that could potentially reduce d_j.But wait, in the original graph, the distances are already computed via Floyd-Warshall, so maybe we don't need to recompute them in the ILP. Hmm, that's a point. Because in the first part, we've already found the shortest path using Floyd-Warshall, so the distances d_j are already known. But in the second part, we're adding edges to potentially improve the network, so the distances might be reduced.But in the ILP, we need to model the distances after adding the edges. So, perhaps the initial distances are not directly used, but rather, the ILP will compute the new distances considering the added edges.Wait, no, because the ILP is about deciding which edges to add to minimize the distance from s to t. So, the ILP needs to consider both the original graph and the potential new edges.So, the variables d_j represent the new shortest distances after adding the edges. The original edges are still present, so their constraints must hold, and the new edges, if added, provide additional constraints.Therefore, the ILP formulation should include:- The original edge constraints: for each (i, j) in E, d_j <= d_i + A_{ij}- The new edge constraints: for each potential (i, j), d_j <= d_i + w_{ij}^new + M*(1 - x_{ij})- The count constraint: Σ x_{ij} <= k- The source constraint: d_s = 0- The non-negativity: d_j >= 0But wait, the potential new edges are not limited to just the ones not present in the original graph. The problem says "adding up to k new edges with given weights." So, the new edges can be any possible directed edges, including those that already exist, but with possibly different weights. Or, maybe the new edges have given weights, which could be better than the existing ones.But in the problem statement, it's not specified whether the new edges can replace existing ones or just add to them. I think it's the latter; we can add edges in addition to the existing ones. So, the original edges remain, and we can add new ones on top.Therefore, the potential new edges are all possible directed edges (i, j) where i ≠ j, and for each, we can decide to add it or not, with a given weight w_{ij}^new.But wait, the problem says "with given weights." So, for each potential new edge, the weight is given, and we can choose whether to add it or not.So, in the ILP, for each potential new edge (i, j), we have a variable x_{ij} which is 1 if we add it, 0 otherwise. The weight of the new edge is given as w_{ij}^new.Therefore, the constraints for the new edges are as I wrote before.But now, considering that the original graph already has edges, and we're adding new ones, the distance variables d_j must satisfy both the original and the new edge constraints.Additionally, we need to ensure that the distances are consistent, meaning that for each node j, d_j is the shortest distance considering all possible paths, including those that use the new edges.But in ILP, we can't model all possible paths explicitly, so we rely on the edge constraints to enforce the shortest path.Therefore, the ILP formulation is as follows:Decision variables:- x_{ij} ∈ {0, 1} for all i, j ∈ V, i ≠ j, indicating whether to add the edge from i to j.- d_j ∈ ℝ for all j ∈ V, representing the shortest distance from s to j.Objective function:Minimize d_tSubject to:1. For each original edge (i, j) ∈ E: d_j <= d_i + A_{ij}2. For each potential new edge (i, j) ∈ V × V, i ≠ j: d_j <= d_i + w_{ij}^new + M*(1 - x_{ij})3. The number of new edges added: Σ_{i,j} x_{ij} <= k4. The source node distance: d_s = 05. Non-negativity constraints: d_j >= 0 for all j ∈ VBut wait, in the original graph, the edges are already present, so their constraints are always active. The new edges, if added, provide additional constraints that could potentially reduce d_j.However, in the ILP, we need to ensure that the distances are the shortest possible, considering both the original and new edges. So, the constraints for the original edges are necessary to ensure that the distances don't violate the existing paths, and the new edge constraints, when activated, can provide shorter paths.But I think the way it's formulated, the distances will automatically be the shortest because the constraints are enforcing that d_j is at most the minimum of all possible paths. However, in ILP, the solver will find the minimal d_t by choosing which edges to add to minimize it, subject to the constraints.But there's a potential issue here: the big-M constraints can sometimes lead to weak formulations, especially if M is not chosen carefully. M should be large enough to not interfere when x_{ij} = 0, but not so large that it causes numerical issues. Typically, M can be set to the sum of all edge weights or something similar, but in practice, it's a bit tricky.Alternatively, another approach is to use indicator constraints, which some ILP solvers support, where you can specify that a constraint is only active if a binary variable is 1. But since the problem asks for a general ILP formulation, I think the big-M approach is acceptable.Another consideration is that adding an edge from i to j might allow for a shorter path from s to j, but also, if j is on the path to t, it could help reduce the overall distance. So, the ILP is considering all possible ways the new edges can affect the shortest path.But I also need to think about the direction of the edges. Since the graph is directed, adding an edge from i to j doesn't necessarily help if the path from s to i is longer than the path from s to j. So, the ILP needs to consider all possible directions.Wait, but in the constraints, for each new edge (i, j), we have d_j <= d_i + w_{ij}^new + M*(1 - x_{ij}). This correctly models that if we add the edge (i, j), then the distance to j can be improved by going through i. But if we don't add it, the constraint is relaxed.However, this only considers the direct effect of adding the edge (i, j). It doesn't consider the possibility that adding multiple edges could create a shorter path through multiple new edges. For example, adding edge (i, j) and then (j, t) could create a shorter path s -> ... -> i -> j -> t. But the constraints as written would capture this because adding both edges would allow the distance to t to be updated through both steps.Wait, no, because the constraints are for each edge individually. So, if we add (i, j) and (j, t), the constraints would be:d_j <= d_i + w_{ij}^newd_t <= d_j + w_{jt}^newBut in the ILP, these are separate constraints, and the solver would enforce them together, effectively allowing the path s -> ... -> i -> j -> t.So, the formulation does account for multiple new edges being added and their combined effect on the shortest path.Another point to consider is that the original graph's edges are fixed, so their constraints are always active. The new edges, if added, provide additional constraints that can potentially reduce the distances.But in the ILP, the distances d_j are variables, so the solver will find the minimal possible d_t by choosing which edges to add and adjusting the distances accordingly.I think this formulation is correct, but let me double-check.The objective is to minimize d_t, the distance from s to t.The constraints ensure that:- The original edges provide their respective path lengths.- The new edges, if added, provide additional possible path lengths.- The total number of new edges added is at most k.- The distance from s is zero.- All distances are non-negative.Yes, this seems to cover all necessary aspects.But wait, in the original graph, the distances are already computed via Floyd-Warshall. So, in the ILP, are we considering the original distances as a starting point, or are we recomputing them? I think we're recomputing them because the ILP is considering the augmented graph (original plus new edges). So, the distances d_j in the ILP are the new shortest distances after adding the edges, not the original ones.Therefore, the ILP is correctly formulated to find the minimal d_t by potentially adding up to k new edges.In summary, the ILP formulation includes variables for adding edges and variables for the shortest distances, with constraints that enforce the shortest path properties considering both the original and new edges.</think>

🧠:<think>Okay, so I have this problem about a wedding car hire business. They have 10 luxury cars, and they can hire them out either as a standalone service or as a package deal. The standalone costs 500, and the package is 800. Last year, they made 72,000 and did 100 weddings in total. First, I need to set up a system of linear equations. Let me think. They mentioned two variables: x is the number of standalone services, and y is the number of package deals. So, the total number of weddings is x plus y, which is 100. That gives me the first equation: x + y = 100.Next, the total revenue is 72,000. Each standalone brings in 500, and each package brings in 800. So, the total revenue equation would be 500x + 800y = 72,000. That seems right.So now I have two equations:1. x + y = 1002. 500x + 800y = 72,000I need to solve for x and y. Let me use substitution or elimination. Maybe elimination is easier here. If I can make the coefficients of x or y the same, I can subtract the equations.Looking at the first equation, I can express x as 100 - y. Then substitute that into the second equation.So, substituting x = 100 - y into the second equation:500(100 - y) + 800y = 72,000Let me compute that step by step.First, multiply out 500*(100 - y):500*100 = 50,000500*(-y) = -500ySo, 50,000 - 500y + 800y = 72,000Combine like terms:(-500y + 800y) = 300ySo, 50,000 + 300y = 72,000Now, subtract 50,000 from both sides:300y = 72,000 - 50,000 = 22,000So, 300y = 22,000Divide both sides by 300:y = 22,000 / 300Let me compute that. 22,000 divided by 300. Well, 300*70 = 21,000, so 22,000 - 21,000 = 1,000. So, 1,000 / 300 is 10/3, which is about 3.333. So, y = 73.333? Wait, that can't be. Wait, 22,000 divided by 300 is actually 73.333... Hmm, that's a decimal. But the number of weddings should be a whole number. Did I make a mistake?Wait, let me check my calculations again.Starting from substitution:x + y = 100 => x = 100 - y500x + 800y = 72,000Substituting x:500*(100 - y) + 800y = 72,000500*100 = 50,000500*(-y) = -500ySo, 50,000 - 500y + 800y = 72,000Combine like terms:(-500y + 800y) = 300ySo, 50,000 + 300y = 72,000Subtract 50,000:300y = 22,000Divide by 300:y = 22,000 / 30022,000 divided by 300: Let's see, 300*70 = 21,000, so 22,000 - 21,000 = 1,000. Then 1,000 / 300 is 10/3, which is approximately 3.333. So, y = 73.333... Hmm, that's a fractional number of weddings, which doesn't make sense. Did I do something wrong?Wait, maybe I made a mistake in setting up the equations. Let me double-check.Total weddings: x + y = 100. That seems correct.Total revenue: 500x + 800y = 72,000. That also seems correct.Wait, maybe the numbers are such that y is not an integer. But in reality, the number of weddings should be whole numbers. Maybe the given total revenue is not compatible with the number of weddings? Or perhaps I made a calculation error.Wait, let me compute 22,000 divided by 300 again. 22,000 / 300. 300 goes into 22,000 how many times? 300*70=21,000, as before. 22,000 -21,000=1,000. 1,000/300=10/3≈3.333. So, y≈73.333. Hmm.Wait, maybe I should express y as 73 and 1/3, but that doesn't make sense for the number of weddings. So, perhaps the problem is designed in such a way that fractional weddings are acceptable for the sake of the problem, or maybe I made a mistake in the setup.Alternatively, maybe the total revenue is 72,000, but 100 weddings at an average of 720 each. Wait, 72,000 divided by 100 is 720. So, the average revenue per wedding is 720. Since standalone is 500 and package is 800, the average is somewhere in between.Let me think of it as a weighted average. Let me denote the proportion of standalone as p and package as (1-p). Then, 500p + 800(1-p) = 720.So, 500p + 800 - 800p = 720Combine like terms: (500p - 800p) + 800 = 720 => (-300p) + 800 = 720Subtract 800: -300p = -80Divide by -300: p = (-80)/(-300) = 80/300 = 8/30 = 4/15 ≈ 0.2667So, p ≈ 26.67%, meaning that about 26.67% of the weddings are standalone, and the rest are packages.So, x = 100*(4/15) ≈ 26.6667, and y = 100 - x ≈ 73.3333.So, that's consistent with my earlier result. So, even though the numbers are fractional, perhaps the problem allows for that, or maybe it's an approximation.Alternatively, maybe the total revenue is not exactly 72,000, but close. But the problem states it's exactly 72,000, so perhaps we have to accept fractional weddings for the sake of the problem.So, moving forward, x ≈ 26.6667 and y ≈ 73.3333.But since the problem asks to solve for x and y, perhaps we can express them as fractions.22,000 / 300 = 220 / 3 = 73 and 1/3.So, y = 73 1/3, and x = 100 - 73 1/3 = 26 2/3.So, x = 80/3 ≈26.6667 and y=220/3≈73.3333.Hmm, that seems a bit odd, but maybe that's the answer.Alternatively, perhaps I made a mistake in the setup. Let me double-check.Wait, another way: Maybe the total revenue is 500x + 800y =72,000, and x + y=100.So, if I solve for x and y, I can express x=100 - y, then plug into the revenue equation:500*(100 - y) +800y=72,000Which is 50,000 -500y +800y=72,000So, 50,000 +300y=72,000300y=22,000y=22,000/300=73.333...Yes, same result. So, perhaps the answer is fractional, which is acceptable in the context of the problem, even though in reality, you can't have a third of a wedding.So, moving on to part 2.They are considering expanding the fleet by adding 5 more cars, making it 15 cars. The demand follows a Poisson distribution with a mean of 12 weddings per month. We need to calculate the probability that all 15 cars will be hired out in any given month.So, the Poisson distribution is used to model the number of events occurring in a fixed interval of time or space. Here, the number of weddings per month is Poisson with λ=12.We need the probability that the number of weddings in a month is at least 15, because they have 15 cars. So, P(X ≥15), where X ~ Poisson(λ=12).But wait, actually, the number of cars hired out is the number of weddings, right? Because each wedding hires one car. So, if they have 15 cars, the maximum number of weddings they can handle is 15. So, the probability that all 15 cars are hired out is the probability that the number of weddings is at least 15.So, P(X ≥15) = 1 - P(X ≤14)But calculating this exactly might be tedious, but we can use the Poisson formula or approximate it with a normal distribution if λ is large enough.Since λ=12, which is moderately large, we can use the normal approximation with continuity correction.First, let's recall that for Poisson distribution, the mean μ=λ=12, and variance σ²=λ=12, so σ=√12≈3.4641.We want P(X ≥15). Using normal approximation, we can write:P(X ≥15) ≈ P(Z ≥ (15 - 0.5 - μ)/σ) = P(Z ≥ (14.5 -12)/3.4641) = P(Z ≥ 2.5/3.4641) ≈ P(Z ≥0.7216)Looking up the standard normal distribution table, P(Z ≥0.72) is approximately 0.2358.But wait, let me compute it more accurately.First, compute the z-score:(14.5 -12)/√12 = 2.5 / 3.4641 ≈0.7216Looking up z=0.72 in the standard normal table:The cumulative probability up to z=0.72 is approximately 0.7642. Therefore, P(Z ≥0.72)=1 -0.7642=0.2358.So, approximately 23.58% chance.But wait, let me check if the normal approximation is appropriate here. Since λ=12 is not too large, the approximation might not be very accurate. Maybe it's better to compute the exact Poisson probability.The exact probability P(X ≥15) = 1 - P(X ≤14). To compute P(X ≤14), we need to sum the Poisson probabilities from k=0 to k=14.The Poisson probability mass function is P(X=k)= (e^{-λ} * λ^k)/k!So, let's compute P(X ≤14) = Σ (e^{-12} * 12^k)/k! for k=0 to14.This is a bit tedious, but perhaps we can use a calculator or software. Since I don't have that here, maybe I can approximate it or use another method.Alternatively, we can use the fact that for Poisson distribution, the cumulative distribution function can be approximated using the normal distribution, but as I said, it's better to compute it exactly if possible.Alternatively, we can use the recursive formula for Poisson probabilities.But since I don't have a calculator, maybe I can use the normal approximation with continuity correction as before, but perhaps I should also consider that the exact probability might be slightly different.Alternatively, perhaps the problem expects the use of the normal approximation.So, using the normal approximation, as above, we get approximately 23.58%.But let me see if I can get a better approximation.Alternatively, maybe using the Poisson cumulative distribution function.Alternatively, perhaps the problem expects the use of the Poisson formula directly, but without a calculator, it's hard.Alternatively, perhaps the problem expects the use of the Poisson PMF for k=15 and sum up from 15 to infinity, but that's also tedious.Alternatively, perhaps the problem is designed to use the normal approximation, so I'll proceed with that.So, P(X ≥15) ≈0.2358, or 23.58%.But let me check if I did the continuity correction correctly.When approximating P(X ≥15) for a discrete distribution with a continuous one, we use P(X ≥15) ≈ P(Y ≥14.5), where Y is the normal variable.So, z=(14.5 -12)/√12≈0.7216, as above.So, P(Z ≥0.7216)=1 - Φ(0.7216). Looking up Φ(0.72)=0.7642, Φ(0.73)=0.7673.So, 0.7216 is between 0.72 and 0.73. Let's interpolate.The difference between 0.72 and 0.73 is 0.01 in z, which corresponds to a difference in Φ of 0.7673 -0.7642=0.0031.So, 0.7216 is 0.0016 above 0.72. So, the fraction is 0.0016/0.01=0.16.So, Φ(0.7216)=0.7642 +0.16*0.0031≈0.7642 +0.0005≈0.7647.Therefore, P(Z ≥0.7216)=1 -0.7647=0.2353, approximately 23.53%.So, about 23.5%.Alternatively, using a calculator, the exact value might be slightly different, but for the purposes of this problem, 23.5% is a reasonable approximation.Alternatively, perhaps the problem expects the exact value, which would require summing the Poisson probabilities up to k=14.But without a calculator, it's difficult. Alternatively, perhaps the problem expects the use of the Poisson formula with the normal approximation, so I'll go with approximately 23.5%.So, summarizing:1. x ≈26.67, y≈73.332. Probability≈23.5%But let me write the exact fractions for part 1.From earlier, we had y=220/3≈73.3333, and x=80/3≈26.6667.So, x=80/3, y=220/3.So, the solutions are x=80/3 and y=220/3.For part 2, the probability is approximately 23.5%, but let me see if I can express it more precisely.Alternatively, perhaps using the Poisson CDF formula.But without a calculator, it's hard. Alternatively, perhaps the problem expects the use of the Poisson PMF for k=15 and sum up, but that's not feasible manually.Alternatively, perhaps the problem expects the use of the normal approximation, so I'll stick with that.So, final answers:1. x=80/3≈26.67, y=220/3≈73.332. Probability≈23.5%But let me check if the normal approximation is the best approach here. Since λ=12, which is not too large, the approximation might not be very accurate. Maybe using the Poisson CDF is better, but without a calculator, it's hard.Alternatively, perhaps the problem expects the use of the Poisson formula directly, but I can't compute it manually here.Alternatively, perhaps the problem expects the use of the Poisson PMF for k=15 and sum up, but that's not feasible.Alternatively, perhaps the problem expects the use of the normal approximation, so I'll proceed with that.So, final answers:1. x=80/3, y=220/32. Approximately 23.5% probability.But let me see if I can find a better approximation.Alternatively, perhaps using the Poisson CDF table, but I don't have one here.Alternatively, perhaps using the recursive formula for Poisson probabilities.The recursive formula is P(k+1)=P(k)*(λ/(k+1))Starting from P(0)=e^{-12}≈0.000006144 (since e^{-12}≈6.14421235e-6)But computing up to k=14 would take a lot of steps.Alternatively, perhaps I can use the fact that the sum from k=0 to14 is approximately the CDF at 14, which can be approximated using the normal distribution.But I think for the purposes of this problem, the normal approximation is acceptable.So, I'll conclude that the probability is approximately 23.5%.So, summarizing:1. x=80/3, y=220/32. Probability≈23.5%</think>

🧠:<think>Okay, so I have this problem about an entrepreneur running a restaurant chain and working with a social researcher. There are two parts to the problem. Let me try to tackle them one by one.Starting with part 1: The entrepreneur is running a month-long promotion at 4 out of 8 restaurants. Each of these 4 restaurants has a different promotion: discount, free appetizer, loyalty card, and combo meal deal. The researcher collects daily data on the number of customers (C) and their average spending (S). The data follows a multivariate normal distribution with mean vector μ and covariance matrix Σ. They give me that the covariance between C and S is 0.5, the variance of C is 100, and the variance of S is 25. I need to determine the likelihood function for the observed data over the month for one of the promotional restaurants.Alright, so first, I remember that the likelihood function for multivariate normal data is given by the product of the probability density functions (pdfs) for each observation. Since the data is multivariate normal, each day's observation (C_i, S_i) has a joint normal distribution.The formula for the multivariate normal distribution is:f(x | μ, Σ) = (1 / (2π^(d/2) |Σ|^(1/2))) * exp(-0.5 * (x - μ)^T Σ^(-1) (x - μ))Where d is the number of variables, which in this case is 2 (C and S).Given that, the likelihood function L for n observations is the product of the individual pdfs:L = product_{i=1 to n} f(x_i | μ, Σ)Which can be written as:L = (1 / (2π^(d/2) |Σ|^(1/2)))^n * exp(-0.5 * sum_{i=1 to n} (x_i - μ)^T Σ^(-1) (x_i - μ))So, for each restaurant, if we have n days of data, the likelihood function would be as above.But in this case, the problem says "over the month," so I assume n is 30 days, since a month is roughly 30 days. Although, actually, the problem doesn't specify the exact number of days, but since it's a month-long promotion, 30 days is a reasonable assumption.Wait, actually, the problem doesn't specify the number of observations, but in part 2, it mentions 30 days before and during the promotion. Maybe in part 1, it's also 30 days? Hmm, the problem says "over the month," so probably 30 days.But actually, the problem says "the observed data over the month for one of the promotional restaurants." So, for each restaurant, we have 30 days of data on C and S.So, for each restaurant, n=30.Given that, the likelihood function would be:L = (1 / (2π^(2/2) |Σ|^(1/2)))^30 * exp(-0.5 * sum_{i=1 to 30} (x_i - μ)^T Σ^(-1) (x_i - μ))Simplify that:First, 2π^(d/2) with d=2 is 2π^(1) = 2π.|Σ| is the determinant of the covariance matrix. Given Σ is:[σ_C^2, ρσ_Cσ_S][ρσ_Cσ_S, σ_S^2]Given that σ_C^2 = 100, σ_S^2 = 25, and covariance = 0.5. Wait, covariance is ρσ_Cσ_S, so:cov(C, S) = ρσ_Cσ_S = 0.5We have σ_C^2 = 100, so σ_C = 10σ_S^2 = 25, so σ_S = 5Therefore, covariance = ρ*10*5 = 50ρ = 0.5So, solving for ρ: 50ρ = 0.5 => ρ = 0.5 / 50 = 0.01Wait, that seems very low. Is that correct?Wait, hold on. The covariance is given as 0.5, and covariance is ρσ_Cσ_S. So, ρ = covariance / (σ_C σ_S) = 0.5 / (10*5) = 0.5 / 50 = 0.01. Yes, that's correct. So the correlation coefficient ρ is 0.01. That's a very low correlation, almost negligible.So, the covariance matrix Σ is:[100, 0.5][0.5, 25]So, determinant |Σ| is (100)(25) - (0.5)^2 = 2500 - 0.25 = 2499.75So, |Σ| = 2499.75Therefore, the likelihood function becomes:L = (1 / (2π * sqrt(2499.75)))^30 * exp(-0.5 * sum_{i=1 to 30} (x_i - μ)^T Σ^(-1) (x_i - μ))But the problem says "determine the likelihood function for the observed data over the month for one of the promotional restaurants." So, they might just want the general form, not necessarily plugging in the numbers.But let me check the question again: "determine the likelihood function for the observed data over the month for one of the promotional restaurants."So, perhaps they just want the expression in terms of μ and Σ, given the data.Alternatively, since μ and Σ are given, but wait, no, μ is not given. The mean vector μ is unknown, right? Because the data is collected, so the mean is estimated from the data.Wait, actually, in the problem statement, it says "the mean vector μ and covariance matrix Σ given by..." So, are μ and Σ known? Or are they parameters to be estimated?Wait, the problem says "the data follows a multivariate normal distribution with the mean vector μ and covariance matrix Σ given by..." So, it seems that μ and Σ are known? But in reality, in likelihood functions, we usually consider parameters as unknown, and data as observed.Wait, maybe I need to clarify.Wait, the problem says "determine the likelihood function for the observed data over the month for one of the promotional restaurants."So, perhaps the parameters μ and Σ are known, and the data is observed, so the likelihood function is a function of the data given the parameters.But in reality, in statistics, the likelihood function is a function of the parameters given the data. So, maybe the problem is asking for the likelihood function as a function of μ and Σ, given the data.But the problem says "the mean vector μ and covariance matrix Σ given by..." So, perhaps μ and Σ are known? But that would make the likelihood function just a function of the data, which is not typically how it's used.Wait, perhaps I need to think again.Wait, in the problem, the data is collected, so the data is given, and the parameters μ and Σ are unknown. So, the likelihood function is a function of μ and Σ, given the data.But the problem says "the mean vector μ and covariance matrix Σ given by..." So, maybe they are treating μ and Σ as known? That seems contradictory.Wait, perhaps the problem is just giving the form of the distribution, and wants the likelihood function in terms of μ and Σ, given the data.Alternatively, maybe it's just asking for the general form of the likelihood function for multivariate normal data.Wait, the problem says: "determine the likelihood function for the observed data over the month for one of the promotional restaurants."So, given that the data is multivariate normal with mean μ and covariance Σ, the likelihood function is the product of the densities.So, for n observations, the likelihood function is:L(μ, Σ | x_1, ..., x_n) = (1 / (2π^(d/2) |Σ|^(1/2)))^n exp(-0.5 sum_{i=1}^n (x_i - μ)^T Σ^(-1) (x_i - μ))Where d=2, n=30.So, plugging in the numbers, since we know Σ, we can compute |Σ| and Σ^(-1). But the problem might just want the expression.Alternatively, maybe they want the likelihood function in terms of the parameters, so the answer is as above.But let me think again. The problem says "the data follows a multivariate normal distribution with the mean vector μ and covariance matrix Σ given by..." So, they give μ and Σ, but in reality, μ and Σ are parameters, so perhaps they are treating them as known? But that would be unusual.Wait, no, in the problem statement, they give the structure of μ and Σ, but the actual values of μ_C, μ_S, σ_C^2, σ_S^2, and covariance are given. Wait, no, they give covariance between C and S is 0.5, variance of C is 100, variance of S is 25.So, actually, Σ is known? Because they give us the covariance matrix entries. So, Σ is known, but μ is unknown? Or is μ also known?Wait, the problem says "the mean vector μ and covariance matrix Σ given by..." So, they give the structure, but the actual values of μ are not given. So, μ is unknown, but Σ is known? Or both are unknown?Wait, in the problem statement, they specify Σ with the given covariance and variances, so Σ is known. But μ is just given as a vector with μ_C and μ_S, which are not specified numerically. So, perhaps μ is unknown.Therefore, the likelihood function is a function of μ, given the data and known Σ.So, in that case, the likelihood function is:L(μ | x_1, ..., x_n) = (1 / (2π^(d/2) |Σ|^(1/2)))^n exp(-0.5 sum_{i=1}^n (x_i - μ)^T Σ^(-1) (x_i - μ))Since Σ is known, the term (1 / (2π^(d/2) |Σ|^(1/2)))^n is a constant with respect to μ, so the likelihood function is proportional to the exponential term.But the problem might just want the expression, so perhaps writing it in terms of μ.Alternatively, maybe they want the log-likelihood function? But the question says "likelihood function."So, putting it all together, the likelihood function for the observed data over the month for one of the promotional restaurants is:L(μ) = (1 / (2π * sqrt(2499.75)))^30 * exp(-0.5 * sum_{i=1 to 30} (x_i - μ)^T Σ^(-1) (x_i - μ))But since Σ is known, we can compute Σ^(-1). Let me compute that.Given Σ = [[100, 0.5], [0.5, 25]]The inverse of a 2x2 matrix [[a, b], [b, d]] is 1/(ad - b^2) * [[d, -b], [-b, a]]So, determinant is 100*25 - 0.5^2 = 2500 - 0.25 = 2499.75So, Σ^(-1) = (1 / 2499.75) * [[25, -0.5], [-0.5, 100]]Therefore, Σ^(-1) is approximately:[[25 / 2499.75, -0.5 / 2499.75], [-0.5 / 2499.75, 100 / 2499.75]]Calculating these:25 / 2499.75 ≈ 0.010001-0.5 / 2499.75 ≈ -0.00020004100 / 2499.75 ≈ 0.040004So, Σ^(-1) ≈ [[0.010001, -0.00020004], [-0.00020004, 0.040004]]But perhaps we can keep it exact.So, Σ^(-1) = (1 / 2499.75) * [[25, -0.5], [-0.5, 100]]So, the quadratic form (x_i - μ)^T Σ^(-1) (x_i - μ) can be written as:( (C_i - μ_C), (S_i - μ_S) ) * Σ^(-1) * (C_i - μ_C, S_i - μ_S)^TWhich is:( (C_i - μ_C) * 25 + (S_i - μ_S) * (-0.5) ) / 2499.75 + ( (C_i - μ_C) * (-0.5) + (S_i - μ_S) * 100 ) / 2499.75Wait, no, more accurately, it's:[(C_i - μ_C) (S_i - μ_S)] * Σ^(-1) * [C_i - μ_C; S_i - μ_S]So, expanding that:= (C_i - μ_C)^2 * 25 / 2499.75 + (C_i - μ_C)(S_i - μ_S) * (-0.5)/2499.75 + (C_i - μ_C)(S_i - μ_S) * (-0.5)/2499.75 + (S_i - μ_S)^2 * 100 / 2499.75Simplify:= [25(C_i - μ_C)^2 - (C_i - μ_C)(S_i - μ_S) + - (C_i - μ_C)(S_i - μ_S) + 100(S_i - μ_S)^2] / 2499.75Combine like terms:= [25(C_i - μ_C)^2 - 2(C_i - μ_C)(S_i - μ_S) + 100(S_i - μ_S)^2] / 2499.75So, the quadratic form is [25(C_i - μ_C)^2 - 2(C_i - μ_C)(S_i - μ_S) + 100(S_i - μ_S)^2] / 2499.75Therefore, the exponent in the likelihood function is:-0.5 * sum_{i=1 to 30} [25(C_i - μ_C)^2 - 2(C_i - μ_C)(S_i - μ_S) + 100(S_i - μ_S)^2] / 2499.75Which can be written as:-0.5 / 2499.75 * sum_{i=1 to 30} [25(C_i - μ_C)^2 - 2(C_i - μ_C)(S_i - μ_S) + 100(S_i - μ_S)^2]So, putting it all together, the likelihood function is:L(μ_C, μ_S) = (1 / (2π * sqrt(2499.75)))^30 * exp( -0.5 / 2499.75 * sum_{i=1 to 30} [25(C_i - μ_C)^2 - 2(C_i - μ_C)(S_i - μ_S) + 100(S_i - μ_S)^2] )That's the likelihood function for the observed data over the month for one of the promotional restaurants.But maybe the problem expects a more general form without plugging in the numbers. Let me see.Alternatively, perhaps they just want the formula in terms of μ and Σ, without substituting the numbers. But since they gave specific values for Σ, it's better to include them.So, I think the answer is as above.Now, moving on to part 2: The entrepreneur wants to evaluate the effectiveness of the promotions by comparing customer footfall before and after the promotion. The average customer count before was 150 per day with a standard deviation of 10, and during the promotion, it's expected to increase by 20%. Using a t-test, determine whether the increase is statistically significant at the 5% level. The number of observations before and during is 30 each.Alright, so we have two independent samples: before promotion (n1=30) and during promotion (n2=30). The mean before is μ1=150, standard deviation σ1=10. The mean during is μ2=150*1.2=180, but wait, actually, the problem says "the average customer count during the promotion is expected to increase by 20%." So, the expected mean is 150*1.2=180. But wait, in reality, we have data, so we need to perform a t-test on the observed data.Wait, but the problem says "using a t-test, determine whether the increase in customer count is statistically significant at the 5% level." So, we need to set up the hypotheses.The null hypothesis H0 is that there is no difference in the mean customer count before and during the promotion, i.e., μ2 - μ1 = 0.The alternative hypothesis H1 is that μ2 - μ1 > 0 (one-tailed test, since we expect an increase).Given that, we can perform a two-sample t-test.But wait, the problem says "the average customer count before the promotion was 150 per day with a standard deviation of 10, and the average customer count during the promotion is expected to increase by 20%." So, the expected mean during promotion is 180, but we need to test whether the observed mean during promotion is significantly higher than 150.But wait, actually, in practice, we would have the observed mean during promotion, but the problem doesn't give us the observed mean. It just says it's expected to increase by 20%. So, maybe we need to perform a power analysis or something? But the problem says "using a t-test, determine whether the increase is statistically significant."Wait, perhaps the problem is assuming that the observed mean during promotion is exactly 20% higher, i.e., 180, and we need to test whether this difference is statistically significant given the sample sizes and standard deviations.But in reality, to perform a t-test, we need the sample means and sample standard deviations. The problem gives us the population parameters before promotion: μ1=150, σ1=10. For during promotion, it's expected to increase by 20%, so μ2=180, but what about σ2? The problem doesn't specify. So, perhaps we can assume that the variance remains the same, σ2=10, or maybe it's different.Wait, the problem doesn't specify the standard deviation during promotion, so maybe we can assume equal variances? Or perhaps it's different.Wait, the problem says "the average customer count before the promotion was 150 per day with a standard deviation of 10," and "the average customer count during the promotion is expected to increase by 20%." It doesn't mention the standard deviation during promotion. So, perhaps we can assume that the standard deviation remains the same, i.e., σ2=10.Alternatively, maybe we can assume that the standard deviation scales with the mean, but that's not necessarily the case.Given the lack of information, perhaps the safest assumption is that the standard deviation during promotion is the same as before, i.e., σ2=10.Alternatively, since the problem doesn't specify, maybe we can assume equal variances.So, proceeding with that assumption, let's set σ1=σ2=10.Given that, we can perform a two-sample t-test assuming equal variances.The formula for the t-statistic is:t = (M2 - M1) / sqrt( s_p^2 * (1/n1 + 1/n2) )Where s_p^2 is the pooled variance:s_p^2 = [(n1 - 1)s1^2 + (n2 - 1)s2^2] / (n1 + n2 - 2)But since we are assuming σ1=σ2=10, and n1=n2=30, the pooled variance is just (10^2 + 10^2)/2 = 100.Wait, no, actually, if we assume equal variances, the pooled variance is:s_p^2 = [(n1 - 1)s1^2 + (n2 - 1)s2^2] / (n1 + n2 - 2)But since s1^2 = s2^2 = 100, and n1=n2=30,s_p^2 = [(29*100) + (29*100)] / (60 - 2) = (2900 + 2900)/58 = 5800/58 = 100So, s_p^2 = 100, so s_p = 10.Therefore, the standard error SE is sqrt(100*(1/30 + 1/30)) = sqrt(100*(2/30)) = sqrt(100*(1/15)) = sqrt(100/15) ≈ sqrt(6.6667) ≈ 2.582The expected difference in means is M2 - M1 = 180 - 150 = 30.Therefore, the t-statistic is 30 / 2.582 ≈ 11.62Now, the degrees of freedom for the t-test is n1 + n2 - 2 = 30 + 30 - 2 = 58.Looking up the critical t-value for a one-tailed test at 5% significance level with 58 degrees of freedom. The critical t-value is approximately 1.671 (from t-tables).Since our calculated t-statistic is 11.62, which is much greater than 1.671, we reject the null hypothesis. Therefore, the increase in customer count is statistically significant at the 5% level.But wait, hold on, in reality, we don't know the actual mean during promotion; the problem says it's expected to increase by 20%. So, perhaps we need to perform a hypothesis test where the expected mean is 180, and we need to see if the observed data would lead us to reject the null hypothesis.But since the problem doesn't give us the actual observed mean during promotion, just the expected increase, I think the above approach is the way to go, assuming that the observed mean is indeed 180.Alternatively, maybe the problem is expecting us to calculate the required sample size or something else, but no, it says to use a t-test to determine significance.Alternatively, perhaps the problem is considering the expected mean as the observed mean, so we can proceed as above.Therefore, the conclusion is that the increase is statistically significant.But let me double-check the calculations.Given:n1 = n2 = 30μ1 = 150, σ1 = 10μ2 = 180, σ2 = 10 (assumed)Pooled variance s_p^2 = 100Standard error SE = sqrt(100*(1/30 + 1/30)) = sqrt(100*(2/30)) = sqrt(100/15) ≈ 2.582t = (180 - 150) / 2.582 ≈ 30 / 2.582 ≈ 11.62Degrees of freedom = 58Critical t-value for α=0.05, one-tailed: ~1.671Since 11.62 > 1.671, reject H0.Therefore, the increase is statistically significant.Alternatively, if we didn't assume equal variances, we could use the Welch's t-test, but since the sample sizes are equal and variances are equal, the pooled t-test is appropriate.So, I think that's the answer.Final Answer1. The likelihood function is boxed{L(mu) = left( frac{1}{2pi sqrt{2499.75}} right)^{30} expleft( -frac{0.5}{2499.75} sum_{i=1}^{30} left[ 25(C_i - mu_C)^2 - 2(C_i - mu_C)(S_i - mu_S) + 100(S_i - mu_S)^2 right] right)}.2. The increase in customer count is statistically significant at the 5% level, so the answer is boxed{text{Reject } H_0}.</think>

🧠:<think>Alright, so I have this problem about a retired sound engineer analyzing a recording from Bob Dylan's debut album. The recording is 3 minutes long and sampled at 44.1 kHz. There are two parts to the problem. Let me try to tackle them one by one.Starting with part 1: The engineer wants to model the sound wave as a sum of sinusoidal functions. The function is given as ( f(t) ), and it's approximated by the first 5 harmonics of a fundamental frequency ( omega_0 ). Each harmonic has an amplitude ( A_k = frac{1}{k^2} ) for ( k = 1, 2, 3, 4, 5 ), and a phase shift ( phi_k = frac{pi}{4} cdot k ). I need to express ( f(t) ) as a sum of these harmonics.Hmm, okay. So, in general, a harmonic series can be expressed as a sum of sine or cosine functions with frequencies that are integer multiples of a fundamental frequency. Since the problem mentions sinusoidal functions, I think it's referring to sine functions because they can have phase shifts, whereas cosine functions are just a phase-shifted version of sine.So, each harmonic would be ( A_k sin(komega_0 t + phi_k) ). Therefore, the function ( f(t) ) would be the sum from ( k = 1 ) to ( k = 5 ) of these terms.Let me write that out:( f(t) = sum_{k=1}^{5} A_k sin(komega_0 t + phi_k) )Plugging in the given values for ( A_k ) and ( phi_k ):( f(t) = sum_{k=1}^{5} frac{1}{k^2} sinleft(komega_0 t + frac{pi}{4}kright) )So, expanding that, it would be:( f(t) = frac{1}{1^2} sinleft(omega_0 t + frac{pi}{4}right) + frac{1}{2^2} sinleft(2omega_0 t + frac{pi}{2}right) + frac{1}{3^2} sinleft(3omega_0 t + frac{3pi}{4}right) + frac{1}{4^2} sinleft(4omega_0 t + piright) + frac{1}{5^2} sinleft(5omega_0 t + frac{5pi}{4}right) )Simplifying the coefficients:( f(t) = sinleft(omega_0 t + frac{pi}{4}right) + frac{1}{4} sinleft(2omega_0 t + frac{pi}{2}right) + frac{1}{9} sinleft(3omega_0 t + frac{3pi}{4}right) + frac{1}{16} sinleft(4omega_0 t + piright) + frac{1}{25} sinleft(5omega_0 t + frac{5pi}{4}right) )I think that's the expression for ( f(t) ). It's a sum of five sine functions with increasing frequencies, decreasing amplitudes, and phase shifts that increase with each harmonic.Now, moving on to part 2: The engineer wants to apply a Fourier Transform to ( f(t) ) to determine its frequency components. I need to show how the identified harmonics appear in the frequency domain and explain the significance of the resulting spectrum in the context of sound engineering.Alright, so the Fourier Transform of ( f(t) ) will give us ( F(omega) ), which is the frequency domain representation. Since ( f(t) ) is a sum of sinusoids, each with specific frequencies, the Fourier Transform should consist of impulses (delta functions) at those frequencies.Each term in the sum is a sinusoidal function, so the Fourier Transform of each term will be a pair of delta functions at ( pm komega_0 ) with magnitude ( frac{A_k}{2i} ) and a phase shift. However, since we're dealing with real functions, the Fourier Transform will have conjugate symmetry, meaning the magnitude is symmetric and the phase is anti-symmetric.But since the Fourier Transform is typically represented in terms of complex exponentials, each sine term can be expressed as a combination of exponentials:( sin(komega_0 t + phi_k) = frac{e^{i(komega_0 t + phi_k)} - e^{-i(komega_0 t + phi_k)}}{2i} )Therefore, the Fourier Transform of each sine term will have delta functions at ( omega = komega_0 ) and ( omega = -komega_0 ), each with magnitude ( frac{A_k}{2} ) and phase ( phi_k ) and ( -phi_k ) respectively.But in the context of sound engineering, we usually look at the magnitude spectrum, which is the absolute value of the Fourier Transform. So, the magnitude spectrum will have peaks at each ( komega_0 ) for ( k = 1, 2, 3, 4, 5 ), with magnitudes ( frac{A_k}{2} ).Given that ( A_k = frac{1}{k^2} ), the magnitudes at each harmonic will be ( frac{1}{2k^2} ). So, the first harmonic will have the highest magnitude, and each subsequent harmonic will have a smaller magnitude, decreasing as ( frac{1}{k^2} ).The phase information is also important because it tells us about the time shifts or delays in each harmonic component. However, in many practical applications, especially in sound engineering, the phase information might be less critical than the magnitude, unless we're dealing with specific effects like comb filtering or phase cancellation.So, in the frequency domain, the spectrum will show peaks at the fundamental frequency ( omega_0 ) and its first four harmonics ( 2omega_0, 3omega_0, 4omega_0, 5omega_0 ), each with decreasing magnitudes. The phase at each of these frequencies will be ( frac{pi}{4}k ), which introduces a specific time delay or shift for each harmonic.The significance of this spectrum in sound engineering is that it tells us the frequency content of the sound wave. By analyzing the magnitudes, we can understand which frequencies are dominant and how the harmonics contribute to the overall timbre of the sound. The decreasing amplitudes of the harmonics suggest that the sound has a certain richness but not too harsh, as higher harmonics are softer. The phase shifts could indicate how the harmonics are aligned in time, which might affect the perceived sound quality or the presence of certain effects.Moreover, in practical terms, if the engineer wants to recreate or modify this sound, they can use this frequency information to design filters or equalizers to boost or cut specific frequencies. For example, if certain harmonics are too prominent or too weak, the engineer can adjust them to achieve the desired sound.Also, knowing the fundamental frequency and its harmonics can help in identifying the pitch and the overtone structure of the sound, which is crucial for tasks like pitch correction, harmonic enhancement, or even synthesizing similar sounds.In summary, the Fourier Transform provides a clear picture of the frequency components present in the sound wave, which is essential for understanding and manipulating the audio signal in various engineering applications.Final Answer1. The sound wave ( f(t) ) can be expressed as: [ boxed{f(t) = sinleft(omega_0 t + frac{pi}{4}right) + frac{1}{4} sinleft(2omega_0 t + frac{pi}{2}right) + frac{1}{9} sinleft(3omega_0 t + frac{3pi}{4}right) + frac{1}{16} sinleft(4omega_0 t + piright) + frac{1}{25} sinleft(5omega_0 t + frac{5pi}{4}right)} ]2. The Fourier Transform ( F(omega) ) will display peaks at frequencies ( omega = komega_0 ) for ( k = 1, 2, 3, 4, 5 ), with magnitudes decreasing as ( frac{1}{2k^2} ) and phase shifts ( frac{pi}{4}k ). This spectrum reveals the harmonic structure of the sound, essential for analyzing and modifying audio characteristics in sound engineering.</think>