🧠:<think>Alright, so I've got this problem about a legal researcher analyzing case distributions over the past decade. It's split into two parts: estimating parameters for a Poisson distribution and calculating a probability for a normal distribution. Let me try to tackle each part step by step.Starting with part 1: The number of cases per year, C(t), follows a Poisson distribution with a time-dependent rate parameter λ(t) = α e^(β t). We know that in 2015, there were 150 cases, and the average annual growth rate over the decade is 5%. We need to estimate α and β.First, let me recall that the Poisson distribution has the property that the mean is equal to the rate parameter λ. So, for each year t, the expected number of cases is λ(t) = α e^(β t). Given that the average annual growth rate is 5%, this suggests that the rate λ(t) is growing exponentially at a rate of 5% per year. So, the growth rate β should be related to this 5%. Wait, but in the Poisson model, λ(t) = α e^(β t). If the growth rate is 5%, then the multiplicative factor each year is 1.05. So, the growth factor per year is e^β = 1.05. Therefore, β = ln(1.05). Let me compute that.Calculating β: ln(1.05) is approximately 0.04879. So, β ≈ 0.0488.Now, we need to find α. We know that in 2015, the number of cases was 150. Assuming that the observed number of cases is close to the mean (since Poisson variance equals the mean), we can set λ(2015) = 150.But wait, the time variable t isn't specified. Is t the year, like 2010, 2011, etc.? If so, we need to define t as the number of years since a certain starting point. Let me assume that t=0 corresponds to 2010. So, 2015 would be t=5.Therefore, λ(5) = α e^(β * 5) = 150.We already have β ≈ 0.0488, so let's plug that in:α e^(0.0488 * 5) = 150.Calculating the exponent: 0.0488 * 5 ≈ 0.244.So, e^0.244 ≈ 1.275.Therefore, α * 1.275 ≈ 150.Solving for α: α ≈ 150 / 1.275 ≈ 117.647.So, α ≈ 117.65 and β ≈ 0.0488.But let me double-check. If we take t=0 as 2010, then in 2015, t=5. So, λ(5) = 117.65 * e^(0.0488*5) ≈ 117.65 * 1.275 ≈ 150. That seems correct.Alternatively, if t is the actual year, like t=2015, then we need to adjust. But since the growth rate is given as an average over the decade, it's more natural to model t as years since 2010. So, I think my initial approach is correct.Moving on to part 2: The proportion of civil cases follows a normal distribution with mean μ = 0.4 and variance σ² = 0.01. We need to find the probability that the proportion is between 35% and 45%, i.e., between 0.35 and 0.45.First, let's note that the normal distribution is defined by its mean and standard deviation. Here, μ = 0.4 and σ² = 0.01, so σ = 0.1.We need to find P(0.35 < X < 0.45), where X ~ N(0.4, 0.1²).To compute this, we'll standardize the variable to Z-scores.Z1 = (0.35 - 0.4) / 0.1 = (-0.05)/0.1 = -0.5Z2 = (0.45 - 0.4)/0.1 = 0.05/0.1 = 0.5So, we need P(-0.5 < Z < 0.5), where Z is the standard normal variable.Looking up these Z-scores in the standard normal table:P(Z < 0.5) ≈ 0.6915P(Z < -0.5) ≈ 0.3085Therefore, P(-0.5 < Z < 0.5) = 0.6915 - 0.3085 = 0.3830So, approximately 38.3% probability.Alternatively, using symmetry, since the normal distribution is symmetric around the mean, the probability between -0.5 and 0.5 is twice the probability from 0 to 0.5 minus 1, but in this case, it's simpler to subtract the two tail probabilities.So, the probability is about 38.3%.Wait, but let me confirm the Z-table values. For Z=0.5, the cumulative probability is indeed about 0.6915, and for Z=-0.5, it's 0.3085. So, subtracting gives 0.383. That seems correct.Alternatively, using a calculator or more precise Z-table, the exact value might be slightly different, but for most purposes, 0.383 is accurate enough.So, summarizing:1. α ≈ 117.65 and β ≈ 0.04882. Probability ≈ 38.3%I think that's it. Let me just make sure I didn't make any calculation errors.For part 1, the growth rate of 5% translates to β = ln(1.05) ≈ 0.04879. Then, using t=5 for 2015, solving for α gives approximately 117.65. That seems right.For part 2, converting 35% and 45% to Z-scores gives -0.5 and 0.5, leading to a probability of about 38.3%. Yes, that makes sense.Final Answer1. The estimated parameters are (alpha approx boxed{117.65}) and (beta approx boxed{0.0488}).2. The probability that the proportion of civil cases is between 35% and 45% is approximately (boxed{0.383}).</think>

🧠:<think>Alright, so I have this problem where a backpacker wants to visit five unique spots in Japan, each with specific coordinates. The goal is to find the optimal route that minimizes the total walking distance, and then calculate the total energy expenditure considering that one location requires extra energy to climb.First, I need to tackle the first part: finding the optimal route. This sounds like the Traveling Salesman Problem (TSP), which is a classic combinatorial optimization problem. The TSP aims to find the shortest possible route that visits each city exactly once and returns to the starting city. Since there are five locations, the number of possible routes is (5-1)! = 24, which is manageable to compute manually or with some assistance.But wait, I don't have a computer here, so maybe I can find a way to approximate or find the optimal route by calculating distances between each pair of points and then trying to find the shortest Hamiltonian cycle.Let me list the coordinates again:A(1, 3)B(4, 7)C(6, 2)D(8, 5)E(3, 8)First, I should calculate the Euclidean distances between each pair of points. The Euclidean distance between two points (x1, y1) and (x2, y2) is sqrt[(x2 - x1)^2 + (y2 - y1)^2].Let me compute all the distances:Distance AB: sqrt[(4-1)^2 + (7-3)^2] = sqrt[9 + 16] = sqrt[25] = 5Distance AC: sqrt[(6-1)^2 + (2-3)^2] = sqrt[25 + 1] = sqrt[26] ≈ 5.099Distance AD: sqrt[(8-1)^2 + (5-3)^2] = sqrt[49 + 4] = sqrt[53] ≈ 7.280Distance AE: sqrt[(3-1)^2 + (8-3)^2] = sqrt[4 + 25] = sqrt[29] ≈ 5.385Distance BA: same as AB, which is 5Distance BC: sqrt[(6-4)^2 + (2-7)^2] = sqrt[4 + 25] = sqrt[29] ≈ 5.385Distance BD: sqrt[(8-4)^2 + (5-7)^2] = sqrt[16 + 4] = sqrt[20] ≈ 4.472Distance BE: sqrt[(3-4)^2 + (8-7)^2] = sqrt[1 + 1] = sqrt[2] ≈ 1.414Distance CA: same as AC, ≈5.099Distance CB: same as BC, ≈5.385Distance CD: sqrt[(8-6)^2 + (5-2)^2] = sqrt[4 + 9] = sqrt[13] ≈ 3.606Distance CE: sqrt[(3-6)^2 + (8-2)^2] = sqrt[9 + 36] = sqrt[45] ≈ 6.708Distance DA: same as AD, ≈7.280Distance DB: same as BD, ≈4.472Distance DC: same as CD, ≈3.606Distance DE: sqrt[(3-8)^2 + (8-5)^2] = sqrt[25 + 9] = sqrt[34] ≈ 5.831Distance EA: same as AE, ≈5.385Distance EB: same as BE, ≈1.414Distance EC: same as CE, ≈6.708Distance ED: same as DE, ≈5.831Okay, so now I have all the pairwise distances. Now, I need to find the shortest possible route that visits all five points and returns to the starting point. Since it's a small number of points, I can try to list possible routes and calculate their total distances, but that might take a while. Alternatively, I can use some heuristics or look for patterns.Alternatively, maybe I can use the nearest neighbor approach as a starting point, although it might not give the optimal solution, but it can be a good approximation.Let me try starting at point A.From A, the nearest neighbor is B (distance 5). Then from B, the nearest unvisited point is E (distance ≈1.414). From E, the nearest unvisited point is D (distance ≈5.831). From D, the nearest unvisited point is C (distance ≈3.606). Then from C back to A (distance ≈5.099). So the total distance would be 5 + 1.414 + 5.831 + 3.606 + 5.099 ≈ 20.95.But maybe there's a better route.Alternatively, starting at A, go to E (≈5.385). From E, nearest is B (≈1.414). From B, nearest is D (≈4.472). From D, nearest is C (≈3.606). From C back to A (≈5.099). Total distance: 5.385 + 1.414 + 4.472 + 3.606 + 5.099 ≈ 20.976. That's slightly worse.Alternatively, starting at A, go to C (≈5.099). From C, nearest is D (≈3.606). From D, nearest is B (≈4.472). From B, nearest is E (≈1.414). From E back to A (≈5.385). Total: 5.099 + 3.606 + 4.472 + 1.414 + 5.385 ≈ 20.976. Same as before.Alternatively, starting at A, go to D (≈7.280). From D, nearest is C (≈3.606). From C, nearest is B (≈5.385). From B, nearest is E (≈1.414). From E back to A (≈5.385). Total: 7.280 + 3.606 + 5.385 + 1.414 + 5.385 ≈ 23.07. That's worse.Alternatively, starting at A, go to E (≈5.385). From E, go to C (≈6.708). From C, go to D (≈3.606). From D, go to B (≈4.472). From B back to A (5). Total: 5.385 + 6.708 + 3.606 + 4.472 + 5 ≈ 25.169. That's worse.Hmm, so the nearest neighbor starting at A gives a total distance of approximately 20.95. Maybe I can try starting at a different point.Let me try starting at B.From B, nearest is E (≈1.414). From E, nearest is A (≈5.385). From A, nearest is C (≈5.099). From C, nearest is D (≈3.606). From D back to B (≈4.472). Total: 1.414 + 5.385 + 5.099 + 3.606 + 4.472 ≈ 20.976.Alternatively, from B, go to D (≈4.472). From D, go to C (≈3.606). From C, go to A (≈5.099). From A, go to E (≈5.385). From E back to B (≈1.414). Total: 4.472 + 3.606 + 5.099 + 5.385 + 1.414 ≈ 20.976.Same as before.Alternatively, from B, go to A (5). From A, go to E (≈5.385). From E, go to D (≈5.831). From D, go to C (≈3.606). From C back to B (≈5.385). Total: 5 + 5.385 + 5.831 + 3.606 + 5.385 ≈ 25.107.Not better.Alternatively, from B, go to C (≈5.385). From C, go to D (≈3.606). From D, go to E (≈5.831). From E, go to A (≈5.385). From A back to B (5). Total: 5.385 + 3.606 + 5.831 + 5.385 + 5 ≈ 25.107.Same as before.So starting at B, the best is still around 20.976.Let me try starting at C.From C, nearest is D (≈3.606). From D, nearest is B (≈4.472). From B, nearest is E (≈1.414). From E, nearest is A (≈5.385). From A back to C (≈5.099). Total: 3.606 + 4.472 + 1.414 + 5.385 + 5.099 ≈ 20.976.Alternatively, from C, go to A (≈5.099). From A, go to B (5). From B, go to E (≈1.414). From E, go to D (≈5.831). From D back to C (≈3.606). Total: 5.099 + 5 + 1.414 + 5.831 + 3.606 ≈ 21.949.Not better.Alternatively, from C, go to E (≈6.708). From E, go to B (≈1.414). From B, go to D (≈4.472). From D, go to A (≈7.280). From A back to C (≈5.099). Total: 6.708 + 1.414 + 4.472 + 7.280 + 5.099 ≈ 24.973.Worse.So starting at C, the best is still 20.976.Now, starting at D.From D, nearest is C (≈3.606). From C, nearest is A (≈5.099). From A, nearest is B (5). From B, nearest is E (≈1.414). From E back to D (≈5.831). Total: 3.606 + 5.099 + 5 + 1.414 + 5.831 ≈ 21.949.Alternatively, from D, go to B (≈4.472). From B, go to E (≈1.414). From E, go to A (≈5.385). From A, go to C (≈5.099). From C back to D (≈3.606). Total: 4.472 + 1.414 + 5.385 + 5.099 + 3.606 ≈ 20.976.Same as before.Alternatively, from D, go to E (≈5.831). From E, go to B (≈1.414). From B, go to A (5). From A, go to C (≈5.099). From C back to D (≈3.606). Total: 5.831 + 1.414 + 5 + 5.099 + 3.606 ≈ 21.95.Not better.So starting at D, the best is 20.976.Finally, starting at E.From E, nearest is B (≈1.414). From B, nearest is D (≈4.472). From D, nearest is C (≈3.606). From C, nearest is A (≈5.099). From A back to E (≈5.385). Total: 1.414 + 4.472 + 3.606 + 5.099 + 5.385 ≈ 20.976.Alternatively, from E, go to A (≈5.385). From A, go to C (≈5.099). From C, go to D (≈3.606). From D, go to B (≈4.472). From B back to E (≈1.414). Total: 5.385 + 5.099 + 3.606 + 4.472 + 1.414 ≈ 20.976.Same as before.Alternatively, from E, go to C (≈6.708). From C, go to D (≈3.606). From D, go to B (≈4.472). From B, go to A (5). From A back to E (≈5.385). Total: 6.708 + 3.606 + 4.472 + 5 + 5.385 ≈ 25.169.Worse.So, regardless of the starting point, the nearest neighbor approach gives a total distance of approximately 20.95 or 20.976. The slight difference is due to rounding.But wait, maybe there's a better route that isn't captured by the nearest neighbor approach. Let me try to find a route manually.Looking at the distances, E is very close to B (≈1.414), which is the shortest distance. So it's beneficial to have E and B adjacent in the route.Similarly, D is close to C (≈3.606), so maybe D and C should be adjacent.Also, B is close to D (≈4.472), which is another relatively short distance.A is connected to B (5), which is a moderate distance, but A is also connected to E (≈5.385) and C (≈5.099).C is connected to A (≈5.099), D (≈3.606), and E (≈6.708).So, perhaps a route that goes through E-B-D-C-A-E or something similar.Wait, let me try the route A-B-E-D-C-A.Compute the total distance:A to B: 5B to E: ≈1.414E to D: ≈5.831D to C: ≈3.606C to A: ≈5.099Total: 5 + 1.414 + 5.831 + 3.606 + 5.099 ≈ 20.95That's the same as the nearest neighbor route starting at A.Alternatively, route A-E-B-D-C-A:A to E: ≈5.385E to B: ≈1.414B to D: ≈4.472D to C: ≈3.606C to A: ≈5.099Total: 5.385 + 1.414 + 4.472 + 3.606 + 5.099 ≈ 20.976Same as before.Alternatively, route A-C-D-B-E-A:A to C: ≈5.099C to D: ≈3.606D to B: ≈4.472B to E: ≈1.414E to A: ≈5.385Total: 5.099 + 3.606 + 4.472 + 1.414 + 5.385 ≈ 20.976Same.Alternatively, route A-D-C-B-E-A:A to D: ≈7.280D to C: ≈3.606C to B: ≈5.385B to E: ≈1.414E to A: ≈5.385Total: 7.280 + 3.606 + 5.385 + 1.414 + 5.385 ≈ 23.07Worse.Alternatively, route A-E-C-D-B-A:A to E: ≈5.385E to C: ≈6.708C to D: ≈3.606D to B: ≈4.472B to A: 5Total: 5.385 + 6.708 + 3.606 + 4.472 + 5 ≈ 25.169Worse.So, it seems that the minimal total distance is approximately 20.95 or 20.976, depending on the route. The slight difference is due to rounding in the distances.But wait, let me check if there's a route that goes through E-B-D-C-A-E with total distance ≈20.95, which is slightly better than the others.Alternatively, maybe there's a route that goes through E-B-D-C-A-E, but that's the same as A-B-E-D-C-A.Wait, actually, the route A-B-E-D-C-A is 5 + 1.414 + 5.831 + 3.606 + 5.099 ≈20.95.But let me check if there's a way to make it even shorter.Is there a way to connect points such that the total distance is less than 20.95?Looking at the distances, the shortest edges are E-B (≈1.414), D-C (≈3.606), B-D (≈4.472), A-B (5), A-C (≈5.099), etc.If I can connect the shortest edges without creating too long detours, maybe I can get a better total.For example, if I connect E-B (1.414), then B-D (4.472), then D-C (3.606), then C-A (5.099), and then A-E (5.385). That's the same as the route A-E-B-D-C-A, which totals ≈20.976.Alternatively, if I connect E-B (1.414), then B-A (5), then A-C (5.099), then C-D (3.606), then D-E (5.831). That would be E-B-A-C-D-E, with total distance 1.414 + 5 + 5.099 + 3.606 + 5.831 ≈21.949.Not better.Alternatively, E-B-D-C-A-E: 1.414 + 4.472 + 3.606 + 5.099 + 5.385 ≈20.976.Same as before.Wait, but earlier I had a route A-B-E-D-C-A with total ≈20.95, which is slightly better. Let me verify the distances again:A to B: 5B to E: ≈1.414E to D: ≈5.831D to C: ≈3.606C to A: ≈5.099Total: 5 + 1.414 + 5.831 + 3.606 + 5.099 ≈20.95Yes, that's correct. So this route is slightly shorter.Is there a way to make it even shorter? Let me think.If I can find a way to connect E to D without going through B, but E to D is ≈5.831, which is longer than E to B (≈1.414). So it's better to go through B.Alternatively, if I go from E to C, which is ≈6.708, which is longer than E to B.So, the route A-B-E-D-C-A seems to be the shortest so far.But let me check another possible route: A-C-D-B-E-A.A to C: ≈5.099C to D: ≈3.606D to B: ≈4.472B to E: ≈1.414E to A: ≈5.385Total: 5.099 + 3.606 + 4.472 + 1.414 + 5.385 ≈20.976Same as before.Alternatively, A-E-B-D-C-A: 5.385 + 1.414 + 4.472 + 3.606 + 5.099 ≈20.976.So, the minimal total distance seems to be approximately 20.95, achieved by the route A-B-E-D-C-A.Wait, but let me check if there's a route that goes through E-B-D-C-A-E, but that's the same as A-B-E-D-C-A.Alternatively, maybe starting at E: E-B-D-C-A-E.E to B: ≈1.414B to D: ≈4.472D to C: ≈3.606C to A: ≈5.099A to E: ≈5.385Total: 1.414 + 4.472 + 3.606 + 5.099 + 5.385 ≈20.976.Same as before.So, it seems that the minimal total distance is approximately 20.95, achieved by the route A-B-E-D-C-A.But wait, let me check if there's a way to connect A to E directly, which is ≈5.385, and then E to B (≈1.414), then B to D (≈4.472), then D to C (≈3.606), then C back to A (≈5.099). That's the same as A-E-B-D-C-A, which totals ≈20.976.So, the route A-B-E-D-C-A is slightly shorter by about 0.026, which is negligible due to rounding.Therefore, the optimal route is either A-B-E-D-C-A or A-E-B-D-C-A, with total distances approximately 20.95 and 20.976 respectively.But since 20.95 is slightly shorter, I'll go with that route.Now, for the second part, the backpacker realizes that location D requires extra energy to climb. The energy required to walk between two points is the Euclidean distance multiplied by an elevation gain factor of 1.5 for D.Wait, does that mean that any segment involving D has its distance multiplied by 1.5? Or is it only when arriving at D?I think it's when moving to D, the energy is multiplied by 1.5. So, for any segment that ends at D, the energy is 1.5 times the distance.Wait, the problem says: "the energy required to walk between two points (x1, y1) and (x2, y2) is given by the Euclidean distance multiplied by the elevation gain factor (which is 1.5 for D)."So, for any segment that includes D, the energy is 1.5 times the distance. So, both when going to D and coming from D, the energy is multiplied by 1.5.Wait, no. Let me read it again: "the energy required to walk between two points (x1, y1) and (x2, y2) is given by the Euclidean distance multiplied by the elevation gain factor (which is 1.5 for D)."So, it's only when D is one of the points. So, for any segment that includes D, the energy is 1.5 times the distance. So, both when going to D and coming from D, the energy is multiplied by 1.5.Therefore, in the route A-B-E-D-C-A, the segments involving D are E-D and D-C. So, the energy for E-D and D-C will be multiplied by 1.5.Similarly, in the route A-E-B-D-C-A, the segments E-B, B-D, D-C, and C-A. Wait, no, in that route, the segments involving D are B-D and D-C.Wait, let me clarify:In the route A-B-E-D-C-A, the segments are A-B, B-E, E-D, D-C, C-A.So, the segments involving D are E-D and D-C.In the route A-E-B-D-C-A, the segments are A-E, E-B, B-D, D-C, C-A.So, the segments involving D are B-D and D-C.Therefore, in both routes, the segments E-D and D-C (or B-D and D-C) involve D, so their energies are multiplied by 1.5.Therefore, to calculate the total energy expenditure, I need to compute the energy for each segment, where segments involving D are multiplied by 1.5.Let me take the route A-B-E-D-C-A.Segments:A-B: distance 5, energy 5B-E: distance ≈1.414, energy ≈1.414E-D: distance ≈5.831, energy ≈5.831 * 1.5 ≈8.7465D-C: distance ≈3.606, energy ≈3.606 * 1.5 ≈5.409C-A: distance ≈5.099, energy ≈5.099Total energy: 5 + 1.414 + 8.7465 + 5.409 + 5.099 ≈25.6685Alternatively, for the route A-E-B-D-C-A:Segments:A-E: distance ≈5.385, energy ≈5.385E-B: distance ≈1.414, energy ≈1.414B-D: distance ≈4.472, energy ≈4.472 * 1.5 ≈6.708D-C: distance ≈3.606, energy ≈3.606 * 1.5 ≈5.409C-A: distance ≈5.099, energy ≈5.099Total energy: 5.385 + 1.414 + 6.708 + 5.409 + 5.099 ≈24.015Wait, that's a significant difference. So, the route A-E-B-D-C-A has a lower total energy expenditure because the segment B-D is shorter than E-D, so even though both are multiplied by 1.5, the shorter distance results in a lower energy cost.Therefore, the route A-E-B-D-C-A has a total energy expenditure of approximately 24.015, while the route A-B-E-D-C-A has approximately 25.6685.Therefore, the optimal route in terms of energy expenditure is A-E-B-D-C-A.Wait, but let me verify the calculations again.For route A-E-B-D-C-A:A-E: ≈5.385E-B: ≈1.414B-D: ≈4.472 *1.5≈6.708D-C: ≈3.606 *1.5≈5.409C-A: ≈5.099Total: 5.385 + 1.414 + 6.708 + 5.409 + 5.099Let me add them step by step:5.385 + 1.414 = 6.7996.799 + 6.708 = 13.50713.507 + 5.409 = 18.91618.916 + 5.099 = 24.015Yes, that's correct.For route A-B-E-D-C-A:A-B: 5B-E: ≈1.414E-D: ≈5.831 *1.5≈8.7465D-C: ≈3.606 *1.5≈5.409C-A: ≈5.099Total: 5 + 1.414 + 8.7465 + 5.409 + 5.099Adding step by step:5 + 1.414 = 6.4146.414 + 8.7465 ≈15.160515.1605 + 5.409 ≈20.569520.5695 + 5.099 ≈25.6685Yes, correct.Therefore, the route A-E-B-D-C-A is better in terms of energy expenditure.But wait, is there a better route? Let me check another possible route.What about A-C-D-B-E-A?Segments:A-C: ≈5.099C-D: ≈3.606 *1.5≈5.409D-B: ≈4.472 *1.5≈6.708B-E: ≈1.414E-A: ≈5.385Total energy: 5.099 + 5.409 + 6.708 + 1.414 + 5.385 ≈23.015Wait, that's even better! Let me check the distances:A-C: ≈5.099C-D: ≈3.606 *1.5≈5.409D-B: ≈4.472 *1.5≈6.708B-E: ≈1.414E-A: ≈5.385Total: 5.099 + 5.409 + 6.708 + 1.414 + 5.385 ≈23.015That's better than the previous 24.015.Is this route valid? Let me see:A-C-D-B-E-A.Yes, it visits all points once and returns to A.So, this route has a total energy expenditure of approximately 23.015, which is better than the previous ones.Wait, can we do even better?Let me check another route: A-D-C-B-E-A.Segments:A-D: ≈7.280 *1.5≈10.92D-C: ≈3.606 *1.5≈5.409C-B: ≈5.385B-E: ≈1.414E-A: ≈5.385Total energy: 10.92 + 5.409 + 5.385 + 1.414 + 5.385 ≈28.513That's worse.Alternatively, route A-E-D-C-B-A.Segments:A-E: ≈5.385E-D: ≈5.831 *1.5≈8.7465D-C: ≈3.606 *1.5≈5.409C-B: ≈5.385B-A: 5Total: 5.385 + 8.7465 + 5.409 + 5.385 + 5 ≈29.925Worse.Alternatively, route A-B-D-C-E-A.Segments:A-B: 5B-D: ≈4.472 *1.5≈6.708D-C: ≈3.606 *1.5≈5.409C-E: ≈6.708E-A: ≈5.385Total: 5 + 6.708 + 5.409 + 6.708 + 5.385 ≈29.21Worse.Alternatively, route A-C-B-E-D-A.Segments:A-C: ≈5.099C-B: ≈5.385B-E: ≈1.414E-D: ≈5.831 *1.5≈8.7465D-A: ≈7.280 *1.5≈10.92Total: 5.099 + 5.385 + 1.414 + 8.7465 + 10.92 ≈31.5645Worse.So, the route A-C-D-B-E-A seems to be better with a total energy of ≈23.015.Wait, let me check if there's a route that goes through E-B-D-C-A-E, but that's the same as A-E-B-D-C-A, which we already calculated as ≈24.015.Alternatively, route A-E-D-C-B-A: as above, ≈29.925.No, worse.Alternatively, route A-E-C-D-B-A.Segments:A-E: ≈5.385E-C: ≈6.708C-D: ≈3.606 *1.5≈5.409D-B: ≈4.472 *1.5≈6.708B-A: 5Total: 5.385 + 6.708 + 5.409 + 6.708 + 5 ≈29.21Same as before.So, the route A-C-D-B-E-A seems to be the best so far with a total energy of ≈23.015.Wait, let me check if there's a way to connect A to C, then C to D, then D to B, then B to E, then E back to A.Yes, that's the route A-C-D-B-E-A.Total energy: 5.099 + 5.409 + 6.708 + 1.414 + 5.385 ≈23.015.Is there a way to make it even better?What if I go A-C-D-E-B-A?Segments:A-C: ≈5.099C-D: ≈3.606 *1.5≈5.409D-E: ≈5.831 *1.5≈8.7465E-B: ≈1.414B-A: 5Total: 5.099 + 5.409 + 8.7465 + 1.414 + 5 ≈25.6685Same as the route A-B-E-D-C-A.Not better.Alternatively, route A-D-C-B-E-A.Segments:A-D: ≈7.280 *1.5≈10.92D-C: ≈3.606 *1.5≈5.409C-B: ≈5.385B-E: ≈1.414E-A: ≈5.385Total: 10.92 + 5.409 + 5.385 + 1.414 + 5.385 ≈28.513Worse.Alternatively, route A-B-D-C-E-A.Segments:A-B: 5B-D: ≈4.472 *1.5≈6.708D-C: ≈3.606 *1.5≈5.409C-E: ≈6.708E-A: ≈5.385Total: 5 + 6.708 + 5.409 + 6.708 + 5.385 ≈29.21Same as before.So, it seems that the route A-C-D-B-E-A is the most efficient in terms of energy expenditure, with a total of approximately 23.015.Wait, but let me check if there's a route that goes through E-B-D-C-A-E, but that's the same as A-E-B-D-C-A, which we saw had a total energy of ≈24.015.So, the route A-C-D-B-E-A is better.Wait, but let me check another route: A-C-B-E-D-A.Segments:A-C: ≈5.099C-B: ≈5.385B-E: ≈1.414E-D: ≈5.831 *1.5≈8.7465D-A: ≈7.280 *1.5≈10.92Total: 5.099 + 5.385 + 1.414 + 8.7465 + 10.92 ≈31.5645Worse.Alternatively, route A-E-C-D-B-A.Segments:A-E: ≈5.385E-C: ≈6.708C-D: ≈3.606 *1.5≈5.409D-B: ≈4.472 *1.5≈6.708B-A: 5Total: 5.385 + 6.708 + 5.409 + 6.708 + 5 ≈29.21Same as before.So, the minimal energy expenditure is achieved by the route A-C-D-B-E-A with a total of approximately 23.015.Wait, but let me check if there's a route that goes through C-D first, then D-B, then B-E, then E-A, and back to C.Wait, no, that would not be a closed route.Alternatively, starting at A, going to C, then D, then B, then E, then back to A.Yes, that's the route A-C-D-B-E-A.So, to confirm, the segments are:A-C: ≈5.099C-D: ≈3.606 *1.5≈5.409D-B: ≈4.472 *1.5≈6.708B-E: ≈1.414E-A: ≈5.385Total energy: 5.099 + 5.409 + 6.708 + 1.414 + 5.385 ≈23.015Yes, that's correct.Therefore, the optimal route in terms of energy expenditure is A-C-D-B-E-A, with a total energy of approximately 23.015.Wait, but let me check if there's a route that goes through E-B-D-C-A-E, but that's the same as A-E-B-D-C-A, which we saw had a total energy of ≈24.015.So, the route A-C-D-B-E-A is better.Therefore, the answer to part 1 is the route A-B-E-D-C-A with a total distance of approximately 20.95, but considering the energy expenditure, the optimal route is A-C-D-B-E-A with a total energy of approximately 23.015.Wait, but the first part asks for the optimal route that minimizes the total walking distance, which is the TSP solution, and the second part asks for the total energy expenditure considering the elevation gain at D.So, for part 1, the optimal route is A-B-E-D-C-A with a total distance of approximately 20.95.For part 2, the total energy expenditure is calculated based on the optimal route from part 1, but considering the elevation gain at D.Wait, no, the problem says: "After deciding the optimal route, the backpacker realizes that one of the locations, say D(8, 5), is on an elevation that requires extra energy to climb. If the energy required to walk between two points (x1, y1) and (x2, y2) is given by the Euclidean distance multiplied by the elevation gain factor (which is 1.5 for D), calculate the total energy expenditure for the entire trip considering this new information."So, the optimal route from part 1 is used, and then the energy is calculated based on that route, with segments involving D multiplied by 1.5.Therefore, the optimal route for part 1 is A-B-E-D-C-A with total distance ≈20.95.Then, for part 2, using this route, the energy expenditure is calculated as follows:Segments:A-B: distance 5, energy 5B-E: ≈1.414, energy ≈1.414E-D: ≈5.831, energy ≈5.831 *1.5≈8.7465D-C: ≈3.606, energy ≈3.606 *1.5≈5.409C-A: ≈5.099, energy ≈5.099Total energy: 5 + 1.414 + 8.7465 + 5.409 + 5.099 ≈25.6685But earlier, I found that the route A-C-D-B-E-A has a lower total energy expenditure, but that's a different route. However, the problem specifies that after deciding the optimal route (which is A-B-E-D-C-A), the energy is calculated based on that route.Therefore, the total energy expenditure is approximately 25.6685.But wait, let me check if the route A-C-D-B-E-A is indeed the optimal for energy, but the problem says to use the optimal route from part 1, which is A-B-E-D-C-A.Therefore, the total energy expenditure is based on that route.Therefore, the answer to part 2 is approximately 25.6685.But let me compute it more accurately.First, let's compute the exact distances without rounding:Distance AB: 5Distance BE: sqrt(2) ≈1.41421356Distance ED: sqrt(34) ≈5.83095189Distance DC: sqrt(13) ≈3.60555128Distance CA: sqrt(26) ≈5.09901951Now, compute the energy:A-B: 5B-E: sqrt(2) ≈1.41421356E-D: sqrt(34) *1.5 ≈5.83095189 *1.5 ≈8.746427835D-C: sqrt(13) *1.5 ≈3.60555128 *1.5 ≈5.40832692C-A: sqrt(26) ≈5.09901951Total energy:5 + 1.41421356 + 8.746427835 + 5.40832692 + 5.09901951Let me add them step by step:5 + 1.41421356 = 6.414213566.41421356 + 8.746427835 ≈15.160641415.1606414 + 5.40832692 ≈20.568968320.5689683 + 5.09901951 ≈25.6679878So, approximately 25.668.But let me check if the route A-C-D-B-E-A is indeed better, but the problem says to use the optimal route from part 1, which is A-B-E-D-C-A.Therefore, the total energy expenditure is approximately 25.668.But wait, the problem says "the energy required to walk between two points (x1, y1) and (x2, y2) is given by the Euclidean distance multiplied by the elevation gain factor (which is 1.5 for D)."Does this mean that only when moving to D, the energy is multiplied by 1.5? Or is it when moving from D as well?The problem states: "the energy required to walk between two points (x1, y1) and (x2, y2) is given by the Euclidean distance multiplied by the elevation gain factor (which is 1.5 for D)."So, for any segment that includes D, whether arriving or departing, the energy is multiplied by 1.5.Therefore, in the route A-B-E-D-C-A, both E-D and D-C segments involve D, so their energies are multiplied by 1.5.Therefore, the total energy is as calculated above: ≈25.668.But let me check if the problem specifies that only the elevation gain when going to D is considered, but I think it's for any movement involving D.Therefore, the total energy expenditure is approximately 25.668.But to be precise, let's compute it without rounding:Compute each segment's energy:A-B: 5B-E: sqrt(2) ≈1.41421356E-D: sqrt(34) *1.5 ≈5.83095189 *1.5 = 8.746427835D-C: sqrt(13) *1.5 ≈3.60555128 *1.5 = 5.40832692C-A: sqrt(26) ≈5.09901951Now, sum them up:5 + 1.41421356 = 6.414213566.41421356 + 8.746427835 = 15.16064139515.160641395 + 5.40832692 = 20.56896831520.568968315 + 5.09901951 = 25.667987825So, approximately 25.668.But let me check if the problem expects the answer to be in terms of exact values or decimal places.Alternatively, maybe I can express the total energy as a sum of exact terms.But perhaps it's better to compute it numerically.So, the total energy expenditure is approximately 25.668.But let me check if I can express it more accurately.Alternatively, perhaps the problem expects the answer to be in terms of exact distances multiplied by 1.5 where applicable.But for the sake of the answer, I think 25.67 is acceptable.Therefore, the answers are:1. The optimal route is A-B-E-D-C-A with a total distance of approximately 20.95 units.2. The total energy expenditure is approximately 25.67 units.But wait, let me check if the problem expects the exact total distance for part 1.The exact total distance for the route A-B-E-D-C-A is:AB: 5BE: sqrt(2)ED: sqrt(34)DC: sqrt(13)CA: sqrt(26)Total distance: 5 + sqrt(2) + sqrt(34) + sqrt(13) + sqrt(26)But perhaps the problem expects the numerical value.5 + 1.4142 + 5.8309 + 3.6056 + 5.0990 ≈20.95So, approximately 20.95.But to be precise, let's compute it:5 + 1.41421356 ≈6.414213566.41421356 + 5.83095189 ≈12.2451654512.24516545 + 3.60555128 ≈15.8507167315.85071673 + 5.09901951 ≈20.94973624So, approximately 20.95.Therefore, the answers are:1. The optimal route is A-B-E-D-C-A with a total distance of approximately 20.95 units.2. The total energy expenditure is approximately 25.67 units.But let me check if the problem expects the answer in terms of exact values or decimal places.Alternatively, perhaps the problem expects the answer to be in terms of exact distances multiplied by 1.5 where applicable.But for the sake of the answer, I think 25.67 is acceptable.Therefore, the final answers are:1. The optimal route is A-B-E-D-C-A with a total distance of approximately 20.95 units.2. The total energy expenditure is approximately 25.67 units.</think>

🧠:<think>Okay, so I have this problem about an author and their editor, and they're using these functions to model satisfaction with the manuscript. Let me try to break it down step by step.First, the function given is ( f(x) = ax^3 + bx^2 + cx + d ). This represents the author's initial satisfaction, where ( x ) is the number of chapters. Then, the editor suggests a transformation, changing it to ( g(x) = f(x - 1) - kx ). Here, ( k ) is a constant representing the editor's influence.The first question is asking: If the author wants to maintain the same overall satisfaction level before and after the editing process for a given number of chapters ( n ), find the relationship between the coefficients ( a, b, c, d ), and the constant ( k ).Alright, so I think this means that for a specific number of chapters ( n ), the satisfaction before editing ( f(n) ) should equal the satisfaction after editing ( g(n) ). So, ( f(n) = g(n) ).Given that ( g(x) = f(x - 1) - kx ), substituting ( x = n ) gives ( g(n) = f(n - 1) - kn ).So, setting ( f(n) = f(n - 1) - kn ). Let me write that equation out:( f(n) = f(n - 1) - kn )Now, let's substitute ( f(n) ) and ( f(n - 1) ) with their expressions in terms of ( a, b, c, d ).( f(n) = a n^3 + b n^2 + c n + d )( f(n - 1) = a (n - 1)^3 + b (n - 1)^2 + c (n - 1) + d )So, plugging these into the equation:( a n^3 + b n^2 + c n + d = [a (n - 1)^3 + b (n - 1)^2 + c (n - 1) + d] - kn )Let me expand ( f(n - 1) ):First, ( (n - 1)^3 = n^3 - 3n^2 + 3n - 1 )Then, ( (n - 1)^2 = n^2 - 2n + 1 )So, substituting back:( f(n - 1) = a(n^3 - 3n^2 + 3n - 1) + b(n^2 - 2n + 1) + c(n - 1) + d )Let me distribute the coefficients:( f(n - 1) = a n^3 - 3a n^2 + 3a n - a + b n^2 - 2b n + b + c n - c + d )Now, combine like terms:- ( n^3 ) term: ( a n^3 )- ( n^2 ) terms: ( (-3a + b) n^2 )- ( n ) terms: ( (3a - 2b + c) n )- Constants: ( (-a + b - c + d) )So, ( f(n - 1) = a n^3 + (-3a + b) n^2 + (3a - 2b + c) n + (-a + b - c + d) )Now, going back to the equation:( f(n) = f(n - 1) - kn )Substitute the expressions:( a n^3 + b n^2 + c n + d = [a n^3 + (-3a + b) n^2 + (3a - 2b + c) n + (-a + b - c + d)] - kn )Let me subtract the right-hand side from both sides to bring everything to the left:( a n^3 + b n^2 + c n + d - [a n^3 + (-3a + b) n^2 + (3a - 2b + c) n + (-a + b - c + d)] + kn = 0 )Simplify term by term:- ( a n^3 - a n^3 = 0 )- ( b n^2 - (-3a + b) n^2 = b n^2 + 3a n^2 - b n^2 = 3a n^2 )- ( c n - (3a - 2b + c) n = c n - 3a n + 2b n - c n = (-3a + 2b) n )- ( d - (-a + b - c + d) = d + a - b + c - d = a - b + c )- Then, we have the ( + kn ) term.Putting it all together:( 3a n^2 + (-3a + 2b) n + (a - b + c) + kn = 0 )Now, let me combine like terms for ( n ):The ( n ) terms are ( (-3a + 2b) n + kn = (-3a + 2b + k) n )So, the equation becomes:( 3a n^2 + (-3a + 2b + k) n + (a - b + c) = 0 )Since this equation must hold true for a specific ( n ), it's a quadratic equation in terms of ( n ). However, since ( n ) is a given number of chapters, it's a specific value, not a variable. Therefore, for the equation to hold, each coefficient must be zero.Wait, hold on. If ( n ) is a specific value, then the equation is a quadratic in ( n ), but since ( n ) is fixed, the coefficients must be zero for the equation to hold for that specific ( n ). Hmm, actually, no. If ( n ) is fixed, then the equation is a scalar equation, not a polynomial identity. So, the entire expression must equal zero for that specific ( n ). Therefore, the equation is:( 3a n^2 + (-3a + 2b + k) n + (a - b + c) = 0 )Which is a linear equation in terms of ( a, b, c, d, k ). So, to find the relationship between these coefficients, we can express this as:( 3a n^2 + (-3a + 2b + k) n + (a - b + c) = 0 )But since this is a single equation with multiple variables, we can express one variable in terms of the others. Let me try to solve for ( k ):First, let's expand the equation:( 3a n^2 - 3a n + 2b n + k n + a - b + c = 0 )Let me group the terms with ( a ), ( b ), ( c ), and ( k ):- Terms with ( a ): ( 3a n^2 - 3a n + a = a(3n^2 - 3n + 1) )- Terms with ( b ): ( 2b n - b = b(2n - 1) )- Terms with ( c ): ( c )- Terms with ( k ): ( k n )So, putting it all together:( a(3n^2 - 3n + 1) + b(2n - 1) + c + k n = 0 )Now, solving for ( k ):( k n = - [a(3n^2 - 3n + 1) + b(2n - 1) + c] )Therefore,( k = - frac{a(3n^2 - 3n + 1) + b(2n - 1) + c}{n} )Assuming ( n neq 0 ), which makes sense because the number of chapters can't be zero.So, that's the relationship between the coefficients and the constant ( k ). Let me just write that more neatly:( k = - frac{a(3n^2 - 3n + 1) + b(2n - 1) + c}{n} )I think that's the answer for part 1.Moving on to part 2:Suppose the author writes a new series, and the editor’s influence is described by a different transformation ( h(x) = f(x) + mx^2 + nx ). Determine the conditions under which the transformation ( h(x) ) results in a polynomial with no real roots, given that ( a, b, c, d, m, ) and ( n ) are real numbers.Alright, so ( h(x) = f(x) + mx^2 + nx ). Since ( f(x) = ax^3 + bx^2 + cx + d ), substituting gives:( h(x) = ax^3 + bx^2 + cx + d + mx^2 + nx )Combine like terms:- ( ax^3 )- ( (b + m) x^2 )- ( (c + n) x )- ( d )So, ( h(x) = ax^3 + (b + m) x^2 + (c + n) x + d )We need to determine the conditions under which this cubic polynomial has no real roots.Hmm, wait. A cubic polynomial always has at least one real root because of the Intermediate Value Theorem. As ( x ) approaches positive infinity, the leading term dominates, so if ( a > 0 ), ( h(x) ) tends to positive infinity, and if ( a < 0 ), it tends to negative infinity. Similarly, as ( x ) approaches negative infinity, the behavior is opposite. Therefore, a cubic polynomial must cross the x-axis at least once. So, it's impossible for a cubic polynomial to have no real roots.But wait, the problem says "results in a polynomial with no real roots". Maybe I'm misunderstanding. Is ( h(x) ) a quadratic? Wait, no, ( h(x) ) is a cubic because ( f(x) ) is a cubic. So, unless the cubic is somehow reduced to a lower degree, but in this case, unless ( a = 0 ), but ( a ) is a coefficient of the cubic term.Wait, but in the original function ( f(x) ), ( a ) is the coefficient of ( x^3 ). If ( a = 0 ), then ( f(x) ) becomes a quadratic, and so ( h(x) ) would also be a quadratic. So, perhaps the problem is considering the case when ( a = 0 ), making ( h(x) ) a quadratic, which can have no real roots.But the problem says "given that ( a, b, c, d, m, ) and ( n ) are real numbers." It doesn't specify that ( a neq 0 ). So, maybe we need to consider both cases: when ( a = 0 ) and when ( a neq 0 ).But if ( a neq 0 ), as I thought earlier, ( h(x) ) is a cubic and must have at least one real root. So, the only way for ( h(x) ) to have no real roots is if it's a quadratic with no real roots. Therefore, ( a ) must be zero, and the resulting quadratic must have a negative discriminant.So, let's formalize that.Case 1: ( a neq 0 ). Then ( h(x) ) is a cubic polynomial, which must have at least one real root. Therefore, no real roots is impossible.Case 2: ( a = 0 ). Then ( h(x) ) becomes a quadratic:( h(x) = (b + m) x^2 + (c + n) x + d )For this quadratic to have no real roots, its discriminant must be negative.The discriminant ( D ) of ( h(x) ) is:( D = [(c + n)]^2 - 4 (b + m) d )So, for no real roots, we need:( (c + n)^2 - 4 (b + m) d < 0 )Therefore, the conditions are:1. ( a = 0 )2. ( (c + n)^2 - 4 (b + m) d < 0 )So, putting it all together, the transformation ( h(x) ) results in a polynomial with no real roots if and only if ( a = 0 ) and the discriminant of the quadratic is negative.Let me just double-check:If ( a = 0 ), then ( h(x) ) is quadratic. For quadratics, no real roots when discriminant is negative. If ( a neq 0 ), it's a cubic, which always has at least one real root. So yes, that seems correct.Therefore, the conditions are ( a = 0 ) and ( (c + n)^2 < 4 (b + m) d ).I think that's the answer for part 2.Final Answer1. The relationship is ( boxed{k = -dfrac{a(3n^2 - 3n + 1) + b(2n - 1) + c}{n}} ).2. The conditions are ( boxed{a = 0} ) and ( boxed{(c + n)^2 < 4(b + m)d} ).</think>

🧠:<think>Alright, so I've got this problem about a lawmaker evaluating a new digital solution for distributing welfare funds. It's divided into two parts, and I need to tackle both. Let me start with the first part.Problem 1: Formulate the Distribution Equation and Prove FairnessOkay, the city has (n) eligible recipients, each with a need score (s_i). The total welfare fund is (W), and the goal is to distribute (a_i) to each recipient such that (a_i) is proportional to (s_i). So, I need to come up with an equation that distributes (W) among the (n) recipients proportionally to their need scores.First, proportionality usually means that each (a_i) is a multiple of (s_i). So, I can write (a_i = k cdot s_i), where (k) is the constant of proportionality. But since the total amount distributed must be (W), I can sum all (a_i) and set it equal to (W).So, summing over all recipients:[sum_{i=1}^{n} a_i = W]Substituting (a_i = k cdot s_i):[sum_{i=1}^{n} k cdot s_i = W]Factor out (k):[k cdot sum_{i=1}^{n} s_i = W]Therefore, solving for (k):[k = frac{W}{sum_{i=1}^{n} s_i}]So, the amount each recipient gets is:[a_i = frac{W cdot s_i}{sum_{i=1}^{n} s_i}]That seems right. Now, I need to prove that the ratio (frac{a_i}{s_i}) is constant for all (i). Let's compute that ratio:[frac{a_i}{s_i} = frac{frac{W cdot s_i}{sum_{i=1}^{n} s_i}}{s_i} = frac{W}{sum_{i=1}^{n} s_i}]Which is indeed a constant because it doesn't depend on (i). So, this shows that the ratio is the same for all recipients, meaning the distribution is fair in terms of proportionality.Problem 2: Define Efficiency Gain and Compare MethodsNow, the second part is about comparing the new digital solution to the previous method, which distributed funds equally. The previous method gave each recipient an average amount (bar{a}). I need to define the efficiency gain (E) and derive an expression for it.First, let's recall that efficiency in this context might relate to how well the funds are targeted to those who need them more. The previous method was equal distribution, so each recipient got (bar{a} = frac{W}{n}).The new method distributes (a_i = frac{W cdot s_i}{sum s_i}). So, to measure efficiency gain, perhaps we can look at how much more (or less) the new method gives to each recipient compared to the old method, but in a way that aggregates across all recipients.Alternatively, efficiency could be measured by how much better the funds are aligned with the need scores. Maybe we can use some sort of metric, like the sum of the products of need scores and the amounts distributed, but I need to think carefully.Wait, the problem says to define efficiency gain (E) in terms of (W), (n), and (s_i). So, perhaps (E) is the difference in some measure between the new and old distributions.Let me consider what the previous method does: it gives each recipient (bar{a} = frac{W}{n}). So, the total welfare is still (W), but distributed equally.The new method gives more to those with higher (s_i) and less to those with lower (s_i). So, the efficiency gain could be how much better the new distribution is at targeting the funds to those with higher needs.One way to measure this is to compute the sum of (a_i cdot s_i) for both methods and see which one is higher. A higher sum would mean that more funds are going to those with higher need scores, which could be considered more efficient.Let me define (E) as the difference in the total "need-weighted" welfare between the new and old methods.So, for the new method, the total need-weighted welfare is:[sum_{i=1}^{n} a_i cdot s_i = sum_{i=1}^{n} left( frac{W s_i}{sum s_j} right) s_i = frac{W}{sum s_j} sum_{i=1}^{n} s_i^2]For the old method, each (a_i = frac{W}{n}), so the total need-weighted welfare is:[sum_{i=1}^{n} frac{W}{n} cdot s_i = frac{W}{n} sum_{i=1}^{n} s_i]Therefore, the efficiency gain (E) can be defined as:[E = frac{W}{sum s_j} sum_{i=1}^{n} s_i^2 - frac{W}{n} sum_{i=1}^{n} s_i]Simplify this expression:Factor out (W):[E = W left( frac{sum s_i^2}{sum s_i} - frac{sum s_i}{n} right )]Combine the terms:[E = W left( frac{sum s_i^2}{sum s_i} - frac{sum s_i}{n} right ) = W left( frac{n sum s_i^2 - (sum s_i)^2}{n sum s_i} right )]So, that's the expression for (E). Now, to discuss under what conditions the new method is more efficient, we need (E > 0).So, when is (E > 0)?Looking at the numerator:[n sum s_i^2 - (sum s_i)^2 > 0]This is equivalent to:[n sum s_i^2 > (sum s_i)^2]Which can be rewritten as:[sum s_i^2 > frac{(sum s_i)^2}{n}]This is the condition that the average of the squares is greater than the square of the average, which is always true unless all (s_i) are equal. Because of the Cauchy-Schwarz inequality, we know that:[sum s_i^2 geq frac{(sum s_i)^2}{n}]With equality if and only if all (s_i) are equal.Therefore, unless all need scores are equal, the new method will have a higher efficiency gain (E > 0). So, the new digital solution is more efficient than the previous method as long as there is some variation in the need scores (s_i). If all (s_i) are the same, then both methods are equally efficient.Wait, but in reality, if all (s_i) are equal, then the new method would actually distribute equally as well, because (a_i) would be the same for all. So, in that case, both methods are the same, so efficiency gain is zero. But if there is any variation, the new method is more efficient.So, summarizing:- The efficiency gain (E) is given by (E = W left( frac{sum s_i^2}{sum s_i} - frac{sum s_i}{n} right )).- The new method is more efficient than the old one when there is variation in the need scores, i.e., when not all (s_i) are equal.Alternatively, since (E) is proportional to the difference between the mean of squares and the square of the mean, which is related to the variance. So, higher variance in need scores leads to higher efficiency gain.Double-Checking My WorkFor Problem 1, I think I did it correctly. The proportionality leads to each (a_i) being a multiple of (s_i), and the constant comes out naturally from the total sum. The ratio (frac{a_i}{s_i}) is indeed constant, which shows fairness.For Problem 2, I considered defining efficiency gain based on the need-weighted welfare. The difference between the new and old method's total need-weighted welfare gives the efficiency gain. The derivation seems correct, and the condition for (E > 0) makes sense because if all need scores are equal, the distributions are the same, so no gain. If there's any variation, the new method is more efficient.I think that's solid. I don't see any mistakes in the reasoning.Final Answer1. The distribution equation is (boxed{a_i = dfrac{W s_i}{sum_{j=1}^{n} s_j}}) and the ratio (dfrac{a_i}{s_i}) is constant for all (i).2. The efficiency gain (E) is (boxed{E = W left( dfrac{sum_{i=1}^{n} s_i^2}{sum_{i=1}^{n} s_i} - dfrac{sum_{i=1}^{n} s_i}{n} right)}), and the new solution is more efficient when there is variation in the need scores (s_i).</think>