🧠:<think>Okay, so I have this problem where an internal auditor is looking at a company's data protection compliance. The company handles a million records each day, and each record has a 0.05% chance of being involved in a data breach because of non-compliance. Each breach costs 10,000. First, I need to calculate the expected cost of data breaches per day. Hmm, expected value. I remember that expected value is like the average outcome if we were to repeat an experiment many times. In this case, each record has a probability of 0.05% of being breached, and each breach costs 10,000. So, maybe I can model this using the expected value formula. For each record, the expected cost would be the probability of a breach multiplied by the cost. That would be 0.05% times 10,000. Let me write that down: Expected cost per record = 0.0005 * 10,000 = 5.But wait, that's per record. Since there are a million records, I need to multiply this by 1,000,000. So, total expected cost per day would be 1,000,000 * 5 = 5,000,000. Wait, that seems really high. Is that right? Let me double-check. Each record has a 0.05% chance, so 0.0005 probability. Multiply by 10,000 gives 5 per record. Multiply by a million records, yeah, that's 5,000,000. Hmm, that does seem high, but maybe it's correct because even though the probability per record is low, the number of records is enormous. So, the expected number of breaches would be 1,000,000 * 0.0005 = 500 breaches per day. Then, 500 breaches times 10,000 each is indeed 5,000,000. Okay, that seems consistent.So, for part 1, the expected cost is 5,000,000 per day.Now, moving on to part 2. The auditor wants to ensure that the probability of at least one data breach occurring in a day doesn't exceed 25%. So, I need to calculate the probability of at least one breach and see if it's less than or equal to 25%.Hmm, how do I calculate the probability of at least one breach? Well, it's often easier to calculate the probability of the complementary event, which is the probability of no breaches occurring, and then subtract that from 1.So, the probability of no breach for a single record is 1 - 0.0005 = 0.9995. Since the records are independent, the probability that none of the million records are breached is (0.9995)^1,000,000.Calculating that might be tricky because it's a very small number raised to a large exponent. I remember that for small probabilities, we can approximate (1 - p)^n ≈ e^(-np), where e is the base of the natural logarithm. So, let's use that approximation.Here, p is 0.0005 and n is 1,000,000. So, np = 1,000,000 * 0.0005 = 500. Therefore, the probability of no breaches is approximately e^(-500). Wait, e^(-500) is an extremely small number, practically zero. So, the probability of at least one breach is 1 - e^(-500), which is almost 1. That means the probability is almost 100%, which is way higher than the 25% threshold.But wait, that seems contradictory to the expected number of breaches. Earlier, we found that the expected number of breaches is 500 per day, which is a lot. So, the probability of at least one breach is extremely high, almost certain. Therefore, the current compliance level does not meet the requirement because the probability is way above 25%.But let me think again. Maybe I made a mistake in the approximation. Let me verify.The exact probability of no breaches is (0.9995)^1,000,000. Taking the natural logarithm, ln(0.9995) is approximately -0.000500125. So, ln((0.9995)^1,000,000) = 1,000,000 * (-0.000500125) = -500.125. Therefore, (0.9995)^1,000,000 = e^(-500.125), which is indeed approximately e^(-500). So, my approximation was correct.Therefore, the probability of at least one breach is 1 - e^(-500), which is practically 1. So, the probability is almost 100%, which is way above 25%. Therefore, the current compliance level does not meet the requirement.Alternatively, if the probability of at least one breach is 1 - e^(-λ), where λ is the expected number of breaches, which is 500. So, 1 - e^(-500) is practically 1, as e^(-500) is negligible.So, conclusion: The probability of at least one breach is almost 100%, which exceeds the 25% threshold. Therefore, the current compliance level does not meet the requirement.Wait, but maybe I should present it more formally. Let me structure it step by step.First, define the probability of a breach for a single record: p = 0.0005.Number of records: n = 1,000,000.The probability of no breach for one record: q = 1 - p = 0.9995.Probability of no breaches in all records: q^n = (0.9995)^1,000,000.As n is large and p is small, we can approximate this using Poisson approximation. The Poisson distribution with λ = n*p = 500.The probability of zero events in Poisson is e^(-λ) = e^(-500), which is practically zero.Therefore, the probability of at least one breach is 1 - e^(-500) ≈ 1.Since 1 > 0.25, the probability exceeds the threshold.So, yes, the current compliance level does not meet the requirement.Alternatively, if I didn't use the approximation, but tried to compute (0.9995)^1,000,000 directly, it's computationally intensive, but we can use logarithms as I did earlier.ln(0.9995) ≈ -0.000500125, so multiplying by 1,000,000 gives -500.125, exponentiate gives e^(-500.125) ≈ 1.33 * 10^(-217), which is effectively zero. Therefore, 1 - 1.33 * 10^(-217) ≈ 1.So, regardless of the method, the probability is practically 1, which is way above 25%.Therefore, the company's current compliance level does not meet the auditor's requirement.I think that's solid. Maybe I should also consider if there's another way to model this, but I think the Poisson approximation is appropriate here because n is large and p is small, which are the conditions for using the Poisson approximation to the binomial distribution.So, summarizing:1. Expected cost per day: 5,000,000.2. Probability of at least one breach: ≈100%, which exceeds 25%, so compliance is insufficient.</think>

🧠:<think>Alright, I need to tackle these two problems related to organizing a film exhibition. Let me start by understanding each problem and figuring out how to approach them.Problem 1: Maximizing Historical Accuracy ScoreSo, I have 120 films, each with a historical accuracy score ( H_i ) ranging from 1 to 10. I can select up to 20 films, and I want to maximize the total historical accuracy. This sounds like a classic optimization problem where I need to pick the best subset under a constraint.Hmm, integer linear programming (ILP) is mentioned. I remember ILP involves variables that are integers, and it's used for optimization with linear constraints. So, I need to define variables, an objective function, and constraints.Let me think about the variables. I can define a binary variable ( x_i ) for each film ( i ), where ( x_i = 1 ) if the film is selected, and ( x_i = 0 ) otherwise. That makes sense because each film is either in or out.The objective is to maximize the total historical accuracy. So, the objective function would be the sum of ( H_i times x_i ) for all films ( i ). That way, if a film is selected, its score contributes to the total.Now, the constraints. The main constraint is that I can select at most 20 films. So, the sum of all ( x_i ) should be less than or equal to 20. Also, each ( x_i ) must be either 0 or 1, which is the binary constraint.Putting it all together, the ILP formulation would look something like:Maximize ( sum_{i=1}^{120} H_i x_i )Subject to:( sum_{i=1}^{120} x_i leq 20 )( x_i in {0, 1} ) for all ( i )I think that's the basic structure. Maybe I should write it more formally with proper notation.Problem 2: Maximizing Thematic CoherenceNow, after selecting the films, I need to arrange them in a sequence to maximize the thematic coherence. The coherence score ( C_{ij} ) is given between any two films ( i ) and ( j ), ranging from 1 to 5.This seems like a problem of finding the optimal permutation of the selected films to maximize the sum of consecutive coherence scores. I recall that arranging things to maximize the sum of adjacent values can be modeled as a graph problem, specifically finding a Hamiltonian path with the maximum total weight.In graph theory terms, if I model each film as a node, and the coherence score ( C_{ij} ) as the weight of the edge from node ( i ) to node ( j ), then the problem becomes finding the path that visits each node exactly once (a Hamiltonian path) with the maximum total edge weight.But wait, Hamiltonian path problems are generally NP-hard, which means they can be computationally intensive, especially as the number of nodes increases. Since we're dealing with up to 20 films, that's a manageable number, but still, I need to think about how to model this.Alternatively, maybe I can use the Traveling Salesman Problem (TSP) approach, where the goal is to find the shortest possible route that visits each city exactly once and returns to the origin city. But in this case, we want the longest path instead of the shortest, so it's more like the Traveling Salesman Problem with maximization.But TSP is usually for minimization. So, perhaps I can invert the coherence scores to turn it into a minimization problem, but that might complicate things. Alternatively, I can directly model it as a maximization problem.Let me think about the variables. If I have a set ( S ) of selected films, I need to arrange them in an order ( pi ) where ( pi ) is a permutation of ( S ). The total coherence score would be the sum of ( C_{pi(k), pi(k+1)} ) for ( k ) from 1 to ( |S| - 1 ).To model this with graph theory, I can represent the films as nodes in a complete graph where each edge has a weight ( C_{ij} ). Then, the problem reduces to finding the Hamiltonian path with the maximum total weight.But how do I construct a mathematical model for this? Maybe using integer variables to represent the order. I remember that permutation matrices or using variables to denote the position of each film can be used, but that might get complicated.Alternatively, I can use dynamic programming for the TSP, but since it's a maximization problem, I need to adjust the DP approach accordingly. However, the user asked for a graph theory model, so perhaps I should stick to that.Wait, another approach is to model this as a directed graph where each edge from ( i ) to ( j ) has weight ( C_{ij} ). Then, finding the path that visits each node exactly once with the maximum total weight is exactly what we need.So, in summary, the mathematical model would involve constructing a complete directed graph with nodes representing films and edges weighted by ( C_{ij} ). The goal is to find a Hamiltonian path in this graph with the maximum total weight.But how do I express this formally? Maybe using adjacency matrices or something similar. Alternatively, I can define variables ( y_{ij} ) which are 1 if film ( i ) is immediately followed by film ( j ) in the sequence, and 0 otherwise. Then, the total coherence score is ( sum_{i,j} C_{ij} y_{ij} ).But I need to ensure that each film is included exactly once, except for the start and end. So, the constraints would be:For each film ( i ), the number of outgoing edges ( y_{ij} ) should be 1 (each film is followed by exactly one other film, except the last one). Similarly, the number of incoming edges ( y_{ji} ) should be 1 for each film ( i ), except the first one.But this might get a bit involved. Maybe it's better to stick with the graph theory concept of a Hamiltonian path.Alternatively, since the number of films is up to 20, which is manageable, I can use permutation-based methods or even brute force with some optimizations, but the user wants a mathematical model, not an algorithm.So, perhaps the best way is to model it as a graph where we seek the maximum weight Hamiltonian path.Putting it all together, the model would involve:- Nodes: Selected films- Edges: Directed edges with weights ( C_{ij} )- Objective: Find a path visiting each node exactly once with maximum total edge weight.I think that's the essence of the model.Double-CheckingFor Problem 1, I defined binary variables and set up the ILP correctly. The constraints are straightforward: sum of selected films ≤ 20, and binary variables.For Problem 2, I considered the problem as a graph where nodes are films and edges represent coherence scores. The goal is to find the optimal path through all selected films, which is a Hamiltonian path with maximum total weight.I should make sure that in Problem 2, the model accounts for the sequence, not just the selection. So, it's not just about which films are selected, but also the order in which they are arranged.I think I covered both aspects. The first problem is about selection, the second about ordering. Both are separate but related problems.Final AnswerProblem 1 can be formulated as an integer linear program, and Problem 2 can be modeled using graph theory to find the optimal sequence. Here are the formal formulations:For Problem 1:Maximize ( sum_{i=1}^{120} H_i x_i )Subject to:( sum_{i=1}^{120} x_i leq 20 )( x_i in {0, 1} ) for all ( i )For Problem 2:Construct a complete directed graph where each node represents a selected film, and each directed edge from node ( i ) to node ( j ) has a weight ( C_{ij} ). The goal is to find a Hamiltonian path in this graph that maximizes the sum of the edge weights.So, the final answers are:Problem 1: ILP formulation as above.Problem 2: Graph model as described.But since the user asked for the final answer in boxes, I'll present them accordingly.Final AnswerProblem 1:boxed{begin{aligned}& text{Maximize} & sum_{i=1}^{120} H_i x_i & text{Subject to} & sum_{i=1}^{120} x_i leq 20 & & x_i in {0, 1} quad text{for all } i = 1, 2, ldots, 120end{aligned}}Problem 2:Construct a complete directed graph with nodes representing the selected films and edges weighted by ( C_{ij} ). Find the Hamiltonian path with the maximum total weight.boxed{text{Find a Hamiltonian path in the directed graph with maximum total edge weight } sum_{(i,j) in text{Path}} C_{ij}}</think>

🧠:<think>Alright, so I need to figure out the values of A, ω, and φ for the intensity function I(t) = A sin(ωt + φ) + C. Let's go through the given conditions one by one and see how they can help me find these parameters.First, the baseline intensity level C is given as 50 units. That's straightforward, so I can note that down: C = 50.Next, the peak intensity is 100 units, and the minimum intensity is 0 units. Since the function is a sine wave, the amplitude A should be half the difference between the maximum and minimum values. Let me calculate that. The maximum is 100, and the minimum is 0, so the difference is 100 - 0 = 100. Half of that is 50, so A should be 50. Wait, but the baseline is 50, so if I have a sine wave with amplitude 50, it would go from 0 to 100, which matches the given peak and minimum. So, A = 50.Now, moving on to the angular frequency ω. The problem states that the maximum workout intensity occurs every 10 minutes. That means the period T of the sine function is 10 minutes. The angular frequency ω is related to the period by the formula ω = 2π / T. Plugging in T = 10, we get ω = 2π / 10 = π/5. So, ω = π/5 per minute.Next, the phase shift φ. The condition given is that at t = 0, the intensity I(0) is at its baseline level. Let's plug t = 0 into the function: I(0) = A sin(ω*0 + φ) + C = A sin(φ) + C. Since I(0) is the baseline, which is 50, we have 50 = 50 sin(φ) + 50. Subtracting 50 from both sides gives 0 = 50 sin(φ). Therefore, sin(φ) = 0. The solutions to this are φ = 0, π, 2π, etc. But we need to consider the behavior of the sine function. If φ = 0, then at t = 0, sin(0) = 0, so I(0) = 50, which is correct. However, we also need to consider whether the function is increasing or decreasing at t = 0. Since the maximum occurs every 10 minutes, and at t = 0 we're at the baseline, which is the midpoint between the peak and minimum. So, the function should be increasing at t = 0 to reach the maximum at t = 10. The derivative of I(t) is I’(t) = Aω cos(ωt + φ). At t = 0, I’(0) = Aω cos(φ). Since A and ω are positive, cos(φ) should be positive. If φ = 0, cos(0) = 1, which is positive, so that's good. If φ = π, cos(π) = -1, which would mean the function is decreasing at t = 0, which isn't what we want because we need it to increase from the baseline to the peak. Therefore, φ = 0 is the correct phase shift.So, summarizing the first sub-problem:- A = 50- ω = π/5- φ = 0Now, moving on to the second sub-problem: calculating the total workout intensity over a 30-minute session by integrating I(t) from t = 0 to t = 30.The function is I(t) = 50 sin((π/5)t) + 50. To find the total intensity, we need to compute the integral of I(t) from 0 to 30.Let me set up the integral:Total Intensity = ∫₀³⁰ [50 sin((π/5)t) + 50] dtWe can split this integral into two parts:= ∫₀³⁰ 50 sin((π/5)t) dt + ∫₀³⁰ 50 dtLet's compute each integral separately.First integral: ∫ 50 sin((π/5)t) dtLet me make a substitution to solve this. Let u = (π/5)t, so du/dt = π/5, which means dt = (5/π) du.So, the integral becomes:50 ∫ sin(u) * (5/π) du = (50 * 5 / π) ∫ sin(u) du = (250 / π) (-cos(u)) + C = -250/π cos(u) + CSubstituting back u = (π/5)t:= -250/π cos((π/5)t) + CNow, evaluating from 0 to 30:[-250/π cos((π/5)*30)] - [-250/π cos(0)]Simplify:= [-250/π cos(6π)] - [-250/π * 1]Since cos(6π) = cos(0) = 1, because cosine has a period of 2π, so 6π is 3 full periods.So:= [-250/π * 1] - [-250/π * 1] = (-250/π) - (-250/π) = 0Interesting, the first integral evaluates to 0. That makes sense because the sine function is symmetric over its period, and over an integer number of periods, the area cancels out. Since 30 minutes is 3 periods (each period is 10 minutes), the integral over 3 periods is zero.Now, the second integral: ∫₀³⁰ 50 dtThis is straightforward:= 50t evaluated from 0 to 30 = 50*30 - 50*0 = 1500 - 0 = 1500Therefore, the total intensity is 0 + 1500 = 1500 units.Wait, that seems too straightforward. Let me double-check.Yes, the integral of the sine part over 3 periods is zero because it's symmetric. The integral of the constant part is just the constant times the interval length, which is 50*30 = 1500. So, the total intensity is 1500 units.So, summarizing the second sub-problem: the total workout intensity over 30 minutes is 1500 units.I think that's it. I don't see any mistakes in the calculations. The key was recognizing that the sine component integrates to zero over an integer number of periods, leaving only the constant term's integral.</think>

🧠:<think>First, I need to calculate the total length of the stone walkway around the regular dodecagon-shaped garden. Since a regular dodecagon has 12 equal sides, each measuring 5 meters, the total perimeter can be found by multiplying the number of sides by the length of one side.Next, to determine the area of the regular hexagon inscribed within the dodecagon, I'll start by finding the radius of the dodecagon. The radius can be calculated using the formula for the side length of a regular polygon, which relates the side length to the radius and the number of sides.Once I have the radius of the dodecagon, it will also be the radius of the inscribed hexagon. Using the radius, I can calculate the area of the regular hexagon with the appropriate formula, which involves the radius and the number of sides of the hexagon.By following these steps, I can provide the homeowner with the necessary measurements for both the walkway and the new gazebo.</think>