answer：First, I need to calculate the total time the cyclist takes to complete the 120 km route. The route is divided into two parts: 70 km of flat terrain and 50 km of hilly terrain.For the flat terrain, the cyclist's speed is 25 km/h. To find the time taken for this part, I'll divide the distance by the speed: 70 km ÷ 25 km/h = 2.8 hours.For the hilly terrain, the cyclist's speed is 15 km/h. Similarly, the time taken for this part is 50 km ÷ 15 km/h ≈ 3.333 hours.Adding both times together gives the total time: 2.8 hours + 3.333 hours ≈ 6.133 hours, which is approximately 6 hours and 8 minutes.Next, the cyclist wants to complete the entire route in 5 hours. The time taken on the flat terrain remains the same at 2.8 hours. Therefore, the time available for the hilly terrain is 5 hours - 2.8 hours = 2.2 hours.To find the new required speed on the hilly terrain, I'll divide the hilly distance by the available time: 50 km ÷ 2.2 hours ≈ 22.73 km/h. This means the cyclist needs to increase their average speed on the hilly terrain to approximately 22.73 km/h to achieve the desired total time.

answer：Alright, so I have this problem about designing a new park in the shape of an ellipse. The town has a history of geometric gardens, specifically elliptical ones with a major to minor axis ratio of 3:2. The perimeter of the ellipse needs to be exactly 500 meters, and there's a formula provided to approximate the circumference. Then, there's a second part about a circular fountain that should take up 15% of the park's area, leaving 85% free. Hmm, okay, let's break this down step by step.Starting with part 1: I need to find the lengths of the major and minor axes given the perimeter is 500 meters and the ratio of major to minor is 3:2. The formula given is ( P approx pi left( 3(a + b) - sqrt{(3a + b)(a + 3b)} right) ), where ( a ) and ( b ) are the semi-major and semi-minor axes. First, since the ratio of major to minor is 3:2, that means if the major axis is 3x, the minor axis is 2x. But wait, in the formula, ( a ) and ( b ) are the semi-axes, so actually, the major axis length is 2a and minor is 2b. So, the ratio of major to minor is (2a)/(2b) = a/b = 3/2. Therefore, a = (3/2)b. Okay, that makes sense.So, let's denote the semi-major axis as a and semi-minor as b. Given a/b = 3/2, so a = (3/2)b. So, if I can express everything in terms of b, maybe I can solve for b first.The perimeter is given as 500 meters, so plugging into the formula:500 ≈ π [3(a + b) - sqrt{(3a + b)(a + 3b)}]But since a = (3/2)b, let's substitute that in:500 ≈ π [3((3/2)b + b) - sqrt{(3*(3/2)b + b)( (3/2)b + 3b)} ]Let me compute each part step by step.First, compute 3(a + b):a + b = (3/2)b + b = (5/2)bSo, 3(a + b) = 3*(5/2)b = (15/2)bNext, compute the square root term: sqrt{(3a + b)(a + 3b)}First, compute 3a + b:3a = 3*(3/2)b = (9/2)bSo, 3a + b = (9/2)b + b = (11/2)bThen, compute a + 3b:a = (3/2)b, so a + 3b = (3/2)b + 3b = (3/2 + 6/2)b = (9/2)bSo, the product inside the square root is (11/2)b * (9/2)b = (99/4)b²Therefore, sqrt{(99/4)b²} = (sqrt(99)/2)bSimplify sqrt(99): sqrt(9*11) = 3*sqrt(11). So, sqrt(99)/2 = (3*sqrt(11))/2So, putting it all back into the formula:500 ≈ π [ (15/2)b - (3*sqrt(11)/2)b ]Factor out b/2:500 ≈ π [ (15 - 3*sqrt(11))/2 * b ]So, 500 ≈ (π/2)(15 - 3*sqrt(11)) * bLet me compute the numerical value of (15 - 3*sqrt(11)):sqrt(11) is approximately 3.3166, so 3*sqrt(11) ≈ 9.9498Therefore, 15 - 9.9498 ≈ 5.0502So, 500 ≈ (π/2)*5.0502 * bCompute (π/2)*5.0502:π ≈ 3.1416, so π/2 ≈ 1.57081.5708 * 5.0502 ≈ Let's compute 1.5708*5 = 7.854, and 1.5708*0.0502 ≈ 0.0788Adding together: 7.854 + 0.0788 ≈ 7.9328So, 500 ≈ 7.9328 * bTherefore, solving for b:b ≈ 500 / 7.9328 ≈ Let's compute that.500 divided by 7.9328. Let me see, 7.9328 * 63 ≈ 7.9328*60=475.968, plus 7.9328*3≈23.7984, total ≈ 499.7664. So, 7.9328*63 ≈ 499.7664, which is very close to 500.So, b ≈ 63 meters.Therefore, since a = (3/2)b, a = (3/2)*63 = 94.5 meters.But wait, hold on. a and b are semi-axes, so the major axis is 2a and minor axis is 2b.So, major axis length is 2a = 2*94.5 = 189 meters.Minor axis length is 2b = 2*63 = 126 meters.Wait, let me verify if that makes sense.But before I proceed, let me check my calculations again because sometimes approximations can lead to errors.So, starting from the formula:500 ≈ π [3(a + b) - sqrt{(3a + b)(a + 3b)}]With a = 94.5 and b = 63.Compute 3(a + b): 3*(94.5 + 63) = 3*(157.5) = 472.5Compute sqrt{(3a + b)(a + 3b)}:3a + b = 3*94.5 + 63 = 283.5 + 63 = 346.5a + 3b = 94.5 + 3*63 = 94.5 + 189 = 283.5So, the product is 346.5 * 283.5. Let me compute that.346.5 * 283.5: Let's compute 346.5 * 283.5First, 346.5 * 200 = 69,300346.5 * 80 = 27,720346.5 * 3.5 = let's compute 346.5 * 3 = 1,039.5 and 346.5 * 0.5 = 173.25, so total 1,039.5 + 173.25 = 1,212.75Adding all together: 69,300 + 27,720 = 97,020; 97,020 + 1,212.75 = 98,232.75So, sqrt(98,232.75). Let me compute that.What's sqrt(98,232.75)? Let's see, 313^2 = 97,969, 314^2 = 98,596. So, sqrt(98,232.75) is between 313 and 314.Compute 313.5^2: (313 + 0.5)^2 = 313^2 + 2*313*0.5 + 0.5^2 = 97,969 + 313 + 0.25 = 98,282.25But 98,282.25 is larger than 98,232.75, so the square root is less than 313.5.Compute 313.25^2: 313 + 0.25(313 + 0.25)^2 = 313^2 + 2*313*0.25 + 0.25^2 = 97,969 + 156.5 + 0.0625 = 98,125.5625Still less than 98,232.75. So, 313.25^2 = 98,125.5625Difference: 98,232.75 - 98,125.5625 = 107.1875Each increment of x by 0.1 increases x^2 by roughly 2*313.25*0.1 + (0.1)^2 ≈ 62.65 + 0.01 ≈ 62.66So, to cover 107.1875, need approximately 107.1875 / 62.66 ≈ 1.71 increments of 0.1, so about 0.171So, approximate sqrt ≈ 313.25 + 0.171 ≈ 313.421So, sqrt ≈ 313.421Therefore, the term inside the formula is 472.5 - 313.421 ≈ 159.079Multiply by π: 159.079 * π ≈ 159.079 * 3.1416 ≈ Let's compute:159.079 * 3 = 477.237159.079 * 0.1416 ≈ Let's compute 159.079 * 0.1 = 15.9079159.079 * 0.04 = 6.36316159.079 * 0.0016 ≈ 0.2545Adding together: 15.9079 + 6.36316 ≈ 22.271, plus 0.2545 ≈ 22.5255So, total π*159.079 ≈ 477.237 + 22.5255 ≈ 499.7625 metersWow, that's really close to 500 meters. So, the approximation gives us 499.7625, which is almost 500. So, our values of a = 94.5 and b = 63 are correct.Therefore, the semi-major axis is 94.5 meters, semi-minor is 63 meters. So, the major axis is 189 meters, minor axis is 126 meters.So, that's part 1 done.Moving on to part 2: The planner wants a circular fountain at the center such that the area not occupied by the fountain is 85% of the total park area. So, the area of the ellipse is A = πab, and the area of the fountain is πr². So, the area not occupied is A - πr² = 0.85A. Therefore, πr² = A - 0.85A = 0.15A.So, πr² = 0.15πab. Dividing both sides by π: r² = 0.15ab. Therefore, r = sqrt(0.15ab).We already have a and b from part 1: a = 94.5, b = 63. So, compute ab first.ab = 94.5 * 63. Let me compute that.94.5 * 60 = 5,67094.5 * 3 = 283.5Total ab = 5,670 + 283.5 = 5,953.5So, ab = 5,953.5Therefore, r² = 0.15 * 5,953.5 = Let's compute 0.15 * 5,953.50.1 * 5,953.5 = 595.350.05 * 5,953.5 = 297.675Adding together: 595.35 + 297.675 = 893.025So, r² = 893.025Therefore, r = sqrt(893.025). Let's compute that.What's sqrt(893.025)?Well, 29^2 = 841, 30^2=900. So, sqrt(893.025) is between 29 and 30.Compute 29.8^2: 29^2 + 2*29*0.8 + 0.8^2 = 841 + 46.4 + 0.64 = 888.0429.8^2 = 888.0429.9^2: 29.8^2 + 2*29.8*0.1 + 0.1^2 = 888.04 + 5.96 + 0.01 = 894.01But 893.025 is between 888.04 and 894.01.Compute 29.8 + x)^2 = 893.025Let me set x as the decimal beyond 29.8.(29.8 + x)^2 = 893.025Expanding: 29.8² + 2*29.8*x + x² = 893.025We know 29.8² = 888.04So, 888.04 + 59.6x + x² = 893.025Subtract 888.04: 59.6x + x² = 4.985Assuming x is small, x² is negligible, so approximately 59.6x ≈ 4.985Therefore, x ≈ 4.985 / 59.6 ≈ 0.0836So, x ≈ 0.0836Therefore, sqrt(893.025) ≈ 29.8 + 0.0836 ≈ 29.8836 metersSo, approximately 29.88 meters.But let me check 29.88^2:29.88 * 29.88:First, compute 30*30 = 900Subtract 0.12*30 + 0.12*30 - (0.12)^2Wait, maybe better to compute directly:29.88 * 29.88:= (30 - 0.12)^2= 30² - 2*30*0.12 + 0.12²= 900 - 7.2 + 0.0144= 892.8144But we have 893.025, so it's a bit higher.Difference: 893.025 - 892.8144 = 0.2106So, we need a little more.Compute the derivative: d/dx (x²) = 2x. At x=29.88, derivative is 59.76.So, to get an additional 0.2106, need delta_x ≈ 0.2106 / 59.76 ≈ 0.00352So, total x ≈ 29.88 + 0.00352 ≈ 29.8835 metersSo, approximately 29.8835 meters.So, rounding to a reasonable decimal place, maybe 29.88 meters.But let's see, the problem says "the radius of the fountain." It doesn't specify the precision, so perhaps we can leave it as sqrt(0.15ab), but since we have the numerical value, 29.88 meters is fine.Alternatively, if we compute sqrt(893.025) more accurately:Let me use a calculator approach.We have 29.88^2 = 892.814429.89^2 = (29.88 + 0.01)^2 = 892.8144 + 2*29.88*0.01 + 0.0001 = 892.8144 + 0.5976 + 0.0001 = 893.4111But 893.4111 is higher than 893.025, so the square root is between 29.88 and 29.89.Compute 893.025 - 892.8144 = 0.2106The difference between 29.89^2 and 29.88^2 is 893.4111 - 892.8144 = 0.5967So, 0.2106 / 0.5967 ≈ 0.3527So, the square root is approximately 29.88 + 0.3527*(0.01) ≈ 29.88 + 0.003527 ≈ 29.8835 meters, which is the same as before.So, approximately 29.8835 meters, which we can round to 29.88 meters.Therefore, the radius of the fountain is approximately 29.88 meters.But let me check if I did everything correctly.We had the area of the ellipse: A = πab = π*94.5*63 ≈ π*5953.5 ≈ 18,706.7 square meters (since π ≈ 3.1416, 5953.5*3.1416 ≈ 18,706.7)Then, the area of the fountain is 0.15*A ≈ 0.15*18,706.7 ≈ 2,806.005 square meters.Compute πr² = 2,806.005So, r² = 2,806.005 / π ≈ 2,806.005 / 3.1416 ≈ 893.025, which matches our earlier calculation.Therefore, r ≈ sqrt(893.025) ≈ 29.88 meters.So, that seems consistent.Therefore, the radius of the fountain is approximately 29.88 meters.But the problem might expect an exact form or a fractional form? Let me see.Wait, 0.15ab = 0.15*94.5*63 = 0.15*5953.5 = 893.025, so r = sqrt(893.025). Since 893.025 is 893.025, which is 893.025 = 893025/1000. Let me see if that can be simplified.But 893025 divided by 25 is 35721, and 1000 divided by 25 is 40. So, 893025/1000 = 35721/40. So, sqrt(35721/40) = sqrt(35721)/sqrt(40). sqrt(35721) is 189, because 189^2 = 35721. So, sqrt(35721/40) = 189 / sqrt(40) = 189 / (2*sqrt(10)) = (189*sqrt(10))/20.Simplify 189/20: 189 divided by 20 is 9.45. So, 9.45*sqrt(10). But sqrt(10) is approximately 3.1623, so 9.45*3.1623 ≈ 29.88, which matches our earlier decimal.So, exact form is (189*sqrt(10))/20 meters, which is approximately 29.88 meters.But since the problem doesn't specify, either form is acceptable, but likely expects the approximate decimal.So, summarizing:1. Major axis: 189 meters, minor axis: 126 meters.2. Radius of fountain: approximately 29.88 meters.I think that's it.**Final Answer**1. The lengths of the major and minor axes are boxed{189} meters and boxed{126} meters, respectively.2. The radius of the fountain is boxed{29.88} meters.

answer：Okay, so I'm trying to help this project manager optimize their team's task allocation and ensure quality. Let's break this down step by step.First, the problem is about resource allocation. We have five developers, each can handle up to three tasks simultaneously. Each task has different completion times based on the developer's expertise. The goal is to minimize the total time taken to complete all tasks, T. Hmm, so each developer can work on up to three tasks at the same time, but the tasks have different completion times. I think this is a scheduling problem where we need to assign tasks to developers in such a way that the makespan (the total time taken) is minimized. Since each developer can handle multiple tasks, the makespan would be the maximum completion time across all developers.Wait, but each developer has three different task completion times. So, for example, Developer A has tasks that take 2x, 3x, and 4x time. I guess each developer can work on up to three tasks, each with their own time. So, if a developer is assigned multiple tasks, the total time they take would be the sum of their individual task times? Or is it the maximum time among the tasks they are handling?Wait, no, because tasks are done simultaneously. So if a developer is assigned multiple tasks, they can work on them at the same time, so the total time for that developer would be the maximum of the individual task times. That makes more sense because they can multitask. So, for a developer, if they are assigned tasks with times t1, t2, t3, their total time is max(t1, t2, t3). But wait, actually, in reality, if a developer is handling multiple tasks, they might not be able to work on all simultaneously. Maybe each task requires their full attention, so they have to do them sequentially. Hmm, the problem says "up to three different tasks simultaneously." So, if they can do them simultaneously, then the total time is the maximum of the individual task times. If they can't, then it's the sum. But the problem says "simultaneously," so I think it's the maximum. So, for each developer, if they are assigned k tasks (k <=3), their total time is the maximum of the k task times. Wait, but each developer has three different task completion times. So, for each developer, they have three tasks with different times. So, if a developer is assigned all three tasks, their total time is the maximum of t1, t2, t3. If they are assigned two tasks, it's the maximum of the two, and if one task, just that time.But the problem is about assigning tasks to developers. Wait, but how many tasks are there? The problem doesn't specify the number of tasks. It just says each developer can handle up to three tasks. So, perhaps the total number of tasks is 15 (5 developers * 3 tasks each). But that might not be the case. Maybe the number of tasks is variable, but each developer can take up to three tasks.Wait, the problem says "each developer can work on up to three different tasks simultaneously." So, each developer can handle up to three tasks, but the tasks are not specified. So, perhaps the number of tasks is more than 5, and we need to assign them to the developers, each developer can take up to three tasks, and the total time is the maximum over all developers' maximum task times.But without knowing the number of tasks, it's hard to proceed. Wait, maybe the tasks are the same for each developer? Like, each developer has three tasks, but the tasks are different across developers. So, total tasks are 15? Or maybe each developer can handle up to three tasks, but the tasks are the same across developers. Hmm, the problem is a bit unclear.Wait, let's read the problem again. It says, "each developer can work on up to three different tasks simultaneously with task completion times varying based on their expertise." So, each developer has three different tasks, each with their own completion time. So, for example, Developer A has tasks A1, A2, A3 with times 2x, 3x, 4x. Similarly for others.So, perhaps each developer has their own set of three tasks, and the project manager needs to assign these tasks to the developers in such a way that the total time is minimized. But wait, the tasks are specific to each developer, so maybe the project manager can assign tasks to different developers to optimize the total time.Wait, no, that might not make sense. If each developer has their own set of tasks, then assigning them to other developers might not be possible because the tasks are specific to their expertise. Hmm, maybe I'm misunderstanding.Alternatively, perhaps the tasks are the same across developers, but each developer has different completion times for each task. So, for example, Task 1 can be done by Developer A in 2x time, by Developer B in 3x, etc. So, the project manager needs to assign tasks to developers, considering that each developer can handle up to three tasks, and each task can be assigned to any developer, but with different completion times.Yes, that makes more sense. So, the tasks are the same, but each developer has different times for each task. So, the project manager needs to assign tasks to developers, with each developer handling up to three tasks, and the goal is to minimize the makespan, which is the maximum completion time across all developers.So, in that case, the problem is similar to a job shop scheduling problem where tasks can be assigned to multiple machines (developers), each machine can handle multiple jobs (tasks), but each job has a different processing time on each machine.So, to model this, we can think of it as a scheduling problem with multiple machines (developers) and multiple jobs (tasks). Each job can be assigned to any machine, and each machine can handle up to three jobs. The goal is to assign jobs to machines such that the makespan (maximum completion time across all machines) is minimized.But the problem is, how many tasks are there? The problem doesn't specify. It just says each developer can handle up to three tasks. So, perhaps the number of tasks is variable, but the project manager needs to assign tasks to developers, each developer can take up to three tasks, and the total time is the maximum of the developers' total times.But without knowing the number of tasks, it's hard to proceed. Wait, maybe the tasks are the same as the number of developers multiplied by three? So, 5 developers * 3 tasks = 15 tasks? That might be the case.Alternatively, perhaps the tasks are the same across developers, but each developer has different times for each task. So, for example, Task 1 can be done by Developer A in 2x, by Developer B in 3x, etc. So, the project manager needs to assign each task to a developer, with each developer handling up to three tasks, and the goal is to minimize the makespan.Yes, that seems to be the case. So, the tasks are the same, but each developer has different times for each task. So, the project manager needs to assign tasks to developers, with each developer handling up to three tasks, and the total time is the maximum of the developers' total times.Wait, but the problem says "each developer can work on up to three different tasks simultaneously." So, each developer can handle up to three tasks at the same time, meaning that the time taken by a developer is the maximum of the times of the tasks they are assigned. Because they can work on them simultaneously.So, for example, if Developer A is assigned Task 1 (2x) and Task 2 (3x), their total time is max(2x, 3x) = 3x. If they are assigned three tasks, their total time is the maximum of the three.Therefore, the problem is to assign tasks to developers, each developer can take up to three tasks, and the total time is the maximum of the maximum task times assigned to each developer. The goal is to minimize this total time.But how many tasks are there? The problem doesn't specify. It just says each developer can handle up to three tasks. So, perhaps the number of tasks is variable, but the project manager needs to assign tasks to developers, each developer can take up to three tasks, and the total time is the maximum of the developers' maximum task times.Wait, but without knowing the number of tasks, it's impossible to model this. Maybe the tasks are the same as the number of developers multiplied by three, so 15 tasks. But that's an assumption.Alternatively, maybe the tasks are the same across developers, but each developer has different times for each task. So, for example, Task 1 can be done by Developer A in 2x, by Developer B in 3x, etc. So, the project manager needs to assign each task to a developer, with each developer handling up to three tasks, and the total time is the maximum of the developers' maximum task times.Yes, that makes sense. So, the tasks are the same, but each developer has different times for each task. So, the project manager needs to assign tasks to developers, with each developer handling up to three tasks, and the total time is the maximum of the developers' maximum task times.Therefore, the problem is to assign tasks to developers such that each developer handles up to three tasks, and the makespan (maximum time across all developers) is minimized.But how many tasks are there? The problem doesn't specify. It just says each developer can handle up to three tasks. So, perhaps the number of tasks is variable, but the project manager needs to assign tasks to developers, each developer can take up to three tasks, and the total time is the maximum of the developers' maximum task times.Wait, maybe the tasks are the same as the number of developers multiplied by three, so 15 tasks. That would make sense because each developer can handle three tasks, so 5 developers * 3 tasks = 15 tasks.So, assuming there are 15 tasks, each developer can handle three tasks, and each task has a different completion time based on the developer. The goal is to assign each task to a developer such that each developer gets exactly three tasks, and the makespan (maximum time across all developers) is minimized.But wait, the problem doesn't specify that each developer must handle exactly three tasks. It just says up to three. So, some developers might handle fewer tasks if needed.But to minimize the makespan, it's better to balance the load, so probably each developer will handle three tasks.But let's proceed with that assumption.So, the problem is to assign 15 tasks to 5 developers, each developer handling up to three tasks, with each task having a specific completion time for each developer, and the goal is to minimize the makespan.This is a classic scheduling problem, specifically a parallel machine scheduling problem with the objective of minimizing makespan.In such problems, the goal is to assign jobs to machines (developers) such that the makespan is minimized. Since each developer can handle multiple tasks simultaneously, the time taken by a developer is the maximum of the task times assigned to them.Therefore, the problem can be formulated as an integer programming problem.Let me define the variables:Let’s denote:- Let T be the total time (makespan) we want to minimize.- Let’s define a binary variable x_{ij} which is 1 if task i is assigned to developer j, 0 otherwise.But wait, in our case, each task can be assigned to only one developer, and each developer can handle up to three tasks. So, the constraints would be:For each task i, sum_{j=1 to 5} x_{ij} = 1 (each task is assigned to exactly one developer).For each developer j, sum_{i=1 to 15} x_{ij} <= 3 (each developer can handle up to three tasks).And the objective is to minimize T, such that for each developer j, T >= sum_{i=1 to 15} t_{ij} * x_{ij} / 3? Wait, no, because the tasks are done simultaneously, so the time for developer j is the maximum of the task times assigned to them.Wait, that complicates things because the objective function would involve maximums, which are non-linear.In integer programming, it's challenging to model maximums directly. One common approach is to use the following formulation:For each developer j, let’s define a variable C_j which represents the completion time of developer j. Then, our objective is to minimize T, such that T >= C_j for all j.And for each developer j, C_j must be greater than or equal to the time of any task assigned to them. So, for each task i assigned to developer j, C_j >= t_{ij}.But since tasks are assigned via x_{ij}, we can model this as:For all i, j: C_j >= t_{ij} * x_{ij}But this is still non-linear because C_j is a variable and x_{ij} is a binary variable.An alternative approach is to use the big-M method. For each task i and developer j, we can write:C_j >= t_{ij} - M(1 - x_{ij})Where M is a large enough constant (larger than the maximum possible t_{ij}).This way, if x_{ij} = 1 (task i is assigned to developer j), then C_j >= t_{ij}. If x_{ij} = 0, the constraint becomes C_j >= -M, which is always true since C_j is non-negative.Then, the objective is to minimize T, subject to T >= C_j for all j.So, putting it all together, the integer programming formulation would be:Minimize TSubject to:For each developer j: sum_{i=1 to 15} x_{ij} <= 3For each task i: sum_{j=1 to 5} x_{ij} = 1For each task i and developer j: C_j >= t_{ij} - M(1 - x_{ij})For each developer j: T >= C_jAnd x_{ij} is binary, C_j and T are continuous variables.But wait, in our case, each developer has their own set of tasks with specific times. Wait, no, actually, the tasks are the same across developers, but each developer has different times for each task. So, for example, Task 1 can be done by Developer A in 2x, by Developer B in 3x, etc.Wait, no, the problem states that each developer has their own set of task completion times. For example, Developer A has t_A1=2x, t_A2=3x, t_A3=4x. So, perhaps each developer has three specific tasks assigned to them, and the project manager can assign these tasks to other developers as well.Wait, this is getting confusing. Let me re-examine the problem statement."Each developer can work on up to three different tasks simultaneously with task completion times varying based on their expertise:- Developer A: t_{A1} = 2x, t_{A2} = 3x, t_{A3} = 4x- Developer B: t_{B1} = 3x, t_{B2} = 2.5x, t_{B3} = 4.5x- Developer C: t_{C1} = 4x, t_{C2} = 3x, t_{C3} = 2x- Developer D: t_{D1} = 2x, t_{D2} = 3.5x, t_{D3} = 4x- Developer E: t_{E1} = 3.5x, t_{E2} = 4.5x, t_{E3} = 2.5x"So, each developer has three tasks with specific times. So, perhaps each developer has their own three tasks, and the project manager can assign these tasks to other developers as well. So, for example, Task A1 can be assigned to Developer B, who would take t_{B1} time, which is 3x.Wait, but the problem doesn't specify that the tasks are the same across developers. It just says each developer has their own set of tasks with different times. So, perhaps the tasks are specific to each developer, meaning that Task A1 is only handled by Developer A, Task B1 by Developer B, etc. But that would mean that each developer must handle their own tasks, and the project manager can't assign tasks across developers. That would make the problem trivial because each developer just handles their own tasks, and the makespan is the maximum of the maximum times of each developer.But that seems too simple, and the problem mentions optimizing the allocation, implying that tasks can be reassigned.Therefore, I think the correct interpretation is that there are 15 tasks in total, each with a specific completion time for each developer. So, Task 1 can be done by Developer A in 2x, by Developer B in 3x, etc. So, the project manager needs to assign each of the 15 tasks to a developer, with each developer handling up to three tasks, and the goal is to minimize the makespan.But wait, the problem doesn't specify that there are 15 tasks. It just says each developer can handle up to three tasks. So, perhaps the number of tasks is variable, but the project manager needs to assign tasks to developers, each developer can handle up to three tasks, and the total time is the maximum of the developers' maximum task times.But without knowing the number of tasks, it's impossible to model this. Therefore, perhaps the tasks are the same as the number of developers multiplied by three, so 15 tasks. That would make sense because each developer can handle three tasks, so 5 developers * 3 tasks = 15 tasks.Therefore, assuming there are 15 tasks, each with a specific completion time for each developer, the project manager needs to assign each task to a developer, with each developer handling up to three tasks, and the goal is to minimize the makespan.So, the problem is to assign 15 tasks to 5 developers, each developer handling up to three tasks, with each task having a specific completion time for each developer, and the goal is to minimize the makespan.This is a classic scheduling problem, specifically a parallel machine scheduling problem with the objective of minimizing makespan.In such problems, the goal is to assign jobs to machines (developers) such that the makespan is minimized. Since each developer can handle multiple tasks, the time taken by a developer is the maximum of the task times assigned to them.Therefore, the problem can be formulated as an integer programming problem.Let me define the variables:- Let T be the total time (makespan) we want to minimize.- Let’s define a binary variable x_{ij} which is 1 if task i is assigned to developer j, 0 otherwise.Constraints:1. Each task is assigned to exactly one developer: For each task i, sum_{j=1 to 5} x_{ij} = 12. Each developer can handle up to three tasks: For each developer j, sum_{i=1 to 15} x_{ij} <= 33. The completion time for each developer j must be at least the time of any task assigned to them: For each task i and developer j, C_j >= t_{ij} * x_{ij}But since C_j is the maximum of the task times assigned to developer j, we can model this using the big-M method as I thought earlier.So, for each task i and developer j:C_j >= t_{ij} - M(1 - x_{ij})Where M is a large constant, larger than the maximum possible t_{ij}.Then, the objective is to minimize T, subject to T >= C_j for all j.So, the integer programming formulation is:Minimize TSubject to:For each task i:sum_{j=1 to 5} x_{ij} = 1For each developer j:sum_{i=1 to 15} x_{ij} <= 3For each task i and developer j:C_j >= t_{ij} - M(1 - x_{ij})For each developer j:T >= C_jAnd x_{ij} is binary, C_j and T are continuous variables.This formulation should allow us to find the optimal assignment of tasks to developers that minimizes the makespan.Now, moving on to the second part, the quality assurance constraint.Each developer has a probability of introducing bugs during development:- Developer A: p_A = 0.02- Developer B: p_B = 0.03- Developer C: p_C = 0.01- Developer D: p_D = 0.025- Developer E: p_E = 0.015We need to define a mathematical model to determine the expected number of bugs in the final software, using the allocation from the previous sub-problem, and calculate the combined probability of bug-free delivery.Assuming that the bugs introduced by each developer are independent events, the expected number of bugs would be the sum of the probabilities of each developer introducing a bug, multiplied by the number of tasks they are assigned.Wait, but each task could introduce a bug independently. So, if a developer is assigned k tasks, the probability that they introduce at least one bug is 1 - (1 - p_j)^k, where p_j is the probability of introducing a bug per task.But the expected number of bugs would be the sum over all developers of (number of tasks assigned to developer j) * p_j.Because for each task, the probability of introducing a bug is p_j, and the expected number of bugs per task is p_j, so for k tasks, it's k * p_j.Therefore, the expected number of bugs E is:E = sum_{j=1 to 5} (number of tasks assigned to j) * p_jBut in our case, the number of tasks assigned to each developer is variable, depending on the allocation from the first part.Wait, but in the first part, each developer is assigned up to three tasks, so the number of tasks assigned to each developer can be 0, 1, 2, or 3.But in the first part, we assumed that each developer is assigned exactly three tasks to minimize the makespan, but actually, the problem allows up to three tasks, so some developers might have fewer tasks.But in the first part, to minimize the makespan, it's likely that each developer is assigned three tasks, as that would balance the load.But regardless, the expected number of bugs would be the sum over all developers of (number of tasks assigned to j) * p_j.So, if we denote k_j as the number of tasks assigned to developer j, then:E = sum_{j=1 to 5} k_j * p_jBut k_j is determined by the allocation from the first part.Alternatively, if we consider that each task has a probability of introducing a bug, and the bugs are independent, then the probability that the entire software is bug-free is the product of (1 - p_j) for each task, but that's not quite right because each task is handled by a developer, and the developer's probability is per task.Wait, actually, for each task assigned to developer j, the probability that it doesn't introduce a bug is (1 - p_j). Therefore, the probability that all tasks are bug-free is the product over all tasks of (1 - p_j), where p_j is the probability of the developer handling that task.But since tasks are assigned to developers, and each developer has a per-task bug probability, the overall probability of bug-free delivery is the product over all tasks of (1 - p_j), where p_j is the developer's bug probability for that task.But in our case, each developer has a fixed probability p_j per task they handle. So, if a developer handles k tasks, the probability that none of their tasks introduce a bug is (1 - p_j)^k.Therefore, the overall probability of bug-free delivery is the product over all developers of (1 - p_j)^{k_j}, where k_j is the number of tasks assigned to developer j.So, the combined probability of bug-free delivery P is:P = product_{j=1 to 5} (1 - p_j)^{k_j}Where k_j is the number of tasks assigned to developer j.Therefore, the mathematical model for the expected number of bugs is E = sum_{j=1 to 5} k_j * p_j, and the probability of bug-free delivery is P = product_{j=1 to 5} (1 - p_j)^{k_j}.But in the first part, we have an allocation of tasks to developers, which determines k_j for each developer. So, once we have the optimal allocation from the first part, we can compute E and P.Therefore, the steps are:1. Formulate the integer programming problem to minimize the makespan T, assigning tasks to developers with each developer handling up to three tasks.2. Solve this problem to get the optimal allocation, which gives k_j for each developer.3. Use k_j to compute the expected number of bugs E = sum_{j=1 to 5} k_j * p_j.4. Compute the probability of bug-free delivery P = product_{j=1 to 5} (1 - p_j)^{k_j}.So, putting it all together, the optimization problem for the first part is as I formulated above, and the quality assurance model is as described.But wait, in the first part, the tasks are assigned to developers, and each task has a specific completion time for each developer. So, the makespan is determined by the maximum completion time across all developers, where each developer's completion time is the maximum of the task times assigned to them.Therefore, the integer programming formulation needs to capture that.But in the formulation I described earlier, we used C_j as the completion time for developer j, which is the maximum of the task times assigned to them. To model this, we used the big-M method to ensure that C_j >= t_{ij} for any task i assigned to developer j.But in reality, C_j should be the maximum of the task times assigned to developer j. So, another way to model this is to use the following constraints:For each developer j, C_j >= t_{ij} for all tasks i assigned to j.But since we don't know which tasks are assigned to which developers in advance, we need to model this in a way that enforces C_j >= t_{ij} if x_{ij}=1.This is where the big-M method comes in, as I mentioned earlier.So, the formulation is correct.Now, for the second part, once we have the optimal allocation, we can compute the expected number of bugs and the probability of bug-free delivery.But in the first part, the number of tasks is 15, each assigned to a developer, so each developer has k_j tasks, where k_j <=3.Therefore, the expected number of bugs is sum_{j=1 to 5} k_j * p_j.And the probability of bug-free delivery is product_{j=1 to 5} (1 - p_j)^{k_j}.So, that's the model.But wait, in the first part, the tasks are assigned to developers, and each task has a specific completion time for each developer. So, the makespan is determined by the maximum completion time across all developers, which is the maximum of the maximum task times assigned to each developer.Therefore, the integer programming formulation is correct.But to solve this, we would need to use an integer programming solver, which can handle the constraints and find the optimal assignment.Once we have the optimal assignment, we can then compute the expected number of bugs and the probability of bug-free delivery.So, in summary, the optimization problem is an integer program with binary variables x_{ij}, continuous variables C_j and T, and the constraints as described.The quality assurance model is a simple calculation based on the number of tasks assigned to each developer, using their respective bug probabilities.Therefore, the final answer is the formulation of the integer programming problem for the first part and the expected number of bugs and probability of bug-free delivery for the second part.

answer：First, I need to determine the number of sentimental items in each of the 7 rooms following the Fibonacci sequence. The sequence starts with 1 and 1, so the number of items in each room will be: 1, 1, 2, 3, 5, 8, and 13.Next, I'll calculate the total number of items by summing these values: 1 + 1 + 2 + 3 + 5 + 8 + 13, which equals 33 items in total.For the second part, the parent wants to distribute these 33 items into boxes where each box contains a different prime number of items. I'll list the prime numbers less than 33: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31.I need to find the smallest set of these primes that sum up to 33. Starting with the largest prime, 31, and adding the smallest prime, 2, gives a total of 33. Therefore, only two boxes are needed: one with 31 items and another with 2 items.

answer：Alright, so I have this problem about team efficiency and communication. Let me try to break it down step by step.First, the problem is divided into two parts. The first part is about deriving a formula for efficiency E as a function of n and Q. The second part is about finding the value of n that maximizes E, given some constraints and a quadratic function for Q.Starting with part 1. It says that the total number of communications per day is given by C(n) = n(n-1)/2. That makes sense because each member communicates with every other member exactly once, so it's the number of combinations of n members taken 2 at a time, which is indeed n(n-1)/2.Then, the efficiency E is given by E(n, Q) = k * Q * C(n). So, they want a formula for E in terms of n and Q. Since C(n) is already given, and Q is another variable, it seems straightforward. I just need to plug in C(n) into the formula.So, substituting C(n) into E(n, Q), we get:E(n, Q) = k * Q * [n(n - 1)/2]Simplifying that, it would be:E(n, Q) = (k * Q * n(n - 1)) / 2I think that's the formula they're asking for. It's just expressing E in terms of n and Q by plugging in the expression for C(n).Moving on to part 2. This seems more involved. They want to maximize efficiency E, given that n is constrained by n ≤ 15. Also, Q is a quadratic function of n: Q(n) = an² + bn + c, where a = -0.1, b = 1.5, and c = 2.So, first, let's write down Q(n):Q(n) = -0.1n² + 1.5n + 2Now, from part 1, we have E(n, Q) = (k * Q * n(n - 1)) / 2. But since k is a constant, and we're looking to maximize E with respect to n, the constant k won't affect the value of n that maximizes E. So, we can ignore k for the purpose of maximization.Therefore, E(n) = Q(n) * C(n) = Q(n) * [n(n - 1)/2]Substituting Q(n) into this:E(n) = (-0.1n² + 1.5n + 2) * [n(n - 1)/2]Let me compute this step by step.First, let's compute the product:(-0.1n² + 1.5n + 2) * (n² - n)/2Wait, actually, n(n - 1) is n² - n, so:E(n) = [(-0.1n² + 1.5n + 2) * (n² - n)] / 2Let me expand the numerator first:Multiply (-0.1n² + 1.5n + 2) by (n² - n):Let's distribute each term:First, multiply -0.1n² by (n² - n):-0.1n² * n² = -0.1n⁴-0.1n² * (-n) = 0.1n³Next, multiply 1.5n by (n² - n):1.5n * n² = 1.5n³1.5n * (-n) = -1.5n²Then, multiply 2 by (n² - n):2 * n² = 2n²2 * (-n) = -2nNow, let's add all these terms together:-0.1n⁴ + 0.1n³ + 1.5n³ - 1.5n² + 2n² - 2nCombine like terms:-0.1n⁴ + (0.1n³ + 1.5n³) + (-1.5n² + 2n²) - 2nCalculating each:-0.1n⁴ + 1.6n³ + 0.5n² - 2nSo, the numerator is -0.1n⁴ + 1.6n³ + 0.5n² - 2nTherefore, E(n) is this divided by 2:E(n) = (-0.1n⁴ + 1.6n³ + 0.5n² - 2n) / 2Simplify each term:-0.05n⁴ + 0.8n³ + 0.25n² - nSo, E(n) = -0.05n⁴ + 0.8n³ + 0.25n² - nNow, we need to find the value of n that maximizes E(n), where n is an integer (since it's the number of team members) and n ≤ 15.To find the maximum, we can take the derivative of E(n) with respect to n, set it equal to zero, and solve for n. However, since n must be an integer, we'll need to check the integer values around the critical point to find the maximum.First, let's compute the derivative E'(n):E(n) = -0.05n⁴ + 0.8n³ + 0.25n² - nE'(n) = dE/dn = -0.2n³ + 2.4n² + 0.5n - 1We need to solve E'(n) = 0:-0.2n³ + 2.4n² + 0.5n - 1 = 0This is a cubic equation. Solving cubic equations analytically can be complex, so perhaps we can use numerical methods or approximate the solution.Alternatively, since n is an integer between 1 and 15, we can compute E(n) for each integer n from, say, 1 to 15, and find which n gives the maximum E(n). That might be simpler given the constraints.But let's see if we can approximate the critical point first.Let me rewrite the derivative equation:-0.2n³ + 2.4n² + 0.5n - 1 = 0Multiply both sides by -5 to eliminate decimals:n³ - 12n² - 2.5n + 5 = 0Hmm, still not very nice coefficients, but maybe we can try to find rational roots using Rational Root Theorem. Possible rational roots are ±1, ±5.Testing n=1:1 - 12 - 2.5 + 5 = -8.5 ≠ 0n=5:125 - 300 - 12.5 + 5 = -182.5 ≠ 0n= -1:-1 - 12 + 2.5 + 5 = -5.5 ≠ 0n= -5:-125 - 300 + 12.5 + 5 = -407.5 ≠ 0So no rational roots. Maybe we can try to approximate.Let me compute E'(n) for some integer values of n to see where it crosses zero.Compute E'(n):E'(n) = -0.2n³ + 2.4n² + 0.5n - 1Let's compute for n=1:-0.2(1) + 2.4(1) + 0.5(1) - 1 = -0.2 + 2.4 + 0.5 - 1 = 1.7Positive.n=2:-0.2(8) + 2.4(4) + 0.5(2) - 1 = -1.6 + 9.6 + 1 - 1 = 8Positive.n=3:-0.2(27) + 2.4(9) + 0.5(3) - 1 = -5.4 + 21.6 + 1.5 - 1 = 16.7Positive.n=4:-0.2(64) + 2.4(16) + 0.5(4) - 1 = -12.8 + 38.4 + 2 - 1 = 26.6Positive.n=5:-0.2(125) + 2.4(25) + 0.5(5) - 1 = -25 + 60 + 2.5 - 1 = 36.5Positive.n=6:-0.2(216) + 2.4(36) + 0.5(6) - 1 = -43.2 + 86.4 + 3 - 1 = 45.2Positive.n=7:-0.2(343) + 2.4(49) + 0.5(7) - 1 = -68.6 + 117.6 + 3.5 - 1 = 51.5Positive.n=8:-0.2(512) + 2.4(64) + 0.5(8) - 1 = -102.4 + 153.6 + 4 - 1 = 54.2Positive.n=9:-0.2(729) + 2.4(81) + 0.5(9) - 1 = -145.8 + 194.4 + 4.5 - 1 = 52.1Positive.n=10:-0.2(1000) + 2.4(100) + 0.5(10) - 1 = -200 + 240 + 5 - 1 = 44Positive.n=11:-0.2(1331) + 2.4(121) + 0.5(11) - 1 = -266.2 + 290.4 + 5.5 - 1 = 28.7Positive.n=12:-0.2(1728) + 2.4(144) + 0.5(12) - 1 = -345.6 + 345.6 + 6 - 1 = 5Positive.n=13:-0.2(2197) + 2.4(169) + 0.5(13) - 1 = -439.4 + 405.6 + 6.5 - 1 = -28.3Negative.Ah, so at n=13, E'(n) becomes negative.So, between n=12 and n=13, the derivative changes from positive to negative. That means the function E(n) has a maximum somewhere between n=12 and n=13.But since n must be an integer, we need to check E(n) at n=12 and n=13 to see which one gives a higher efficiency.But before that, let's compute E(n) for n=12 and n=13.But wait, let's compute E(n) for all n from, say, 10 to 15 to see the trend.But first, let's recall E(n) = -0.05n⁴ + 0.8n³ + 0.25n² - nCompute E(10):-0.05*(10000) + 0.8*(1000) + 0.25*(100) - 10= -500 + 800 + 25 - 10 = 315E(11):-0.05*(14641) + 0.8*(1331) + 0.25*(121) - 11Compute each term:-0.05*14641 = -732.050.8*1331 = 1064.80.25*121 = 30.25So total:-732.05 + 1064.8 + 30.25 - 11 = (-732.05 + 1064.8) = 332.75; 332.75 + 30.25 = 363; 363 - 11 = 352E(11) = 352E(12):-0.05*(20736) + 0.8*(1728) + 0.25*(144) - 12Compute each term:-0.05*20736 = -1036.80.8*1728 = 1382.40.25*144 = 36So total:-1036.8 + 1382.4 = 345.6; 345.6 + 36 = 381.6; 381.6 - 12 = 369.6E(12) = 369.6E(13):-0.05*(28561) + 0.8*(2197) + 0.25*(169) - 13Compute each term:-0.05*28561 = -1428.050.8*2197 = 1757.60.25*169 = 42.25So total:-1428.05 + 1757.6 = 329.55; 329.55 + 42.25 = 371.8; 371.8 - 13 = 358.8E(13) = 358.8E(14):-0.05*(38416) + 0.8*(2744) + 0.25*(196) - 14Compute each term:-0.05*38416 = -1920.80.8*2744 = 2195.20.25*196 = 49So total:-1920.8 + 2195.2 = 274.4; 274.4 + 49 = 323.4; 323.4 - 14 = 309.4E(14) = 309.4E(15):-0.05*(50625) + 0.8*(3375) + 0.25*(225) - 15Compute each term:-0.05*50625 = -2531.250.8*3375 = 27000.25*225 = 56.25So total:-2531.25 + 2700 = 168.75; 168.75 + 56.25 = 225; 225 - 15 = 210E(15) = 210So, compiling the results:n | E(n)---|---10 | 31511 | 35212 | 369.613 | 358.814 | 309.415 | 210Looking at these values, the maximum E(n) occurs at n=12 with E=369.6.Wait, but let's check n=9, n=8, etc., just to make sure we didn't miss a higher value before n=10.Compute E(9):-0.05*(6561) + 0.8*(729) + 0.25*(81) - 9= -328.05 + 583.2 + 20.25 - 9= (-328.05 + 583.2) = 255.15; 255.15 + 20.25 = 275.4; 275.4 - 9 = 266.4E(9)=266.4E(8):-0.05*(4096) + 0.8*(512) + 0.25*(64) - 8= -204.8 + 409.6 + 16 - 8= (-204.8 + 409.6) = 204.8; 204.8 + 16 = 220.8; 220.8 - 8 = 212.8E(8)=212.8E(7):-0.05*(2401) + 0.8*(343) + 0.25*(49) - 7= -120.05 + 274.4 + 12.25 - 7= (-120.05 + 274.4) = 154.35; 154.35 + 12.25 = 166.6; 166.6 - 7 = 159.6E(7)=159.6So, as n increases from 7 to 12, E(n) increases, peaks at n=12, then decreases.Therefore, the maximum efficiency occurs at n=12.Wait, but let me double-check E(12) and E(13):E(12)=369.6E(13)=358.8Yes, so n=12 is higher.Therefore, the value of n that maximizes efficiency E is 12.**Final Answer**The value of ( n ) that maximizes the efficiency ( E ) is boxed{12}.

answer：Okay, so I have this problem about a Canadian hunter who wants to buy a new rifle for the hunting season. He's looking at three models: A, B, and C. Each has different costs and probabilities of hitting the target. The hunter goes on 20 trips, and on each trip, there's a 0.7 chance of encountering a game animal. For each encounter, he takes one shot. The probabilities of hitting with each model are 0.85 for A, 0.9 for B, and 0.95 for C. First, I need to figure out which model will give him the highest expected number of successful shots over the season. Then, considering the costs, determine which model is the best value for money in terms of cost per expected successful shot.Alright, let's break this down step by step.Starting with part 1: Expected number of successful shots.So, the hunter goes on 20 trips. Each trip has a 0.7 probability of encountering an animal. So, the expected number of encounters is 20 trips multiplied by 0.7. Let me calculate that:20 * 0.7 = 14.So, on average, he can expect to encounter 14 game animals during the season.Now, for each encounter, he takes one shot. The probability of hitting depends on the model. So, the expected number of successful shots would be the expected number of encounters multiplied by the probability of hitting with each model.So, for Model A: 14 * 0.85.Let me compute that: 14 * 0.85. Hmm, 10*0.85 is 8.5, and 4*0.85 is 3.4, so total is 8.5 + 3.4 = 11.9.Similarly, for Model B: 14 * 0.9.14 * 0.9 is straightforward: 12.6.And for Model C: 14 * 0.95.14 * 0.95. Let me compute that: 10*0.95 is 9.5, 4*0.95 is 3.8, so total is 9.5 + 3.8 = 13.3.So, summarizing:- Model A: 11.9 expected successful shots.- Model B: 12.6 expected successful shots.- Model C: 13.3 expected successful shots.Therefore, to maximize the expected number of successful shots, he should choose Model C, as it gives the highest expected value of 13.3.Wait, but let me double-check my calculations to make sure I didn't make a mistake.For Model A: 14 * 0.85. 14 * 0.8 is 11.2, and 14 * 0.05 is 0.7, so 11.2 + 0.7 = 11.9. That seems correct.Model B: 14 * 0.9. 10*0.9 is 9, 4*0.9 is 3.6, so 9 + 3.6 = 12.6. Correct.Model C: 14 * 0.95. 14 * 1 is 14, so 14 * 0.95 is 14 - 14*0.05 = 14 - 0.7 = 13.3. That's correct as well.So, yes, Model C is the best in terms of expected successful shots.Moving on to part 2: Determining the best value for money in terms of cost per expected successful shot.The costs are:- Model A: 1200- Model B: 1500- Model C: 2000The hunter's budget is 1800, so he can't afford Model C because it costs 2000, which is over his budget. So, he can only consider Models A and B.Wait, hold on. The problem says "the hunter has a budget of 1800." So, he can't spend more than that. So, Model C is out of the question because it's 2000.But let me check: is the budget a hard limit, or can he go over? The problem says "he has a budget of 1800," so I think it's a hard limit. So, he can't choose Model C.Therefore, he has to choose between Model A and Model B.Now, to determine which provides the best value, we need to compute the cost per expected successful shot for each model.First, for Model A: cost is 1200, expected successful shots are 11.9.So, cost per expected shot is 1200 / 11.9.Similarly, for Model B: 1500 / 12.6.Let me compute these.Starting with Model A: 1200 / 11.9.Let me compute 1200 divided by 11.9.11.9 goes into 1200 how many times?Well, 11.9 * 100 = 1190.So, 1200 - 1190 = 10.So, 100 + (10 / 11.9) ≈ 100 + 0.8403 ≈ 100.8403.So, approximately 100.84 per expected successful shot.For Model B: 1500 / 12.6.12.6 goes into 1500.12.6 * 100 = 1260.1500 - 1260 = 240.12.6 * 19 = 239.4.So, 100 + 19 = 119, and 240 - 239.4 = 0.6.So, 0.6 / 12.6 ≈ 0.0476.Therefore, total is approximately 119.0476.So, approximately 119.05 per expected successful shot.Comparing the two:- Model A: ~100.84 per shot.- Model B: ~119.05 per shot.Therefore, Model A is more cost-efficient, as it has a lower cost per expected successful shot.But wait, let me verify my calculations because sometimes when dealing with division, it's easy to make a mistake.For Model A: 1200 / 11.9.Let me compute 11.9 * 100 = 1190.1200 - 1190 = 10.So, 10 / 11.9 ≈ 0.8403.So, total is 100 + 0.8403 ≈ 100.8403, so approximately 100.84.For Model B: 1500 / 12.6.Let me compute 12.6 * 100 = 1260.1500 - 1260 = 240.240 / 12.6.Let me compute 12.6 * 19 = 239.4.240 - 239.4 = 0.6.0.6 / 12.6 = 0.0476.So, total is 100 + 19 + 0.0476 ≈ 119.0476, which is approximately 119.05.So, yes, Model A is cheaper per expected successful shot.But wait, hold on. Is there another way to compute this? Maybe using decimals more accurately.Alternatively, let me use decimal division.For Model A: 1200 / 11.9.Let me write this as 1200 ÷ 11.9.Multiply numerator and denominator by 10 to eliminate the decimal: 12000 ÷ 119.Now, 119 goes into 12000 how many times?119 * 100 = 11900.12000 - 11900 = 100.So, 100 / 119 ≈ 0.8403.So, total is 100 + 0.8403 ≈ 100.8403, same as before.Similarly, for Model B: 1500 / 12.6.Multiply numerator and denominator by 10: 15000 ÷ 126.126 goes into 15000.126 * 100 = 12600.15000 - 12600 = 2400.126 * 19 = 2394.2400 - 2394 = 6.So, 6 / 126 = 0.0476.So, total is 100 + 19 + 0.0476 ≈ 119.0476, same as before.So, the calculations hold.Therefore, Model A is more cost-efficient.But wait, let me think again. Is there a better way to represent this? Maybe compute the expected shots per dollar?Alternatively, compute the ratio of expected shots to cost.For Model A: 11.9 / 1200.For Model B: 12.6 / 1500.Compute these:Model A: 11.9 / 1200 ≈ 0.009917.Model B: 12.6 / 1500 ≈ 0.0084.So, Model A gives approximately 0.009917 expected shots per dollar, while Model B gives 0.0084. So, Model A is better.Alternatively, the reciprocal is cost per expected shot, which we already calculated.So, yes, Model A is better in terms of cost per expected shot.But wait, the hunter's budget is 1800. So, he can choose Model A or Model B, but not Model C. So, between A and B, A is cheaper and gives a better rate.But hold on, can he buy Model C if he goes over budget? The problem says he has a budget of 1800, so I think he can't exceed that. So, Model C is out.Therefore, the best value is Model A.But let me just think again: is there a way to get more expected shots within the budget? For example, if he buys Model A, he spends 1200, and has 600 left. Could he use that 600 to buy something else? But the problem doesn't mention any other purchases, so I think he's just choosing one rifle.Therefore, he can only choose between A and B, with C being too expensive.So, in terms of cost per expected shot, Model A is better.But let me just compute the exact cost per expected shot:For Model A: 1200 / 11.9 ≈ 100.8403.For Model B: 1500 / 12.6 ≈ 118.58.Wait, earlier I had 119.05, but actually, 1500 / 12.6 is exactly 118.58 approximately.Wait, let me compute 12.6 * 118.58.12.6 * 100 = 1260.12.6 * 18.58 ≈ 12.6 * 18 = 226.8, and 12.6 * 0.58 ≈ 7.288.So, total ≈ 1260 + 226.8 + 7.288 ≈ 1494.088, which is less than 1500. Hmm, so perhaps my earlier calculation was a bit off.Wait, maybe I should compute 1500 / 12.6 more accurately.12.6 goes into 1500.12.6 * 118 = 12.6 * 100 + 12.6 * 18 = 1260 + 226.8 = 1486.8.1500 - 1486.8 = 13.2.13.2 / 12.6 ≈ 1.0476.So, total is 118 + 1.0476 ≈ 119.0476.So, approximately 119.05.Wait, so 12.6 * 119.0476 ≈ 1500.Yes, because 12.6 * 119 = 12.6*(120 -1) = 1512 -12.6=1499.4.So, 12.6*119=1499.4, which is 0.6 less than 1500. So, 119 + (0.6 /12.6)=119 + 0.0476≈119.0476.So, 1500 /12.6≈119.0476.Similarly, 1200 /11.9≈100.8403.So, the exact values are approximately 100.84 and 119.05.Therefore, Model A is cheaper per expected shot.So, the conclusion is:1. To maximize expected successful shots: Model C with 13.3.2. Considering budget, Model C is too expensive, so between A and B, Model A is better value.But wait, the problem says "determine which model provides the best value for money in terms of cost per expected successful shot."But Model C is more expensive than the budget, so it's not an option. So, the hunter can only choose between A and B.But just to make sure, is there a way to buy Model C with the budget? The budget is 1800, and Model C is 2000. So, no, he can't.Therefore, the best value is Model A.But hold on, let me think again. Maybe the hunter can buy Model B, which is 1500, and have 300 left. But he can't buy another rifle because the other models are more expensive. So, he can't get more rifles. So, he can only buy one rifle, either A or B.Therefore, in terms of cost per expected shot, Model A is better.But let me just think about the exact numbers:Model A: 11.9 expected shots for 1200.Model B: 12.6 expected shots for 1500.So, the hunter can compute the cost per shot:A: 1200 /11.9≈100.84.B: 1500 /12.6≈118.58.So, A is cheaper per shot.Alternatively, if we compute the shots per dollar:A: 11.9 /1200≈0.009917.B:12.6 /1500≈0.0084.So, A gives more shots per dollar.Therefore, Model A is better value.But wait, another way to think about it: if he buys Model A, he can have 11.9 shots for 1200, and if he buys Model B, he can have 12.6 shots for 1500.But since he has 1800, could he buy Model A and have 600 left, but he can't buy another rifle because the next cheapest is Model B at 1500, which he can't afford with 600. So, he can't buy multiple rifles.Therefore, he can only buy one rifle, so he has to choose between A and B.So, in that case, Model A is better.But let me just think about the exact cost per shot:Model A: ~100.84 per shot.Model B: ~118.58 per shot.So, Model A is cheaper per shot.Therefore, the best value is Model A.But wait, the problem says "determine which model provides the best value for money in terms of cost per expected successful shot."So, it's possible that even though Model C is more expensive, it's still the best in terms of cost per shot, but since it's over the budget, it's not an option.Therefore, the answer is Model A.But just to make sure, let me compute the exact cost per shot:For Model A: 1200 /11.9.Let me compute this more accurately.11.9 * 100 = 1190.1200 -1190 =10.So, 10 /11.9 = approx 0.8403.So, total is 100 +0.8403=100.8403.So, approximately 100.84 per shot.For Model B: 1500 /12.6.12.6 *100=1260.1500 -1260=240.240 /12.6= approx 19.0476.So, total is 100 +19.0476=119.0476.So, approximately 119.05 per shot.So, yes, Model A is cheaper per shot.Therefore, the hunter should choose Model A for the best value.But wait, just to make sure, is there a way to calculate the expected shots per dollar?Yes, that's another way.For Model A: 11.9 /1200≈0.009917 shots per dollar.For Model B:12.6 /1500≈0.0084 shots per dollar.So, Model A gives more shots per dollar, which is better.Therefore, Model A is the best value.So, summarizing:1. To maximize expected successful shots: Model C with 13.3.2. Considering budget, Model A is the best value at ~100.84 per shot.But wait, the problem says "considering the expected number of successful shots from part 1 and the costs, determine which model provides the best value for money in terms of cost per expected successful shot."So, the answer is Model A.But just to make sure, let me think if there's any other angle.Is there a way to combine the two aspects? Like, maybe some kind of ratio or something else?But the problem specifically says "cost per expected successful shot," so it's just cost divided by expected shots.So, yes, that's straightforward.Therefore, the conclusion is:1. Model C gives the highest expected shots: 13.3.2. Model A gives the best value: ~100.84 per shot.But since the hunter can't afford Model C, he has to choose between A and B, and A is better.Therefore, the answers are:1. Model C.2. Model A.But let me just make sure I didn't make any calculation errors.For expected shots:20 trips, 0.7 chance each trip, so 14 expected encounters.Each model's hit probability:A:0.85, B:0.9, C:0.95.So, expected hits:A:14*0.85=11.9.B:14*0.9=12.6.C:14*0.95=13.3.Correct.Costs:A:1200, B:1500, C:2000.Budget:1800.So, C is too expensive.Compute cost per expected shot:A:1200 /11.9≈100.84.B:1500 /12.6≈119.05.So, A is better.Therefore, the answers are:1. Model C.2. Model A.Yes, that seems correct.