🧠:<think>Okay, so I have two problems to solve here. Let me start with the first one about seating arrangements.Problem 1: Seating ArrangementMs. Clara has 12 women educators, 3 each from four different subjects: Math, Science, Literature, and Arts. She wants to arrange them in a circle such that no two women who teach the same subject sit next to each other. I need to find the number of distinct ways to arrange them.Hmm, circular arrangements can be tricky because rotations are considered the same. So, for linear arrangements, we have n! ways, but for circular, it's (n-1)! because rotating the entire circle doesn't create a new arrangement. But here, we have additional constraints about not having the same subject next to each other.This seems similar to arranging people around a table with restrictions. Since there are four subjects, each with three women, we need to ensure that no two adjacent seats have the same subject.I remember something called the "round table problem" or "circular permutations with restrictions." Maybe I can use the principle of inclusion-exclusion or some combinatorial methods here.Wait, another thought: since each subject has exactly three women, and there are four subjects, it's like a problem of arranging four different types of objects around a circle with no two identical objects adjacent. But in this case, each type has three objects.This might be a problem related to graph coloring or derangements, but I'm not sure. Let me think step by step.First, in a circular arrangement, fixing one person's position can help eliminate rotational symmetry. So, let's fix one person's seat. Let's say we fix a Math teacher at a specific seat. Then, we have 11 seats left to arrange the remaining 11 women.But we have to ensure that no two same subjects are adjacent. So, after fixing a Math teacher, the next seat can't be Math. It can be Science, Literature, or Arts. Similarly, the seat after that can't be the same as the previous one, and so on.This seems similar to arranging colored beads around a necklace with no two adjacent beads of the same color. But in our case, each color has exactly three beads, and we have four colors.Wait, actually, this is a problem of counting the number of circular arrangements of objects with repetition, where no two identical objects are adjacent. The formula for this is a bit complicated, but maybe I can use the principle of inclusion-exclusion.Alternatively, I recall that for circular arrangements with no two identical adjacent objects, the formula is similar to linear arrangements but adjusted for circularity.But in our case, it's more specific because each subject has exactly three women. So, it's not just any arrangement, but one where each subject is equally represented.Wait, maybe I can model this as a permutation problem with constraints. Let's think about arranging the 12 seats in a circle, with 3 Ms, 3 Sc, 3 L, and 3 A (for Math, Science, Literature, Arts). We need to arrange them so that no two Ms are adjacent, no two Scs, etc.But since it's a circle, the first and last seats are also adjacent, so we have to ensure that the first and last seats are not the same subject either.This is similar to arranging colored beads in a circle with no two adjacent beads of the same color, but with exactly three beads of each color.I think this is a problem that can be approached using the concept of derangements or using the principle of inclusion-exclusion.Alternatively, maybe I can use the formula for the number of circular arrangements with no two identical adjacent objects, which is:For n objects with n1, n2, ..., nk of each type, the number is:[frac{(n-1)!}{n1!n2!...nk!} - text{adjustments for circularity and adjacency}]But I'm not sure. Maybe I need to use the concept of graph theory here, considering each subject as a node and arranging them in a circle such that no two same nodes are adjacent.Wait, another approach: Since each subject has three women, and there are four subjects, the arrangement must alternate between different subjects. But with four subjects, it's possible to have a repeating pattern every four seats, but since we have three of each, it's not straightforward.Wait, maybe it's impossible? Let me check.If we have four subjects, each with three women, arranging them in a circle without two same subjects adjacent. Each subject must be separated by at least one different subject.But in a circle, each subject will have two neighbors. So, for each subject, the number of seats between two same subjects must be at least one.But since each subject has three women, the circle will have 12 seats. So, the arrangement must have each subject appearing every four seats? Wait, 12 divided by 3 is 4, so each subject would be spaced every four seats. But since it's a circle, that would mean each subject is equally spaced.Wait, if we fix one Math teacher, then the next Math teacher would be four seats away, and the third Math teacher would be eight seats away. But since it's a circle, eight seats away is equivalent to four seats in the other direction. Hmm, but that would mean that the Math teachers are equally spaced, but the other subjects would have to fill in the gaps.But then, if we have three Math teachers spaced every four seats, the remaining seats would have to be filled with Science, Literature, and Arts teachers, each also spaced every four seats. But since each of those also has three teachers, they would also need to be spaced every four seats.But wait, 12 seats divided by four seats per subject would mean each subject is placed every four seats. But that would require that each subject is placed exactly three times, which is the case here.So, this seems like a possible arrangement. But how many such arrangements are there?Wait, maybe this is a problem of counting the number of ways to arrange the subjects around the circle such that each subject is equally spaced, and then permuting the teachers within each subject.But first, let's think about arranging the subjects. Since it's a circle, the number of distinct arrangements is (4-1)! = 6, because for circular permutations of four subjects, it's (4-1)!.But wait, no, because each subject has three identical teachers, but actually, the teachers are distinct individuals, so maybe we need to consider permutations differently.Wait, no, the subjects are four types, each with three identical objects? No, the teachers are distinct, so each subject is represented by three distinct individuals.Wait, so perhaps first, we need to arrange the subjects around the circle, ensuring that no two same subjects are adjacent, and then assign the specific teachers to each seat.But arranging the subjects: since each subject must appear three times, and no two same subjects can be adjacent, is that even possible?Wait, in a circle with 12 seats, each subject appears three times. So, the arrangement must be such that between any two same subjects, there are at least two different subjects.But with four subjects, each appearing three times, is that possible?Wait, let me think about the spacing. If each subject appears three times, the minimum number of seats between two same subjects would be (12 / 4) - 1 = 2. So, each subject would be separated by two other subjects.Wait, that might work. So, the arrangement would be like M, S, L, A, M, S, L, A, M, S, L, A, and then back to M. But in a circle, this would create a repeating pattern every four seats.But in this case, each subject is separated by two other subjects, so no two same subjects are adjacent. That works.So, the number of such arrangements would be the number of ways to arrange the four subjects around the circle, considering rotational symmetry, and then permuting the teachers within each subject.But wait, arranging four subjects around a circle with rotational symmetry is (4-1)! = 6 ways. But since each subject is repeated three times, we have to consider the permutations of the subjects.Wait, actually, no. Because each subject is repeated three times, the arrangement is more complex.Wait, perhaps it's better to model this as a necklace with beads of four colors, each appearing three times, with no two adjacent beads of the same color.The number of such necklaces can be calculated using combinatorial methods, but it's non-trivial.Alternatively, maybe I can use the concept of derangements for circular permutations.Wait, another thought: since each subject must be equally spaced, the arrangement must be a rotation of the sequence M, S, L, A repeated three times.But in a circle, this would fix the arrangement up to rotation and reflection.Wait, but the problem is that Ms. Clara wants to arrange the women educators, who are distinct individuals, so the specific teachers matter, not just their subjects.So, perhaps first, we fix the subject arrangement, and then assign the teachers to each seat.But how many distinct subject arrangements are there?If we fix one Math teacher's position, then the other Math teachers must be placed four seats apart. Similarly, the Science, Literature, and Arts teachers must be placed in the remaining seats.But since it's a circle, fixing one position removes rotational symmetry. So, fixing one Math teacher, the next Math teacher is four seats away, and the third is eight seats away (which is equivalent to four seats in the other direction).Then, the remaining seats are for Science, Literature, and Arts. Each of these subjects also has three teachers, so they must be placed in the remaining seats with no two same subjects adjacent.But wait, if we fix the Math teachers, the remaining seats are 12 - 3 = 9 seats. These 9 seats need to be filled with Science, Literature, and Arts, each appearing three times, with no two same subjects adjacent.But in this case, the remaining seats are arranged in a circle as well, but with fixed positions relative to the Math teachers.Wait, no, because the Math teachers are fixed at positions 1, 5, and 9 (assuming a 12-seat circle). So, the remaining seats are 2,3,4,6,7,8,10,11,12.But these are not in a circle anymore because seat 12 is adjacent to seat 1 (which is Math), so seat 12 can't be Math, but it can be any other subject.Wait, actually, the remaining seats are in a circle because seat 12 is adjacent to seat 1, which is Math, so seat 12 can be any subject except Math.But the remaining seats (2,3,4,6,7,8,10,11,12) form a circle? No, because seat 12 is adjacent to seat 1, which is fixed as Math, so seat 12 is adjacent to Math, but the other seats are adjacent to each other.Wait, maybe it's better to think of the remaining seats as a linear arrangement because seat 12 is adjacent to seat 1, which is fixed, so seat 12 can't be the same as seat 11 or seat 1.But this is getting complicated. Maybe I need a different approach.Alternatively, perhaps I can model this as arranging the 12 women in a circle with the given constraints. Since each subject has three women, and no two same subjects can be adjacent, it's similar to arranging them such that each subject is equally spaced.Wait, in a circle, if each subject is equally spaced, the number of seats between each same subject would be 12 / 4 = 3. So, each subject would be separated by three seats.But wait, that would mean each subject is three seats apart, but with three women per subject, that would require 4 * 3 = 12 seats, which is exactly the number we have.So, the arrangement would be like M, X, X, X, M, X, X, X, M, X, X, X, and then back to M. But in this case, the X's would have to be filled with the other subjects, each appearing three times.But this seems to create a problem because the X's would have to be arranged such that no two same subjects are adjacent.Wait, maybe it's better to think of the arrangement as a repetition of a four-seat block, each containing one of each subject.So, the sequence would be M, S, L, A, M, S, L, A, M, S, L, A, and then back to M. This way, each subject is separated by three seats, and no two same subjects are adjacent.But in this case, the arrangement is fixed up to rotation and reflection. So, how many distinct arrangements are there?Since it's a circle, fixing one position removes rotational symmetry. So, if we fix one Math teacher at position 1, the rest of the subjects are determined in a repeating pattern.But the subjects can be permuted in different orders. For example, instead of M, S, L, A, it could be M, L, S, A, etc.So, the number of distinct subject arrangements is the number of ways to arrange the four subjects in the repeating block, considering rotational symmetry.Since we fixed one subject (Math) at position 1, the remaining three subjects (Science, Literature, Arts) can be arranged in 3! = 6 ways around the circle.But wait, in a circle, fixing one position, the number of distinct arrangements is (4-1)! = 6, which matches the 3! for the remaining three subjects.So, there are 6 distinct subject arrangements.But wait, each subject arrangement corresponds to a specific repeating pattern. So, for each of these 6 arrangements, we can assign the specific teachers to each seat.Since each subject has three distinct teachers, the number of ways to assign them is 3! for each subject.So, for each subject arrangement, the number of ways to assign the teachers is (3!)^4, because there are four subjects, each with 3! permutations.Therefore, the total number of arrangements is 6 * (3!)^4.But wait, let me verify this.First, fixing one Math teacher at position 1. Then, the subject arrangement around the circle is determined by the permutation of the other three subjects (Science, Literature, Arts). There are 3! = 6 ways to arrange these three subjects in the repeating block.For each such arrangement, the specific teachers can be assigned to each seat. Since each subject has three distinct teachers, for each subject, there are 3! ways to assign the teachers to their respective seats.Since there are four subjects, the total number of assignments is (3!)^4.Therefore, the total number of arrangements is 6 * (6)^4 = 6 * 1296 = 7776.Wait, but hold on. Is this correct? Because we fixed one Math teacher, which accounts for rotational symmetry, and then considered the permutations of the other subjects.But does this account for all possible distinct arrangements?Wait, another thought: in circular permutations, fixing one position is a common technique to avoid counting rotations multiple times. So, by fixing one Math teacher, we're effectively considering arrangements up to rotation, which is correct.Then, the number of distinct subject arrangements is 3! = 6, as the other three subjects can be permuted in any order.Then, for each subject, the teachers can be assigned in 3! ways, so (3!)^4 for all subjects.Therefore, the total number of arrangements is 6 * (6)^4 = 6^5 = 7776.But wait, 3! is 6, so (3!)^4 is 6^4 = 1296, and 6 * 1296 = 7776.Yes, that seems correct.But let me think again. Is there a possibility that some arrangements are being overcounted or undercounted?Wait, no, because by fixing one Math teacher, we eliminate rotational symmetry. Then, permuting the other subjects gives us distinct arrangements, and permuting the teachers within each subject also gives distinct arrangements.Therefore, the total number of distinct ways is 7776.Wait, but I'm not entirely sure. Let me check with a smaller case.Suppose we have 4 subjects, each with 1 teacher, arranged in a circle. The number of arrangements should be (4-1)! = 6. Which matches our formula: fixing one subject, then permuting the other three, which is 3! = 6.Similarly, if each subject has two teachers, arranged in a circle with no two same subjects adjacent. Wait, that would be 8 seats, each subject appearing twice. The number of arrangements would be more complex, but our approach would be similar.But in our case, with three teachers per subject, the approach seems consistent.Therefore, I think the answer is 7776.Wait, but let me think about reflections. In circular permutations, sometimes reflections are considered distinct, sometimes not. The problem doesn't specify whether arrangements that are mirror images are considered the same or different.In the problem statement, it says "distinct ways," which usually considers rotations as the same, but reflections as different. So, I think we don't need to divide by 2 for reflections.Therefore, the total number of arrangements is 7776.Problem 2: Event Schedule OptimizationNow, the second problem is about scheduling five activities (A, B, C, D, E) in sequence, each taking a positive integer number of hours, with the total time being 10 hours. Additionally, A and B combined must take exactly 4 hours, and D and E combined must take exactly 5 hours. We need to find the number of ways to schedule the activities, considering that the order matters and each activity's duration is a positive integer.Okay, so let's break this down.We have five activities: A, B, C, D, E.Total time: 10 hours.Constraints:- A + B = 4 hours.- D + E = 5 hours.Each activity takes a positive integer number of hours, so A, B, C, D, E ≥ 1.We need to find the number of sequences (permutations) of these activities, considering their durations, such that the total time is 10 hours.Wait, but the problem says "the order of activities matters," so we need to consider the sequence in which the activities are scheduled.But also, each activity's duration is a positive integer, so we have to find the number of ordered sequences where the sum of durations is 10, with the given constraints on A+B and D+E.Wait, actually, the problem is about scheduling the activities in sequence, meaning the order matters, but the durations are variables subject to the constraints.So, it's a problem of finding the number of ordered sequences (permutations) of the activities, where the sum of their durations is 10, with A + B = 4 and D + E = 5.But wait, actually, the activities are distinct, and the order matters, so it's about finding the number of permutations of the activities, each assigned a positive integer duration, such that A + B = 4, D + E = 5, and the total sum is 10.Wait, but the total sum is A + B + C + D + E = 10.Given that A + B = 4 and D + E = 5, then C must be 10 - 4 - 5 = 1.So, C must be exactly 1 hour.Therefore, the problem reduces to assigning durations to A, B, D, E such that A + B = 4, D + E = 5, and C = 1, with each of A, B, D, E being positive integers.Then, the number of ways to assign these durations is equal to the number of ways to choose A and B such that A + B = 4, multiplied by the number of ways to choose D and E such that D + E = 5.Additionally, since the order of activities matters, we need to consider all possible permutations of the activities A, B, C, D, E, with the given durations.Wait, no, actually, the problem is about scheduling the activities in sequence, meaning the order matters, but the durations are determined by the constraints.Wait, perhaps I need to clarify.Each activity has a duration, which is a positive integer. The total duration is 10 hours. The durations must satisfy A + B = 4 and D + E = 5. So, C must be 1.Therefore, the durations are fixed as follows:- A and B are positive integers such that A + B = 4.- D and E are positive integers such that D + E = 5.- C is 1.So, first, we need to find the number of ways to assign durations to A, B, D, E.For A + B = 4, the number of positive integer solutions is 3: (1,3), (2,2), (3,1).Similarly, for D + E = 5, the number of positive integer solutions is 4: (1,4), (2,3), (3,2), (4,1).Therefore, the total number of ways to assign durations is 3 * 4 = 12.But now, since the order of activities matters, we need to consider all possible sequences (permutations) of the activities A, B, C, D, E, with the assigned durations.However, the problem is that the activities are distinct, so each permutation is unique regardless of the durations. But the durations are determined by the constraints, so for each permutation, the durations are fixed based on the assignments.Wait, no, actually, the durations are assigned to the activities, so for each permutation, the durations are determined by the assignments of A, B, D, E.Wait, perhaps I need to think differently.The problem is to schedule the activities in sequence, meaning the order matters, and each activity has a duration. The durations are subject to the constraints A + B = 4, D + E = 5, and total duration 10.So, the number of ways is equal to the number of possible assignments of durations to A, B, D, E, multiplied by the number of permutations of the activities.Wait, but the durations are fixed once we choose A, B, D, E. So, for each assignment of durations, we can permute the activities in any order, but the durations are fixed per activity.But wait, actually, no. Because the problem says "the order of activities matters," so each permutation of the activities is a different schedule, regardless of the durations.But the durations are determined by the constraints, so for each permutation, the durations are fixed as per the assignments.Wait, I'm getting confused.Let me rephrase.We have five activities: A, B, C, D, E.Each has a duration:- A: a hours- B: b hours- C: 1 hour- D: d hours- E: e hoursWith constraints:- a + b = 4- d + e = 5- a, b, d, e ≥ 1Total duration: a + b + c + d + e = 10, which is satisfied since 4 + 1 + 5 = 10.So, the problem reduces to:1. Assign durations to A, B, D, E such that a + b = 4 and d + e = 5.2. Schedule the five activities in some order, considering that each activity has a fixed duration.But since the order of activities matters, each permutation of the five activities is a distinct schedule.However, the durations are fixed once we assign a, b, d, e.Wait, but the problem is asking for the number of ways to schedule the activities, considering their durations, which are subject to the constraints.So, perhaps the number of ways is equal to the number of possible assignments of durations multiplied by the number of permutations of the activities.But no, because the permutations are independent of the duration assignments.Wait, actually, the duration assignments are separate from the scheduling order.So, first, we choose the durations for A, B, D, E, which can be done in 3 * 4 = 12 ways.Then, for each such assignment, we can arrange the five activities in any order, which is 5! = 120 ways.Therefore, the total number of ways is 12 * 120 = 1440.But wait, is that correct?Wait, no, because the duration assignments are fixed per activity, so for each assignment, the activities have fixed durations, and the order can be any permutation.Therefore, the total number of schedules is the number of duration assignments multiplied by the number of permutations.So, 12 * 120 = 1440.But let me think again.Alternatively, perhaps the problem is considering the duration assignments as part of the scheduling.Meaning, for each permutation of the activities, we need to assign durations to A, B, D, E such that A + B = 4 and D + E = 5.So, for each permutation, the number of ways to assign durations is 3 * 4 = 12.Therefore, the total number of schedules is 5! * 12 = 120 * 12 = 1440.Yes, that seems correct.But wait, another thought: the problem says "each activity’s duration is a positive integer," and "the order of activities matters." So, it's about sequences where each activity is assigned a duration, and the order is considered.So, for each permutation of the activities, we can assign durations to A, B, D, E such that A + B = 4 and D + E = 5.Therefore, for each of the 5! permutations, there are 3 * 4 = 12 ways to assign durations.Hence, total number of ways is 120 * 12 = 1440.Yes, that seems correct.But let me verify with a smaller case.Suppose we have two activities, A and B, with A + B = 2, and total duration 2. Each activity must take at least 1 hour.The number of duration assignments is 1: A=1, B=1.The number of permutations is 2! = 2.So, total number of schedules is 2 * 1 = 2.Which makes sense: AB and BA, each with durations 1 and 1.Similarly, if we have A + B = 3, then duration assignments are 2: (1,2) and (2,1). For each permutation of A and B, which is 2, we have 2 duration assignments, so total 4 schedules.Which is correct: A=1, B=2; A=2, B=1; B=1, A=2; B=2, A=1.Therefore, in our original problem, the total number of ways is 5! * (number of duration assignments) = 120 * 12 = 1440.Therefore, the answer is 1440.Final Answer1. The number of distinct seating arrangements is boxed{7776}.2. The number of ways to schedule the activities is boxed{1440}.</think>

🧠:<think>Okay, so I have this problem about a college football coach who wants to build a strong team. The coach relies on a talent evaluator who uses a rating system based on four performance metrics: speed, strength, agility, and game intelligence. Each metric is scored from 1 to 10, and there's a weighted formula to calculate each player's total score. Then, there's a budget constraint where the coach can only recruit up to two players, and the cost is inversely proportional to their total score. Alright, let me break this down step by step.First, part 1: calculating the total scores for three players. The formula given is ( P_i = 0.3S_i + 0.25T_i + 0.2A_i + 0.25G_i ). So, for each player, I need to plug in their respective scores into this formula.Starting with Player 1: S1 = 8, T1 = 7, A1 = 9, G1 = 6.So, P1 = 0.3*8 + 0.25*7 + 0.2*9 + 0.25*6.Let me compute each term:0.3*8 = 2.40.25*7 = 1.750.2*9 = 1.80.25*6 = 1.5Adding them up: 2.4 + 1.75 = 4.15; 4.15 + 1.8 = 5.95; 5.95 + 1.5 = 7.45.So, P1 is 7.45.Next, Player 2: S2 = 6, T2 = 8, A2 = 7, G2 = 9.P2 = 0.3*6 + 0.25*8 + 0.2*7 + 0.25*9.Calculating each term:0.3*6 = 1.80.25*8 = 2.00.2*7 = 1.40.25*9 = 2.25Adding them up: 1.8 + 2.0 = 3.8; 3.8 + 1.4 = 5.2; 5.2 + 2.25 = 7.45.Wait, so P2 is also 7.45? Hmm, same as Player 1.Now, Player 3: S3 = 9, T3 = 6, A3 = 8, G3 = 7.P3 = 0.3*9 + 0.25*6 + 0.2*8 + 0.25*7.Calculating each term:0.3*9 = 2.70.25*6 = 1.50.2*8 = 1.60.25*7 = 1.75Adding them up: 2.7 + 1.5 = 4.2; 4.2 + 1.6 = 5.8; 5.8 + 1.75 = 7.55.So, P3 is 7.55.So, summarizing:Player 1: 7.45Player 2: 7.45Player 3: 7.55Therefore, Player 3 has the highest total score and should be prioritized.Wait, but both Player 1 and Player 2 have the same score. So, if the coach can only choose one, Player 3 is the top choice. If they can choose two, maybe Player 3 and either Player 1 or Player 2.But part 1 just asks to determine which player should be prioritized based on the highest score, so that's Player 3.Moving on to part 2: the coach has a budget constraint of 30,000 and can recruit up to two players. The cost to recruit each player is inversely proportional to their total score, given by ( C_i = frac{k}{P_i} ), where k is a constant. We need to find the maximum value of k such that the sum of the costs for two players does not exceed 30,000.So, first, let's note the total scores:Player 1: 7.45Player 2: 7.45Player 3: 7.55Assuming the coach wants to maximize the total score within the budget, they would likely choose the two highest-scoring players. Since Player 3 is the highest, and then either Player 1 or Player 2, who are tied.So, the two players to recruit would be Player 3 and Player 1 (or Player 2, since they are the same). So, let's compute the total cost for recruiting Player 3 and Player 1.Given ( C_i = frac{k}{P_i} ), so total cost is ( C_3 + C_1 = frac{k}{7.55} + frac{k}{7.45} ).This total cost must be less than or equal to 30,000.So, ( frac{k}{7.55} + frac{k}{7.45} leq 30,000 ).We can factor out k:( k left( frac{1}{7.55} + frac{1}{7.45} right) leq 30,000 ).So, let's compute ( frac{1}{7.55} + frac{1}{7.45} ).First, compute each reciprocal:1/7.55 ≈ 0.132451/7.45 ≈ 0.13422Adding them together: 0.13245 + 0.13422 ≈ 0.26667.So, approximately 0.26667.Therefore, ( k * 0.26667 leq 30,000 ).To find the maximum k, we can set the inequality to equality:( k * 0.26667 = 30,000 )So, solving for k:k = 30,000 / 0.26667 ≈ ?Let me compute that.First, 30,000 divided by 0.26667.Well, 0.26667 is approximately 4/15, since 4 divided by 15 is approximately 0.266666...So, 30,000 divided by (4/15) is 30,000 * (15/4) = 30,000 * 3.75 = 112,500.Wait, let me verify that:4/15 is approximately 0.266666...So, 30,000 / (4/15) = 30,000 * (15/4) = (30,000 / 4) * 15 = 7,500 * 15 = 112,500.Yes, that seems correct.Alternatively, using the approximate decimal:30,000 / 0.26667 ≈ 30,000 / 0.26667 ≈ 112,500.So, k ≈ 112,500.But let me check with more precise calculations.Compute 1/7.55:7.55 goes into 1 how many times? 7.55 * 0.132 = approx 1.7.55 * 0.132 = 0.9996, which is roughly 1. So, 1/7.55 ≈ 0.13245.Similarly, 1/7.45:7.45 * 0.134 = approx 1.7.45 * 0.134 = 0.9993, so 1/7.45 ≈ 0.13422.Adding them: 0.13245 + 0.13422 = 0.26667.So, 30,000 / 0.26667 ≈ 112,500.Therefore, k ≈ 112,500.But let me verify with exact fractions.7.55 is equal to 151/20, and 7.45 is equal to 149/20.So, 1/7.55 = 20/151 ≈ 0.132451/7.45 = 20/149 ≈ 0.13422So, 20/151 + 20/149 = 20*(1/151 + 1/149) = 20*(149 + 151)/(151*149) = 20*(300)/(22549) = 6000/22549 ≈ 0.26667.So, 6000/22549 is exactly equal to 0.26667 approximately.Therefore, k = 30,000 / (6000/22549) = 30,000 * (22549/6000) = (30,000 / 6000) * 22549 = 5 * 22549 = 112,745.Wait, that's a bit different. Wait, 30,000 divided by (6000/22549) is 30,000 * (22549/6000) = (30,000 / 6000) * 22549 = 5 * 22549 = 112,745.Hmm, so 112,745 is more precise.Wait, but 6000/22549 is approximately 0.26667, so 30,000 / 0.26667 is approximately 112,500, but exact calculation gives 112,745.So, which one is correct?Wait, let's compute 30,000 divided by (6000/22549):30,000 / (6000/22549) = 30,000 * (22549/6000) = (30,000 / 6000) * 22549 = 5 * 22549 = 112,745.Yes, that's exact.But 6000/22549 is approximately 0.26667, and 30,000 / 0.26667 is approximately 112,500, which is close but not exact.So, the exact value is 112,745.But let me check 6000/22549:Compute 22549 * 0.26667:22549 * 0.26667 ≈ 22549 * (2/7.5) ≈ 22549 * 0.266666...Wait, 22549 * 0.266666... = 22549 * (4/15) = (22549 * 4)/15 = 90,196 / 15 ≈ 6,013.07.But 6000 is less than that. Hmm, maybe I made a miscalculation.Wait, 6000/22549 is approximately 0.26667.Yes, 22549 * 0.26667 ≈ 6,013, which is more than 6,000. So, 6000/22549 is slightly less than 0.26667.So, 30,000 / (6000/22549) = 30,000 * (22549/6000) = 5 * 22549 = 112,745.So, the exact value is 112,745.But since k is a constant, it can be a decimal, so perhaps we can write it as 112,745.But let me see if 112,745 is correct.Alternatively, perhaps it's better to use exact fractions.Let me compute 1/7.55 + 1/7.45.Expressed as fractions:7.55 = 151/20, so 1/7.55 = 20/151.7.45 = 149/20, so 1/7.45 = 20/149.Thus, 20/151 + 20/149 = 20*(149 + 151)/(151*149) = 20*300/(151*149).Compute denominator: 151*149.151*150 = 22,650, so 151*149 = 22,650 - 151 = 22,499.So, 20*300 = 6,000.Thus, 6,000 / 22,499 ≈ 0.26667.So, 6,000 / 22,499 is the exact value.Thus, k = 30,000 / (6,000 / 22,499) = 30,000 * (22,499 / 6,000) = (30,000 / 6,000) * 22,499 = 5 * 22,499 = 112,495.Wait, wait, 22,499 * 5 = 112,495.Wait, earlier I thought 22549, but actually, 151*149 is 22,499, not 22,549.Yes, 151*149: 150*150 = 22,500, so 151*149 = 22,500 - 150 - 149 + 1 = 22,500 - 299 + 1 = 22,202? Wait, no, that's not correct.Wait, actually, 151*149 = (150 + 1)(150 - 1) = 150² - 1² = 22,500 - 1 = 22,499. Yes, that's correct.So, 22,499.Thus, k = 5 * 22,499 = 112,495.So, k is 112,495.But let me verify:If k = 112,495, then the cost for Player 3 is 112,495 / 7.55, and for Player 1 is 112,495 / 7.45.Compute 112,495 / 7.55:7.55 * 14,900 = ?Wait, 7.55 * 14,900 = 7.55 * 10,000 = 75,500; 7.55 * 4,900 = 7.55*4,000=30,200; 7.55*900=6,795. So total is 75,500 + 30,200 = 105,700 + 6,795 = 112,495.Yes, so 7.55 * 14,900 = 112,495.Similarly, 112,495 / 7.45:7.45 * 15,100 = ?7.45 * 15,000 = 111,7507.45 * 100 = 745So, 111,750 + 745 = 112,495.Yes, so 7.45 * 15,100 = 112,495.Therefore, the cost for Player 3 is 14,900 and for Player 1 is 15,100.Adding them together: 14,900 + 15,100 = 30,000.Perfect, so k = 112,495 is the exact value where the total cost is exactly 30,000.Therefore, the maximum value of k is 112,495.But let me think again: the problem says "the sum of the costs for the two chosen players must not exceed this budget." So, the maximum k is such that when you choose the two most expensive players (which are the two with the lowest total scores, but wait, no, the cost is inversely proportional to the total score, so higher total score means lower cost.Wait, hold on, the cost is inversely proportional to the total score, so higher P_i means lower C_i.Therefore, to maximize k, the coach would choose the two players with the lowest costs, which are the two with the highest total scores.Wait, but in our case, the two highest total scores are Player 3 (7.55) and Player 1/2 (7.45). So, recruiting Player 3 and Player 1 (or Player 2) would result in the lowest total cost, allowing k to be as large as possible without exceeding the budget.Wait, but in our calculation, we found that k = 112,495 would make the total cost exactly 30,000 for Player 3 and Player 1.If k were any larger, the total cost would exceed 30,000. If k were smaller, the total cost would be under.Therefore, the maximum k is 112,495.But let me confirm:If k = 112,495, then:C3 = 112,495 / 7.55 = 14,900C1 = 112,495 / 7.45 = 15,100Total cost: 14,900 + 15,100 = 30,000.Perfect.Therefore, the maximum value of k is 112,495.But wait, let me check if there's a possibility that recruiting two other players could allow for a higher k.Wait, if the coach chooses two players with lower total scores, their costs would be higher, so the total cost would be higher, meaning k would have to be smaller to stay within the budget.Therefore, to maximize k, the coach should choose the two players with the highest total scores, which are Player 3 and Player 1 (or Player 2), as they have the lowest costs.Hence, k = 112,495 is indeed the maximum value.But let me see if I can represent this as a fraction.Since k = 30,000 / (1/7.55 + 1/7.45) = 30,000 / (20/151 + 20/149) = 30,000 / (20*(1/151 + 1/149)) = 30,000 / (20*(149 + 151)/(151*149)) = 30,000 / (20*300/22499) = 30,000 / (6000/22499) = 30,000 * (22499/6000) = 5 * 22499 = 112,495.Yes, so 112,495 is the exact value.Alternatively, if the problem expects a decimal, it would be approximately 112,500, but since 112,495 is exact, we should use that.Therefore, the maximum value of k is 112,495.Final Answer1. The total scores are ( P_1 = 7.45 ), ( P_2 = 7.45 ), and ( P_3 = 7.55 ). Player 3 should be prioritized. The answer is boxed{7.55}.2. The maximum value of ( k ) is boxed{112495}.</think>

🧠:<think>Alright, so I have this problem about complex analysis, specifically dealing with residues of a function and relating them to polynomials. Let me try to unpack this step by step.First, the function given is ( f(z) = frac{e^{z^2}}{(z^2 + 1)(z - a)} ). The singularities of this function are where the denominator is zero, so that's at ( z = pm i ) and ( z = a ). Since ( a neq pm i ), these are all isolated singularities, specifically poles. The function is analytic everywhere else, so that's good.I need to find the residues at each of these singularities. I remember that for a function ( f(z) ) with a simple pole at ( z = z_0 ), the residue can be calculated as ( text{Res}(f, z_0) = lim_{z to z_0} (z - z_0)f(z) ). If it's a higher-order pole, the formula is a bit more complicated, but in this case, all the singularities are simple poles because each factor in the denominator is linear and distinct.So, let's start with the residue at ( z = i ). The function can be written as ( f(z) = frac{e^{z^2}}{(z - i)(z + i)(z - a)} ). So, the residue at ( z = i ) is the limit as ( z ) approaches ( i ) of ( (z - i)f(z) ). That simplifies to ( frac{e^{i^2}}{(i + i)(i - a)} ). Since ( i^2 = -1 ), this becomes ( frac{e^{-1}}{(2i)(i - a)} ).Wait, let me double-check that. So, ( (z - i)f(z) = frac{e^{z^2}}{(z + i)(z - a)} ). Plugging in ( z = i ), we get ( frac{e^{(i)^2}}{(i + i)(i - a)} = frac{e^{-1}}{(2i)(i - a)} ). Yeah, that seems right.Similarly, for ( z = -i ), the residue will be ( lim_{z to -i} (z + i)f(z) ). That gives ( frac{e^{(-i)^2}}{(-i + i)(-i - a)} ). Wait, hold on, ( (-i)^2 = (-i)(-i) = i^2 = -1 ), so ( e^{-1} ). The denominator is ( (-i + i) ) which is zero, but that's not right because we're taking the limit as ( z ) approaches ( -i ), so actually, it's ( (z + i)f(z) = frac{e^{z^2}}{(z - i)(z - a)} ). Plugging in ( z = -i ), we get ( frac{e^{-1}}{(-i - i)(-i - a)} = frac{e^{-1}}{(-2i)(-i - a)} ). Simplifying that, the denominator is ( (-2i)(-i - a) = (-2i)(-1)(i + a) = 2i(i + a) ). So, the residue is ( frac{e^{-1}}{2i(i + a)} ).Wait, but hold on, the denominator when ( z = -i ) is ( (-i - a) ), which is ( -(i + a) ), so actually, the denominator is ( (-2i)(-i - a) = (-2i)(-1)(i + a) = 2i(i + a) ). So, the residue is ( frac{e^{-1}}{2i(i + a)} ). Hmm, that seems correct.Now, for the residue at ( z = a ), since ( a neq pm i ), it's another simple pole. So, the residue is ( lim_{z to a} (z - a)f(z) ). That simplifies to ( frac{e^{a^2}}{(a^2 + 1)} ). Because ( (z - a)f(z) = frac{e^{z^2}}{(z^2 + 1)} ), so plugging in ( z = a ), we get ( frac{e^{a^2}}{(a^2 + 1)} ).Let me recap:- Residue at ( z = i ): ( frac{e^{-1}}{2i(i - a)} )- Residue at ( z = -i ): ( frac{e^{-1}}{2i(i + a)} )- Residue at ( z = a ): ( frac{e^{a^2}}{a^2 + 1} )Wait, hold on, let me check the residue at ( z = -i ) again. When I plug in ( z = -i ) into ( (z + i)f(z) ), I get ( frac{e^{(-i)^2}}{( -i - i)(-i - a)} ). So, ( (-i)^2 = (-1)^2 i^2 = 1*(-1) = -1 ), so ( e^{-1} ). The denominator is ( (-2i)(-i - a) ). So, ( (-2i)(-i - a) = (-2i)(-1)(i + a) = 2i(i + a) ). So, the residue is ( frac{e^{-1}}{2i(i + a)} ). Hmm, that seems correct.Wait, but in the denominator, ( (-i - a) = -(i + a) ), so ( (-2i)(-i - a) = (-2i)(-1)(i + a) = 2i(i + a) ). So, the residue is ( frac{e^{-1}}{2i(i + a)} ). Yeah, that's correct.So, that's the first part done. Now, moving on to the second part.Given that ( P(z) ) is a polynomial with roots at the singularities of ( f(z) ). So, the singularities are ( z = i, -i, a ). So, ( P(z) = (z - i)(z + i)(z - a) = (z^2 + 1)(z - a) ). So, that's the polynomial.Then, ( Q(z) = P'(z) ). So, ( Q(z) ) is the derivative of ( P(z) ). So, ( P(z) = (z^2 + 1)(z - a) ). Let's compute ( P'(z) ).Using the product rule: ( P'(z) = (2z)(z - a) + (z^2 + 1)(1) = 2z(z - a) + z^2 + 1 ). Let's expand that:( 2z(z - a) = 2z^2 - 2a z )Adding ( z^2 + 1 ): total is ( 2z^2 - 2a z + z^2 + 1 = 3z^2 - 2a z + 1 ).So, ( Q(z) = 3z^2 - 2a z + 1 ).Now, the question is to find a relationship between the roots of ( P(z) ) and the residues of ( f(z) ) at those roots.Wait, so ( P(z) ) has roots at ( z = i, -i, a ). So, the roots are the singularities of ( f(z) ). So, the residues of ( f(z) ) at these points are related to the roots of ( P(z) ) and ( Q(z) ).But ( Q(z) = P'(z) ), so ( Q(z) ) is the derivative of ( P(z) ). I remember that for a polynomial ( P(z) ), the derivative ( P'(z) ) has roots related to the critical points of ( P(z) ), but how does that relate to the residues?Wait, perhaps we can use the concept of residues in terms of the polynomial and its derivative. Let me think.In complex analysis, there's a formula that relates the residue of a function ( f(z) ) at a root ( z_0 ) of a polynomial ( P(z) ) when ( f(z) ) is expressed as ( frac{g(z)}{P(z)} ). The residue is ( frac{g(z_0)}{P'(z_0)} ). Is that correct?Yes, I think that's the formula. So, in general, if ( f(z) = frac{g(z)}{P(z)} ), and ( P(z) ) has a simple root at ( z_0 ), then ( text{Res}(f, z_0) = frac{g(z_0)}{P'(z_0)} ).In our case, ( f(z) = frac{e^{z^2}}{(z^2 + 1)(z - a)} = frac{e^{z^2}}{P(z)} ). So, ( g(z) = e^{z^2} ). Therefore, the residue at each root ( z_0 ) of ( P(z) ) is ( frac{e^{z_0^2}}{P'(z_0)} ).But ( P'(z_0) ) is exactly ( Q(z_0) ), since ( Q(z) = P'(z) ). Therefore, the residue at each singularity ( z_0 ) is ( frac{e^{z_0^2}}{Q(z_0)} ).So, that gives us a relationship: the residue of ( f(z) ) at each singularity ( z_0 ) is equal to ( frac{e^{z_0^2}}{Q(z_0)} ).But let's verify this with the residues we calculated earlier.For ( z = i ):Residue is ( frac{e^{-1}}{2i(i - a)} ).Compute ( Q(i) = 3i^2 - 2a i + 1 = 3(-1) - 2a i + 1 = -3 - 2a i + 1 = -2 - 2a i = -2(1 + a i) ).So, ( frac{e^{i^2}}{Q(i)} = frac{e^{-1}}{-2(1 + a i)} = frac{e^{-1}}{-2(1 + a i)} ). Hmm, but our residue was ( frac{e^{-1}}{2i(i - a)} ).Wait, let's see if these are equal.Compute ( frac{e^{-1}}{-2(1 + a i)} ) vs. ( frac{e^{-1}}{2i(i - a)} ).Let me manipulate ( frac{1}{2i(i - a)} ). Multiply numerator and denominator by ( i ):( frac{i}{2i^2(i - a)} = frac{i}{2(-1)(i - a)} = frac{-i}{2(i - a)} ).Wait, that's not the same as ( frac{1}{-2(1 + a i)} ). Hmm, maybe I made a mistake.Wait, let's compute ( Q(i) ) again.( Q(z) = 3z^2 - 2a z + 1 ).So, ( Q(i) = 3(i)^2 - 2a(i) + 1 = 3(-1) - 2a i + 1 = -3 - 2a i + 1 = (-3 + 1) - 2a i = -2 - 2a i ).So, ( Q(i) = -2(1 + a i) ).So, ( frac{e^{i^2}}{Q(i)} = frac{e^{-1}}{-2(1 + a i)} ).But earlier, we had the residue as ( frac{e^{-1}}{2i(i - a)} ).Let me see if these are equal.Compute ( frac{1}{-2(1 + a i)} ) vs. ( frac{1}{2i(i - a)} ).Multiply numerator and denominator of the second expression by ( i ):( frac{i}{2i^2(i - a)} = frac{i}{2(-1)(i - a)} = frac{-i}{2(i - a)} ).Hmm, not the same as the first expression. So, perhaps I made a mistake in the relationship.Wait, perhaps I need to consider that ( P(z) = (z - i)(z + i)(z - a) ), so ( P'(z) = (z + i)(z - a) + (z - i)(z - a) + (z - i)(z + i) ). Wait, that's another way to compute ( P'(z) ).Wait, no, actually, ( P(z) = (z^2 + 1)(z - a) ), so ( P'(z) = 2z(z - a) + (z^2 + 1) ), which is what I had before, ( 3z^2 - 2a z + 1 ).Wait, perhaps the formula ( text{Res}(f, z_0) = frac{g(z_0)}{P'(z_0)} ) is correct, but in our case, ( g(z) = e^{z^2} ), so ( g(z_0) = e^{z_0^2} ).So, the residue should be ( frac{e^{z_0^2}}{P'(z_0)} ), which is ( frac{e^{z_0^2}}{Q(z_0)} ).But when I plug in ( z_0 = i ), I get ( frac{e^{-1}}{Q(i)} = frac{e^{-1}}{-2(1 + a i)} ), but our residue was ( frac{e^{-1}}{2i(i - a)} ).Wait, let me compute ( frac{1}{-2(1 + a i)} ) and ( frac{1}{2i(i - a)} ) to see if they are equal.Compute ( frac{1}{-2(1 + a i)} = frac{-1}{2(1 + a i)} ).Compute ( frac{1}{2i(i - a)} = frac{1}{2i(-1)(a - i)} = frac{-1}{2i(a - i)} ).Hmm, these are not obviously equal, but perhaps they are related through complex conjugates or something.Wait, let me compute ( frac{1}{2i(i - a)} ).Multiply numerator and denominator by ( i ):( frac{i}{2i^2(i - a)} = frac{i}{2(-1)(i - a)} = frac{-i}{2(i - a)} ).So, ( frac{-i}{2(i - a)} ) vs. ( frac{-1}{2(1 + a i)} ).Wait, let me see if ( -i/(2(i - a)) = -1/(2(1 + a i)) ).Multiply both sides by 2:( -i/(i - a) = -1/(1 + a i) ).Multiply both sides by denominators:Left side: ( -i(1 + a i) )Right side: ( -1(i - a) )Compute left side: ( -i - a i^2 = -i - a(-1) = -i + a ).Compute right side: ( -i + a ).So, yes, they are equal. Therefore, ( frac{-i}{2(i - a)} = frac{-1}{2(1 + a i)} ).Therefore, ( frac{e^{-1}}{2i(i - a)} = frac{e^{-1}}{-2(1 + a i)} ).So, that means the residue at ( z = i ) is indeed ( frac{e^{i^2}}{Q(i)} ).Similarly, let's check for ( z = -i ).Residue at ( z = -i ) is ( frac{e^{-1}}{2i(i + a)} ).Compute ( Q(-i) = 3(-i)^2 - 2a(-i) + 1 = 3(-1) - 2a(-i) + 1 = -3 + 2a i + 1 = (-3 + 1) + 2a i = -2 + 2a i = -2(1 - a i) ).So, ( frac{e^{(-i)^2}}{Q(-i)} = frac{e^{-1}}{-2(1 - a i)} ).Compare this with our residue ( frac{e^{-1}}{2i(i + a)} ).Let me manipulate ( frac{1}{2i(i + a)} ):Multiply numerator and denominator by ( i ):( frac{i}{2i^2(i + a)} = frac{i}{2(-1)(i + a)} = frac{-i}{2(i + a)} ).So, ( frac{e^{-1}}{2i(i + a)} = frac{-i e^{-1}}{2(i + a)} ).Wait, but ( frac{e^{-1}}{-2(1 - a i)} = frac{-e^{-1}}{2(1 - a i)} ).Hmm, let's see if ( frac{-i}{2(i + a)} = frac{-1}{2(1 - a i)} ).Multiply both sides by 2:( frac{-i}{i + a} = frac{-1}{1 - a i} ).Multiply both sides by denominators:Left side: ( -i(1 - a i) )Right side: ( -1(i + a) )Compute left side: ( -i + a i^2 = -i + a(-1) = -i - a ).Compute right side: ( -i - a ).So, yes, they are equal. Therefore, ( frac{-i}{2(i + a)} = frac{-1}{2(1 - a i)} ).Therefore, ( frac{e^{-1}}{2i(i + a)} = frac{e^{-1}}{-2(1 - a i)} ), which matches ( frac{e^{(-i)^2}}{Q(-i)} ).Similarly, for ( z = a ):Residue is ( frac{e^{a^2}}{a^2 + 1} ).Compute ( Q(a) = 3a^2 - 2a a + 1 = 3a^2 - 2a^2 + 1 = a^2 + 1 ).So, ( frac{e^{a^2}}{Q(a)} = frac{e^{a^2}}{a^2 + 1} ), which matches our residue at ( z = a ).Therefore, the relationship is that the residue of ( f(z) ) at each singularity ( z_0 ) is equal to ( frac{e^{z_0^2}}{Q(z_0)} ), where ( Q(z) = P'(z) ).So, in general, for each root ( z_0 ) of ( P(z) ), the residue of ( f(z) ) at ( z_0 ) is ( frac{e^{z_0^2}}{Q(z_0)} ).Therefore, the relationship is that each residue is equal to ( frac{e^{z_0^2}}{Q(z_0)} ), where ( z_0 ) is a root of ( P(z) ).So, to summarize:1. The residues of ( f(z) ) at ( z = i ), ( z = -i ), and ( z = a ) are ( frac{e^{-1}}{2i(i - a)} ), ( frac{e^{-1}}{2i(i + a)} ), and ( frac{e^{a^2}}{a^2 + 1} ) respectively.2. The relationship is that each residue is equal to ( frac{e^{z_0^2}}{Q(z_0)} ), where ( z_0 ) is a root of ( P(z) ).I think that's the answer. Let me just make sure I didn't make any calculation errors, especially with the signs and the complex numbers.For ( z = i ):- ( P'(i) = Q(i) = -2 - 2a i )- ( e^{i^2} = e^{-1} )- So, ( frac{e^{-1}}{-2 - 2a i} = frac{e^{-1}}{-2(1 + a i)} )- Which simplifies to ( frac{-e^{-1}}{2(1 + a i)} )- Which, as we saw earlier, is equal to ( frac{e^{-1}}{2i(i - a)} )Yes, that's correct.Similarly for ( z = -i ):- ( P'(-i) = Q(-i) = -2 + 2a i )- ( e^{(-i)^2} = e^{-1} )- So, ( frac{e^{-1}}{-2 + 2a i} = frac{e^{-1}}{-2(1 - a i)} )- Which simplifies to ( frac{-e^{-1}}{2(1 - a i)} )- Which is equal to ( frac{e^{-1}}{2i(i + a)} )Yes, that's correct.And for ( z = a ):- ( P'(a) = Q(a) = a^2 + 1 )- ( e^{a^2} ) divided by that is exactly the residue.So, all checks out. Therefore, the relationship is indeed that each residue is ( frac{e^{z_0^2}}{Q(z_0)} ).</think>

🧠:<think>Okay, so I have two problems here that I need to solve. Both are related to optimizing a new system for handling user requests. Let me take them one by one.Starting with the first problem: The department head and IT manager want the system to be beneficial if the probability of receiving more than 25 requests in an hour is less than 5%. I need to find the maximum value of λ that satisfies this requirement. Hmm, okay. So this is a Poisson distribution problem because the number of requests follows a Poisson distribution with mean λ.I remember that the Poisson probability mass function is given by:P(X = k) = (e^{-λ} * λ^k) / k!But here, we're dealing with the probability of receiving more than 25 requests, which is P(X > 25). They want this probability to be less than 5%, or 0.05.So, I need to find the maximum λ such that P(X > 25) < 0.05. That means P(X ≤ 25) should be greater than or equal to 0.95.Calculating cumulative Poisson probabilities can be tricky because it's the sum from k=0 to k=25 of (e^{-λ} * λ^k) / k! and we need this sum to be at least 0.95. This might not have a closed-form solution, so I might need to use some approximation or computational methods.Wait, maybe I can use the normal approximation to the Poisson distribution? I think when λ is large, the Poisson distribution can be approximated by a normal distribution with mean λ and variance λ. But is 25 a large enough number for this approximation? Maybe, but I'm not sure. Alternatively, I can use the cumulative distribution function (CDF) of the Poisson distribution and solve for λ numerically.Let me think about how to approach this. If I can express the CDF in terms of λ, I can set it equal to 0.95 and solve for λ. But since the CDF of Poisson doesn't have a simple formula, I might need to use iterative methods or look up tables.Alternatively, maybe I can use the relationship between Poisson and chi-squared distributions. I recall that the sum of Poisson variables can be related to chi-squared, but I'm not sure if that helps here.Wait, another idea: perhaps I can use the inverse Poisson function if I have access to statistical software or a calculator. But since I'm just brainstorming here, maybe I can use trial and error with different λ values.Let me try to estimate λ. If λ is 25, then the mean is 25, and the probability of more than 25 would be around 0.5, which is way higher than 5%. So we need a much lower λ.Wait, no, actually, for a Poisson distribution, the probability of X being greater than the mean is about 0.5, so if we set λ lower, the probability of X > 25 would decrease. So we need to find a λ such that P(X > 25) < 0.05.Alternatively, maybe using the normal approximation. Let's try that.If I approximate Poisson(λ) with N(λ, λ), then P(X > 25) = P(Z > (25 - λ)/sqrt(λ)) < 0.05.We want (25 - λ)/sqrt(λ) to be greater than the z-score corresponding to 0.05 in the upper tail, which is 1.645.So:(25 - λ)/sqrt(λ) > 1.645Let me denote sqrt(λ) as x, so λ = x^2.Then the inequality becomes:(25 - x^2)/x > 1.645Multiply both sides by x (assuming x > 0):25 - x^2 > 1.645xRearranging:x^2 + 1.645x - 25 < 0This is a quadratic inequality. Let's solve x^2 + 1.645x - 25 = 0.Using quadratic formula:x = [-1.645 ± sqrt(1.645^2 + 100)] / 2Calculate discriminant:1.645^2 ≈ 2.706So sqrt(2.706 + 100) = sqrt(102.706) ≈ 10.134Thus,x = [-1.645 + 10.134]/2 ≈ (8.489)/2 ≈ 4.2445We discard the negative root because x = sqrt(λ) must be positive.So x ≈ 4.2445, so λ ≈ x^2 ≈ 18.02.So approximately λ ≈ 18.02. Let me check if this makes sense.If λ is about 18, then the mean is 18, and we want P(X > 25) < 0.05. Using the normal approximation, we set up the z-score as (25 - 18)/sqrt(18) ≈ 7/4.2426 ≈ 1.65, which is the z-score for 0.05. So that seems consistent.But wait, the normal approximation might not be very accurate here because λ is not extremely large. Maybe I should check with the actual Poisson CDF.Alternatively, perhaps using the chi-squared approximation. I remember that for Poisson, the quantity 2λ can be approximated by a chi-squared distribution with 2k degrees of freedom, but I'm not sure.Alternatively, maybe using the inverse gamma function or something else. But perhaps I can use the relationship with the gamma distribution.Wait, another approach: use the fact that the sum of Poisson variables is Poisson, but that might not help here.Alternatively, perhaps using the Markov inequality? But Markov gives an upper bound, which might not be tight enough.Wait, let me think again. If I use the normal approximation, I get λ ≈ 18.02. But maybe I can get a better estimate.Alternatively, perhaps using the exact Poisson calculation. Let me try to compute P(X ≤ 25) for λ = 18 and see if it's close to 0.95.But calculating the exact Poisson CDF for λ=18 up to 25 is tedious by hand, but maybe I can approximate it.Alternatively, perhaps using the relationship between Poisson and exponential distributions, but that might not help.Wait, another idea: use the fact that for Poisson, the probability P(X > k) can be approximated using the gamma function or the incomplete gamma function. Specifically, P(X > k) = 1 - γ(k+1, λ)/k! where γ is the lower incomplete gamma function.But without computational tools, this might be difficult.Alternatively, perhaps using the Poisson CDF tables. But I don't have tables here.Wait, maybe I can use the fact that for Poisson, the CDF can be approximated using the normal distribution with continuity correction. So instead of P(X > 25), we can use P(X ≥ 25.5) in the normal approximation.So, let's try that.Using continuity correction, we have:P(X > 25) ≈ P(X ≥ 25.5) ≈ P(Z ≥ (25.5 - λ)/sqrt(λ)) < 0.05So,(25.5 - λ)/sqrt(λ) > 1.645Again, let x = sqrt(λ), so λ = x^2.Then,25.5 - x^2 > 1.645xRearranged:x^2 + 1.645x - 25.5 < 0Solve x^2 + 1.645x - 25.5 = 0Discriminant: 1.645^2 + 4*25.5 ≈ 2.706 + 102 ≈ 104.706sqrt(104.706) ≈ 10.23So,x = [-1.645 + 10.23]/2 ≈ (8.585)/2 ≈ 4.2925Thus, x ≈ 4.2925, so λ ≈ x^2 ≈ 18.42.So with continuity correction, λ ≈ 18.42.So the maximum λ is approximately 18.42. But since we need P(X >25) <0.05, we might need to take a slightly lower λ to be safe.Alternatively, perhaps using the exact Poisson calculation.Wait, maybe I can use the relationship between Poisson and chi-squared. I think that for Poisson(λ), P(X ≤ k) = γ(k+1, λ)/k! where γ is the lower incomplete gamma function. And γ(k+1, λ) can be related to the chi-squared distribution.Specifically, 2λ ~ χ²(2k+2). Wait, no, more precisely, if X ~ Poisson(λ), then 2λ ~ χ²(2(k+1)) approximately for large λ.But I'm not sure if that's accurate here.Alternatively, perhaps using the fact that the sum of Poisson variables is Poisson, but that might not help.Alternatively, perhaps using the relationship with the exponential distribution. The waiting time until the next event in a Poisson process is exponential with rate λ.But I'm not sure that helps here.Alternatively, perhaps using the fact that the Poisson CDF can be expressed in terms of the regularized gamma function. Specifically, P(X ≤ k) = Γ(k+1, λ)/k! where Γ is the upper incomplete gamma function. Wait, no, actually, it's the lower incomplete gamma function.Wait, let me double-check. The CDF of Poisson is P(X ≤ k) = e^{-λ} Σ_{i=0}^k (λ^i)/i! which is equal to Γ(k+1, λ)/k! where Γ is the lower incomplete gamma function.But without computational tools, it's hard to compute this exactly.Alternatively, perhaps using the relationship with the chi-squared distribution. I think that for Poisson(λ), the quantity 2λ can be approximated by a chi-squared distribution with 2(k+1) degrees of freedom for the upper tail.Wait, more precisely, for Poisson(λ), P(X ≤ k) ≈ P(χ²_{2(k+1)} > 2λ). So to find λ such that P(X ≤25) ≥0.95, we can set 2λ = χ²_{2*26, 0.95}.Wait, let me think. The chi-squared approximation for Poisson CDF is given by:P(X ≤ k) ≈ P(χ²_{2(k+1)} > 2λ)So to have P(X ≤25) ≥0.95, we need 2λ ≤ χ²_{52, 0.95}Looking up the chi-squared table for 52 degrees of freedom and 0.95 quantile.Wait, actually, the 0.95 quantile of χ²_{52} is the value such that P(χ²_{52} ≤ x) = 0.95. So we need 2λ ≤ x, where x is the 0.95 quantile of χ²_{52}.Looking up chi-squared tables or using a calculator, the 0.95 quantile for χ²_{52} is approximately 67.505.So 2λ ≤ 67.505 => λ ≤ 33.7525.Wait, that can't be right because earlier estimates were around 18. So maybe I got the direction wrong.Wait, actually, the formula is P(X ≤k) ≈ P(χ²_{2(k+1)} > 2λ). So to have P(X ≤25) ≥0.95, we need P(χ²_{52} > 2λ) ≥0.95, which means that 2λ ≤ the 0.05 quantile of χ²_{52}.Wait, no, because P(χ²_{52} > 2λ) ≥0.95 implies that 2λ ≤ the 0.05 quantile of χ²_{52}.Wait, let me clarify. If P(χ²_{52} > 2λ) ≥0.95, then 2λ must be less than or equal to the value x such that P(χ²_{52} > x) =0.95, which is the 0.05 quantile.Looking up the 0.05 quantile for χ²_{52}, which is approximately 36.782.So 2λ ≤36.782 => λ ≤18.391.That's closer to our earlier estimate of around 18.42.So using the chi-squared approximation, λ ≈18.39.So that seems consistent with the normal approximation with continuity correction.Therefore, the maximum λ is approximately 18.39.But since we need P(X >25) <0.05, which is equivalent to P(X ≤25) ≥0.95, and using the chi-squared approximation, we get λ ≈18.39.So rounding to two decimal places, λ ≈18.39.But let me check if this is accurate.Alternatively, perhaps using the exact Poisson calculation with λ=18.39.But without computational tools, it's hard to compute the exact CDF.Alternatively, perhaps using the relationship with the gamma function.Wait, another approach: use the fact that for Poisson, the CDF can be approximated using the normal distribution with continuity correction, which gave us λ≈18.42, and the chi-squared approximation gave us λ≈18.39. These are very close, so I think λ≈18.4 is a good estimate.Therefore, the maximum λ is approximately 18.4.But let me check if λ=18.4 gives P(X >25) <0.05.Using the normal approximation with continuity correction:Z = (25.5 -18.4)/sqrt(18.4) ≈7.1/4.29≈1.655The z-score of 1.655 corresponds to a probability of about 0.049, which is just under 0.05. So that seems to fit.Therefore, λ≈18.4 is the maximum value.So for the first problem, the maximum λ is approximately 18.4.Now, moving on to the second problem: The IT manager predicts that the system will improve efficiency by reducing the average time spent on resolving each request by 30%. The current average time per request is μ hours. The system must handle up to 40 requests per hour with an average resolution time of less than 1 hour. Find the maximum value of μ that meets this condition.Okay, so currently, the average time per request is μ hours. After the system improvement, the average time becomes μ*(1 - 0.30) = 0.7μ hours.The system must handle up to 40 requests per hour. So the total time spent per hour should be less than 1 hour.Wait, let me parse this carefully."the system must handle up to 40 requests per hour with an average resolution time of less than 1 hour"So, if the system handles 40 requests per hour, the average time per request must be less than 1 hour. So the total time per hour is 40 requests * average time per request < 1 hour.But wait, that can't be, because 40 requests * average time per request would be the total time, but total time per hour can't be less than 1 hour if the system is handling 40 requests. Wait, that doesn't make sense. Maybe I misinterpret.Wait, perhaps it's the average time per request must be less than 1 hour, regardless of the number of requests. But the system must handle up to 40 requests per hour.Wait, maybe it's that the system can handle up to 40 requests per hour, and for each request, the average resolution time is less than 1 hour.But that seems a bit conflicting because if the system can handle 40 requests per hour, then the average time per request is 1/40 hours, which is 1.5 minutes. But that seems too fast.Wait, perhaps the total time spent on all requests per hour must be less than 1 hour. So if the system handles up to 40 requests per hour, the total time spent on all requests is 40 * average time per request, and this must be less than 1 hour.So,40 * (0.7μ) < 1Therefore,0.7μ < 1/40μ < (1/40)/0.7 ≈ (0.025)/0.7 ≈0.0357 hours.Wait, 0.0357 hours is about 2.14 minutes. That seems very fast, but maybe that's correct.Wait, let me think again.If the system must handle up to 40 requests per hour, and the average resolution time per request is less than 1 hour, then:Average resolution time per request <1 hour.But the system can handle up to 40 requests per hour, so the total time spent per hour is 40 * average time per request.But if the average time per request is less than 1 hour, then the total time per hour is less than 40 hours, which is not useful.Wait, perhaps I misinterpret the condition.Wait, maybe the system must handle up to 40 requests per hour, meaning that the maximum number of requests it can handle is 40 per hour, and for that maximum load, the average resolution time per request must be less than 1 hour.So, for 40 requests per hour, the average time per request is 1/40 hours, which is 1.5 minutes. But the system improves efficiency by 30%, so the new average time is 0.7μ.Wait, maybe the condition is that the system can handle 40 requests per hour, and the average time per request is less than 1 hour. So,40 requests per hour * average time per request <1 hour.So,40 * (0.7μ) <1Thus,0.7μ <1/40μ < (1/40)/0.7 ≈0.0357 hours.But 0.0357 hours is 2.14 minutes, which seems very fast. Is that correct?Alternatively, perhaps the condition is that the average time per request after improvement is less than 1 hour, regardless of the number of requests. So,0.7μ <1Thus,μ <1/0.7 ≈1.4286 hours.But that seems too lenient because the system must handle up to 40 requests per hour. So if μ is 1.4286 hours, then the total time per hour would be 40 *1.4286≈57.14 hours, which is way more than 1 hour.Wait, that can't be. So perhaps the correct interpretation is that the system must handle up to 40 requests per hour, and for that load, the average resolution time per request is less than 1 hour.So, for 40 requests per hour, the average time per request is 1/40 hours, which is 1.5 minutes. But the system improves the average time by 30%, so the new average time is 0.7μ.Wait, that seems conflicting. Let me try to rephrase.The current average time per request is μ hours. After the system improvement, it becomes 0.7μ hours. The system must be able to handle up to 40 requests per hour, and for that maximum load, the average resolution time per request must be less than 1 hour.Wait, that doesn't make sense because if the system handles 40 requests per hour, the average time per request is 1/40 hours, regardless of μ. So perhaps the condition is that the average time per request after improvement is less than 1 hour, and the system can handle up to 40 requests per hour.So,0.7μ <1Thus,μ <1/0.7≈1.4286 hours.But then, the system can handle up to 40 requests per hour, so the total time per hour is 40 *0.7μ <40*1=40 hours, which is not a constraint.Wait, perhaps the condition is that the system can handle up to 40 requests per hour, and the average time per request is less than 1 hour. So,40 * average time per request <1 hour.But average time per request after improvement is 0.7μ, so:40 *0.7μ <1Thus,28μ <1μ <1/28≈0.0357 hours, which is about 2.14 minutes.That seems very fast, but perhaps that's the correct interpretation.Alternatively, maybe the system must handle up to 40 requests per hour, meaning that the maximum number of requests it can process is 40 per hour, and for each request, the average resolution time is less than 1 hour.But that would mean that the total time per hour is 40 *1=40 hours, which is impossible because the system can only process 40 requests in 1 hour.Wait, that doesn't make sense. So perhaps the correct interpretation is that the system must handle up to 40 requests per hour, and for each request, the average resolution time is less than 1 hour. But that would mean that the total time per hour is 40 *1=40 hours, which is impossible because the system can only process 40 requests in 1 hour.Wait, that can't be. So perhaps the condition is that the average time per request after improvement is less than 1 hour, and the system can handle up to 40 requests per hour.So,0.7μ <1Thus,μ <1/0.7≈1.4286 hours.But then, the system can handle up to 40 requests per hour, so the total time per hour is 40 *0.7μ <40*1=40 hours, which is not a constraint.Wait, I'm getting confused. Let me try to rephrase the problem.The IT manager predicts that the system will improve efficiency by reducing the average time spent on resolving each request by 30%. So, new average time is 0.7μ.The system must handle up to 40 requests per hour with an average resolution time of less than 1 hour.So, for the system to handle 40 requests per hour, the average resolution time per request must be less than 1 hour.So,0.7μ <1Thus,μ <1/0.7≈1.4286 hours.But wait, if the system handles 40 requests per hour, the total time spent per hour is 40 *0.7μ.But the problem says "with an average resolution time of less than 1 hour". So, does that mean the average time per request is less than 1 hour, or the total time per hour is less than 1 hour?I think it's the average time per request. So,0.7μ <1Thus,μ <1/0.7≈1.4286 hours.But then, the system can handle up to 40 requests per hour, so the total time per hour is 40 *0.7μ <40*1=40 hours, which is not a constraint.Wait, but the problem says "the system must handle up to 40 requests per hour with an average resolution time of less than 1 hour". So, perhaps the average resolution time per request is less than 1 hour, and the system can handle up to 40 requests per hour.So, the average resolution time per request is 0.7μ <1, so μ <1.4286 hours.But then, the system can handle up to 40 requests per hour, which is a separate condition. So, the maximum μ is 1.4286 hours.But wait, that seems too high because if μ is 1.4286 hours, then the new average time is 1 hour, which is the threshold. But the problem says "less than 1 hour", so μ must be less than 1.4286.Wait, but let me think again.If the system must handle up to 40 requests per hour, and for each request, the average resolution time is less than 1 hour, then:For each request, 0.7μ <1 => μ <1.4286.But the system can handle 40 requests per hour, so the total time per hour is 40 *0.7μ <40*1=40 hours, which is not a constraint because the system is only operating for 1 hour.Wait, that doesn't make sense. The total time spent per hour can't exceed 1 hour because the system is only running for 1 hour. So, if the system handles 40 requests in 1 hour, the total time spent is 40 * average time per request, which must be less than or equal to 1 hour.Wait, that makes more sense.So,40 *0.7μ <1Thus,28μ <1μ <1/28≈0.0357 hours.Which is approximately 2.14 minutes.So, the maximum μ is approximately 0.0357 hours.But that seems very fast. Let me check.If μ is 0.0357 hours, which is about 2.14 minutes, then the new average time is 0.7*0.0357≈0.025 hours, which is about 1.5 minutes.So, for 40 requests, the total time is 40*0.025=1 hour, which meets the condition.Therefore, the maximum μ is 1/28≈0.0357 hours.So, converting that to minutes, 0.0357*60≈2.14 minutes.Therefore, the maximum μ is approximately 0.0357 hours.But let me express it as a fraction.1/28≈0.0357.So, μ <1/28 hours.Therefore, the maximum μ is 1/28 hours.So, to answer the second problem, the maximum μ is 1/28 hours.But let me double-check.If μ=1/28 hours, then the new average time is 0.7*(1/28)=0.7/28=0.025 hours.Then, 40 requests would take 40*0.025=1 hour, which meets the condition of less than 1 hour.Wait, but the problem says "less than 1 hour", so we need 40*0.7μ <1.Thus, μ <1/(40*0.7)=1/28≈0.0357 hours.So, μ must be less than 1/28 hours.Therefore, the maximum μ is 1/28 hours.So, summarizing:1. The maximum λ is approximately 18.4.2. The maximum μ is 1/28 hours.But let me express them in the required format.For the first problem, I think the exact value can be found using the chi-squared approximation, which gave us λ≈18.39, which is approximately 18.4.For the second problem, the exact value is μ=1/28 hours.So, final answers:1. λ≈18.42. μ=1/28 hours≈0.0357 hours.But let me check if 1/28 is the exact value.Yes, because 40*0.7μ=1 => μ=1/(40*0.7)=1/28.So, exact value is 1/28.Therefore, the answers are:1. λ≈18.42. μ=1/28 hours.But let me write them in boxed form.For the first problem, since it's an approximate value, I can write it as 18.4.For the second problem, it's an exact fraction, so 1/28.So,1. boxed{18.4}2. boxed{dfrac{1}{28}}</think>