🧠:<think>Alright, let's tackle these two problems one by one. Both seem a bit tricky, but I'll try to break them down step by step.Problem 1: Optimal Position for the Cardiologist's TableFirst, the conference hall is arranged in a 4x4 grid of tables, each seating 8 people. The goal is to place the cardiologist at a table such that the sum of the shortest paths from the cardiologist to all other attendees is minimized. Hmm, okay. So, we have a grid of 16 tables (since 4x4 is 16). Each table can seat 8 people, and there are 128 attendees in total (16 tables * 8 people). The cardiologist is one of these attendees, so we need to place their table in a position that minimizes the total distance to all other 127 attendees.Wait, but the problem says the sum of the shortest paths from the cardiologist to all other attendees. So, we need to calculate, for each possible table, the sum of the shortest path distances from that table to every other table, multiplied by the number of people at each table. Since each table has 8 people, except maybe the cardiologist's table, which has 7 other people? Or is the cardiologist just one person, so their table has 7 others?Wait, actually, the total number of attendees is 128, and each table seats 8. So, including the cardiologist, there are 16 tables, each with 8 people. So, the cardiologist is just one person at one of these tables, and the rest are 127 attendees. So, the distance from the cardiologist's table to each other table is the same for all 8 people at that table? Or is it just the distance to the table, and then multiplied by the number of people?Wait, the problem says "the sum of the shortest paths from the cardiologist to all other attendees." So, each attendee is a person, and we need to calculate the shortest path from the cardiologist's table to each attendee's table, and sum all those distances.But since each table has 8 people, except the cardiologist's table, which has 7 other people. Wait, no, the cardiologist is one person, so their table has 7 other people, and the other tables have 8 people each.So, the total sum would be: for each table, the distance from the cardiologist's table to that table multiplied by the number of people at that table. Except for the cardiologist's own table, which would have 7 people (since the cardiologist is one person, so 8 - 1 = 7).So, the formula would be: Sum = (distance from cardiologist's table to itself * 7) + sum over all other tables (distance * 8).But the distance from the cardiologist's table to itself is zero, so that term is zero. Therefore, the total sum is just the sum over all other tables of (distance * 8).Therefore, to minimize the total sum, we need to minimize the sum of distances from the cardiologist's table to all other tables, multiplied by 8. Since 8 is a constant factor, it's equivalent to minimizing the sum of distances.So, the problem reduces to finding the table in the 4x4 grid that minimizes the sum of Manhattan distances to all other tables.Wait, the grid is 4x4, so each table can be identified by its coordinates (i, j), where i and j range from 1 to 4.The Manhattan distance between two tables (i1, j1) and (i2, j2) is |i1 - i2| + |j1 - j2|.So, we need to find the table (i, j) such that the sum over all (k, l) of |i - k| + |j - l| is minimized.This is equivalent to finding the geometric median of the grid points. In a grid, the geometric median is often the center point(s). For a 4x4 grid, the center would be around the middle two rows and middle two columns.Wait, in a 4x4 grid, the center is between the 2nd and 3rd rows and columns. So, the optimal points would be the four center tables: (2,2), (2,3), (3,2), (3,3). Let me verify this.Let me calculate the sum of Manhattan distances for each table.First, let's list all tables as (i, j) where i, j ∈ {1,2,3,4}.For each table (i,j), compute sum_{k=1 to 4} sum_{l=1 to 4} (|i - k| + |j - l|).But since each table has 8 people, except the cardiologist's table, which has 7, but as we saw earlier, the total sum is 8 * sum of distances, so we can ignore the 8 for now and just find the table with the minimal sum of distances.Let's compute the sum for each table.Starting with (1,1):Sum = sum_{k=1 to 4} sum_{l=1 to 4} (|1 - k| + |1 - l|)For each row k:- k=1: sum_{l=1 to 4} (0 + |1 - l|) = 0 + (0 + 1 + 2 + 3) = 6- k=2: sum_{l=1 to 4} (1 + |1 - l|) = 1*4 + (0 + 1 + 2 + 3) = 4 + 6 = 10- k=3: sum_{l=1 to 4} (2 + |1 - l|) = 2*4 + 6 = 8 + 6 = 14- k=4: sum_{l=1 to 4} (3 + |1 - l|) = 3*4 + 6 = 12 + 6 = 18Total sum = 6 + 10 + 14 + 18 = 48Similarly, for (1,2):Sum = sum_{k=1 to 4} sum_{l=1 to 4} (|1 - k| + |2 - l|)For each row k:- k=1: sum_{l=1 to 4} (0 + |2 - l|) = 0 + (1 + 0 + 1 + 2) = 4- k=2: sum_{l=1 to 4} (1 + |2 - l|) = 1*4 + (1 + 0 + 1 + 2) = 4 + 4 = 8- k=3: sum_{l=1 to 4} (2 + |2 - l|) = 2*4 + 4 = 8 + 4 = 12- k=4: sum_{l=1 to 4} (3 + |2 - l|) = 3*4 + 4 = 12 + 4 = 16Total sum = 4 + 8 + 12 + 16 = 40Wait, that's better. Let's do (1,3):Similarly, it should be symmetric to (1,2), so sum would be 40.And (1,4):Sum would be similar to (1,1), so 48.Now, moving to row 2:(2,1):Sum = sum_{k=1 to 4} sum_{l=1 to 4} (|2 - k| + |1 - l|)For each row k:- k=1: sum_{l=1 to 4} (1 + |1 - l|) = 1*4 + (0 + 1 + 2 + 3) = 4 + 6 = 10- k=2: sum_{l=1 to 4} (0 + |1 - l|) = 0 + 6 = 6- k=3: sum_{l=1 to 4} (1 + |1 - l|) = 1*4 + 6 = 4 + 6 = 10- k=4: sum_{l=1 to 4} (2 + |1 - l|) = 2*4 + 6 = 8 + 6 = 14Total sum = 10 + 6 + 10 + 14 = 40Similarly, (2,2):Sum = sum_{k=1 to 4} sum_{l=1 to 4} (|2 - k| + |2 - l|)For each row k:- k=1: sum_{l=1 to 4} (1 + |2 - l|) = 1*4 + (1 + 0 + 1 + 2) = 4 + 4 = 8- k=2: sum_{l=1 to 4} (0 + |2 - l|) = 0 + 4 = 4- k=3: sum_{l=1 to 4} (1 + |2 - l|) = 4 + 4 = 8- k=4: sum_{l=1 to 4} (2 + |2 - l|) = 8 + 4 = 12Total sum = 8 + 4 + 8 + 12 = 32Wait, that's better. Let's check (2,3):It should be symmetric to (2,2), so sum is 32.(2,4):Sum would be similar to (2,1), which was 40.Now, row 3:(3,1):Sum should be same as (2,1), which was 40.(3,2):Same as (2,2), sum is 32.(3,3):Same as (2,2), sum is 32.(3,4):Same as (2,4), sum is 40.Row 4:(4,1):Same as (1,1), sum is 48.(4,2):Same as (1,2), sum is 40.(4,3):Same as (1,3), sum is 40.(4,4):Same as (1,1), sum is 48.So, compiling the sums:- Corners: (1,1), (1,4), (4,1), (4,4) have sum 48- Edge centers: (1,2), (1,3), (2,1), (2,4), (3,1), (3,4), (4,2), (4,3) have sum 40- Center tables: (2,2), (2,3), (3,2), (3,3) have sum 32So, the minimal sum is 32, achieved by the four center tables: (2,2), (2,3), (3,2), (3,3).Therefore, the optimal position for the cardiologist's table is any of these four center tables.But the question asks for "the optimal position", so perhaps any of them is acceptable, but maybe the exact coordinates are needed. Since the grid is 4x4, the center is between 2 and 3 in both rows and columns, so the four center tables are indeed the optimal.So, the answer is that the cardiologist should be placed at one of the four center tables, specifically at positions (2,2), (2,3), (3,2), or (3,3).But since the problem might expect a single answer, perhaps the exact coordinates, but since all four are equally optimal, we can choose any. Maybe the exact answer is the center, but in a 4x4 grid, there isn't a single center, so the four are the optimal.Problem 2: Seating Arrangement for Balanced SpecialtiesNow, the second problem is about seating arrangement to maximize the number of tables where each table has at least one individual from each specialty: cardiologists, surgeons, and general practitioners.Given:- Total attendees: 128- Each table seats 8 people- Number of tables: 16 (since 128 / 8 = 16)- Specialties: - 64 cardiologists - 32 surgeons - 32 general practitionersWe need to arrange them such that as many tables as possible have at least one from each specialty.So, the goal is to maximize the number of tables with at least one cardiologist, one surgeon, and one general practitioner.First, let's note the numbers:- Cardiologists: 64 (half of 128)- Surgeons: 32 (quarter)- General practitioners: 32 (quarter)Each table has 8 people. We need to distribute these specialties across the 16 tables.To maximize the number of tables with all three specialties, we need to ensure that each such table has at least one cardiologist, one surgeon, and one GP.But given the numbers, we have to see how many tables can have all three.Let me think about the constraints.Each table needs at least one of each specialty, so for each such table, we need to allocate at least 3 people: 1 cardiologist, 1 surgeon, 1 GP. The remaining 5 can be any specialty.But we have 64 cardiologists, which is a lot. 32 surgeons and 32 GPs.If we try to maximize the number of tables with all three, let's denote T as the number of tables with all three.Each such table uses 1 cardiologist, 1 surgeon, and 1 GP.So, the total used would be T cardiologists, T surgeons, T GPs.The remaining cardiologists: 64 - TThe remaining surgeons: 32 - TThe remaining GPs: 32 - TThese remaining people need to be distributed across the remaining 16 - T tables.But each of these remaining tables can have any combination, but we want to maximize T, so we need to ensure that the remaining people can be distributed without violating the counts.But also, the remaining tables can have any number of people, but each table must have 8 people.Wait, but the remaining tables can have any specialties, but we need to make sure that the remaining people can fill the tables.Let me think in terms of the maximum possible T.We have 32 surgeons and 32 GPs. Each table with all three uses 1 surgeon and 1 GP. So, the maximum T is limited by the number of surgeons and GPs, since each T requires 1 of each.We have 32 surgeons and 32 GPs, so T cannot exceed 32, but we only have 16 tables. So, T is limited by 16.But wait, each table requires 1 surgeon and 1 GP, so with 32 surgeons and 32 GPs, we could potentially have 32 tables, but we only have 16 tables. So, T can be up to 16, but let's check if the cardiologists can support that.Each table with all three uses 1 cardiologist, so 16 tables would use 16 cardiologists. We have 64, so that's fine.But then, the remaining cardiologists would be 64 - 16 = 48.The remaining surgeons: 32 - 16 = 16The remaining GPs: 32 - 16 = 16These 48 cardiologists, 16 surgeons, and 16 GPs need to be distributed across the remaining 0 tables? Wait, no, because if T=16, then all tables are already used for T, so there are no remaining tables. Wait, no, T is the number of tables with all three, but the total number of tables is 16. So, if T=16, all tables have all three specialties.But let's check if that's possible.Each table would have 1 cardiologist, 1 surgeon, 1 GP, and 5 others. But the total number of cardiologists would be 16, but we have 64. So, we can't have T=16 because we have more cardiologists than that.Wait, no, T=16 would require 16 cardiologists, 16 surgeons, and 16 GPs. But we have 64 cardiologists, which is more than enough. The issue is that after assigning 16 cardiologists, 16 surgeons, and 16 GPs, we still have 48 cardiologists left. These 48 need to be distributed across the 16 tables, each table already has 1 cardiologist, so we can add 3 more cardiologists to each table, making each table have 4 cardiologists, 1 surgeon, 1 GP, and 2 others? Wait, no, each table must have 8 people.Wait, let's recast this.If T=16, each table has 1 cardiologist, 1 surgeon, 1 GP, and 5 others. But the total cardiologists used would be 16, but we have 64, so we have 48 extra cardiologists to distribute.Similarly, surgeons and GPs are exactly used up: 16 each, which matches our counts.So, the remaining 48 cardiologists need to be distributed across the 16 tables. Each table can take up to 7 more cardiologists (since they already have 1), but each table must have 8 people.Wait, no, each table already has 1 cardiologist, 1 surgeon, 1 GP, and 5 others. But the 5 others can be any specialty, including cardiologists.So, to distribute the remaining 48 cardiologists, we can add them to the tables. Each table can take up to 7 more cardiologists (since they already have 1), but we have 48 to distribute.48 / 16 tables = 3 per table. So, each table can have 1 + 3 = 4 cardiologists, 1 surgeon, 1 GP, and 2 others (but wait, 4 +1 +1 +2=8). But the others can be any, but we have no other specialties left except cardiologists, because surgeons and GPs are already used up.Wait, no, because we have 64 cardiologists, 32 surgeons, 32 GPs. If we assign 16 cardiologists, 16 surgeons, 16 GPs to the T=16 tables, the remaining are 48 cardiologists, 16 surgeons, 16 GPs. Wait, no, because 32 -16=16 surgeons and GPs left.Wait, no, if T=16, each table uses 1 surgeon and 1 GP, so total used is 16 surgeons and 16 GPs, leaving 16 surgeons and 16 GPs. But we have 64 cardiologists, so 64 -16=48 cardiologists left.So, the remaining 16 surgeons and 16 GPs need to be distributed across the 16 tables, along with the 48 cardiologists.Each table already has 1 cardiologist, 1 surgeon, 1 GP, and 5 others. So, the 5 others can be filled with the remaining cardiologists, surgeons, and GPs.But we have 48 cardiologists, 16 surgeons, 16 GPs left.Each table can take up to 5 more people. Let's see:Total remaining people: 48 +16 +16=80But we have 16 tables, each needing 5 more people: 16*5=80. Perfect.So, we can distribute the remaining 48 cardiologists, 16 surgeons, and 16 GPs across the 16 tables, 5 per table.But we need to ensure that each table ends up with at least one of each specialty. Wait, no, because we already assigned 1 of each to each table, so even if we add more, each table will have at least one of each.But the problem is, can we distribute the remaining 48 cardiologists, 16 surgeons, and 16 GPs into the 16 tables, 5 per table, without violating any constraints.Yes, because 48 +16 +16=80, and 16 tables *5=80.So, each table can get 3 cardiologists, 1 surgeon, and 1 GP in the remaining 5 seats. That would use up 3*16=48 cardiologists, 1*16=16 surgeons, and 1*16=16 GPs.So, each table would end up with:- 1 (initial) +3=4 cardiologists- 1 (initial) +1=2 surgeons- 1 (initial) +1=2 GPsTotal per table: 4+2+2=8.So, yes, this works.Therefore, it's possible to have all 16 tables have at least one of each specialty.Wait, but that seems too good. Let me double-check.Total cardiologists used: 16 (initial) +48 (additional) =64 ✔️Total surgeons used:16 (initial) +16 (additional)=32 ✔️Total GPs used:16 (initial) +16 (additional)=32 ✔️Yes, that works.But wait, the problem says "maximize the number of tables where each table has at least one individual from each specialty". So, if we can have all 16 tables satisfy this, that's the maximum possible.But is that feasible? Because each table would have 4 cardiologists, 2 surgeons, and 2 GPs, which is fine.So, the answer is that it's possible to have all 16 tables each have at least one cardiologist, one surgeon, and one GP.But wait, the problem might be more complex because the initial assignment is 1 of each, and then adding more. But maybe there's a better way.Alternatively, perhaps the maximum number of tables with all three is limited by the number of surgeons and GPs, which are 32 each. Since each table needs at least one surgeon and one GP, the maximum number of tables that can have all three is 32, but since we only have 16 tables, it's possible to have all 16 tables have all three.Wait, but 32 surgeons and 32 GPs can support 32 tables, but we only have 16 tables, so yes, all 16 can have all three.Therefore, the optimal seating arrangement is to have each table have at least one cardiologist, one surgeon, and one GP, and the remaining seats can be filled with cardiologists, since we have plenty.So, the answer is that all 16 tables can have at least one of each specialty.But let me think again. If we have 16 tables, each needing at least 1 surgeon and 1 GP, we need 16 surgeons and 16 GPs. We have 32 each, so that's fine. The remaining 16 surgeons and 16 GPs can be distributed as needed.But the cardiologists are 64, so each table can have 4 cardiologists, as above.Yes, so the maximum number is 16.But wait, the problem says "maximize the number of tables where each table has at least one individual from each specialty". So, the answer is 16 tables.But let me think if there's a constraint I'm missing. For example, can we actually arrange the seating so that each table has at least one of each specialty?Yes, as shown above.So, the answer is that all 16 tables can have at least one cardiologist, one surgeon, and one GP.But let me think about the exact distribution.Each table:- 4 cardiologists- 2 surgeons- 2 GPsTotal per table: 8.Total across all tables:- 4*16=64 cardiologists ✔️- 2*16=32 surgeons ✔️- 2*16=32 GPs ✔️Perfect.Therefore, the optimal arrangement is to have each table consist of 4 cardiologists, 2 surgeons, and 2 GPs.But the problem says "maximize the number of tables where each table has at least one individual from each specialty". So, the maximum is 16 tables.But perhaps the answer expects the number, which is 16, or the arrangement.But the question says "Find a seating arrangement for each table that maximizes the number of tables where each table has at least one individual from each specialty".So, the answer is that all 16 tables can have at least one of each specialty, with the distribution as above.But maybe the problem expects a different approach. Let me think differently.Suppose we don't assign 1 of each to each table first, but instead try to distribute them optimally.We have 64 cardiologists, which is 4 times the number of tables (16). So, if we spread them evenly, each table can have 4 cardiologists.Similarly, surgeons and GPs are 32 each, which is 2 per table.So, each table can have 4 cardiologists, 2 surgeons, and 2 GPs, which is exactly the distribution we had earlier.Therefore, this is the optimal arrangement, and it allows all 16 tables to have at least one of each specialty.So, the answer is that all 16 tables can have at least one cardiologist, one surgeon, and one GP, with the remaining seats filled with cardiologists.</think>

🧠:<think>First, I'll calculate the total number of hospital visits for each neighborhood by multiplying the number of residents by the percentage of residents with the illness and then by the average number of hospital visits per resident.For Neighborhood A:1200 residents * 10% = 120 residents with the illness.120 residents * 3 visits = 360 visits.For Neighborhood B:800 residents * 15% = 120 residents with the illness.120 residents * 2 visits = 240 visits.For Neighborhood C:1000 residents * 12% = 120 residents with the illness.120 residents * 4 visits = 480 visits.For Neighborhood D:600 residents * 20% = 120 residents with the illness.120 residents * 5 visits = 600 visits.For Neighborhood E:1400 residents * 8% = 112 residents with the illness.112 residents * 1 visit = 112 visits.Next, I'll sum the visits from all neighborhoods to find the total number of hospital visits:360 (A) + 240 (B) + 480 (C) + 600 (D) + 112 (E) = 1,792 visits.For prioritizing interventions, I'll calculate the score for each neighborhood by multiplying the percentage of residents with the illness by the average number of hospital visits per resident.Neighborhood A: 10% * 3 = 0.3Neighborhood B: 15% * 2 = 0.3Neighborhood C: 12% * 4 = 0.48Neighborhood D: 20% * 5 = 1.0Neighborhood E: 8% * 1 = 0.08Neighborhood D has the highest score of 1.0, indicating it should be prioritized for intervention.</think>

🧠:<think>Alright, so I've got this problem about designing a mural on a curved wall. Let me try to break it down step by step. First, the wall is part of a circular garden with a radius of 20 meters. The mural spans an arc of 90 degrees. I need to calculate the length of this arc. Hmm, I remember that the formula for the arc length is something like (θ/360) * 2πr, where θ is the central angle in degrees. Let me confirm that. Yeah, that sounds right because the circumference is 2πr, and if you take a fraction of that circumference based on the angle, it should give the arc length.So, plugging in the numbers: θ is 90 degrees, radius r is 20 meters. So, arc length L = (90/360) * 2π*20. Let me compute that. 90 divided by 360 is 1/4. So, 1/4 of 2π*20. 2π*20 is 40π. Then, 1/4 of 40π is 10π. So, the arc length is 10π meters. That's approximately 31.42 meters, but I should keep it as 10π for exactness.Now, the next part is to determine the dimensions of a rectangular canvas for the sketch at a scale of 1:50. So, the sketch is a scaled-down version. Since the mural is on a curved wall, but the sketch is rectangular, I think the length of the canvas should correspond to the arc length, and the width should correspond to the height of the wall. Wait, but the problem doesn't specify the height of the wall. Hmm, maybe I need to assume it's the same as the arc length? Or perhaps the mural is just a flat rectangle on the curved wall, so the width is the chord length?Wait, hold on. The wall is a section of the outer circumference, which is curved. So, the mural is on this curved wall, but when you sketch it on a canvas, you have to represent that curve on a flat surface. Hmm, but the problem says it's a rectangular canvas. Maybe they just want the length of the canvas to be the arc length, and the width to be the height of the mural. But since the height isn't given, maybe the mural is just a flat rectangle on the curved wall, so the width is the same as the chord length?Wait, I'm getting confused. Let me reread the problem. It says, "the mural spans an arc of 90 degrees on this circle." So, the mural is along the arc, which is 10π meters long. But it's on a curved wall, so the height of the mural would be the same as the radius? Or maybe the height is the same as the chord length?Wait, no, the chord length is the straight line connecting the two ends of the arc. Let me calculate that. The chord length formula is 2r sin(θ/2). So, θ is 90 degrees, so θ/2 is 45 degrees. So, chord length is 2*20*sin(45°). Sin(45°) is √2/2, so chord length is 2*20*(√2/2) = 20√2 meters. That's approximately 28.28 meters.But the problem doesn't specify the height of the mural. Hmm. Maybe the mural is just a flat rectangle on the curved wall, so the width is the chord length, and the length is the arc length? But that would make the canvas a rectangle with length 10π and width 20√2. Then, scaling that down by 1:50 would give the sketch dimensions.Wait, but the problem says "the sketch is a scaled-down version of the actual mural." So, if the mural is on a curved wall, but the sketch is rectangular, perhaps the sketch just represents the flat dimensions, which would be the chord length as the width and the arc length as the length? Or maybe the mural is a flat rectangle on the curved wall, so it's a rectangle with length equal to the arc length and height equal to the radius? I'm not sure.Wait, maybe I'm overcomplicating. The problem says the mural spans an arc of 90 degrees, so the length is the arc length, which is 10π. The width of the mural, since it's on a circular wall, would be the same as the radius? Or perhaps the height is the radius? Wait, the radius is 20 meters, but the mural is on the outer circumference, so the height of the mural would be the same as the radius? Or maybe the height is the same as the chord length?I think I need to clarify. If the mural is on a circular wall, the height of the mural would be the same as the radius, because the wall curves outwards. So, the mural would be a rectangle with length equal to the arc length (10π) and height equal to the radius (20 meters). So, the dimensions of the mural are 10π meters by 20 meters.Therefore, the sketch at a scale of 1:50 would have dimensions (10π)/50 by 20/50. Let me compute that. 10π divided by 50 is π/5, which is approximately 0.628 meters, or 62.8 centimeters. 20 divided by 50 is 0.4 meters, or 40 centimeters. So, the sketch canvas would be approximately 62.8 cm by 40 cm. But since the problem asks for exact dimensions, I should keep it in terms of π. So, the length is π/5 meters, and the width is 0.4 meters.Wait, but the problem says "rectangular canvas that the mural artist should use to sketch the design at a scale of 1:50." So, maybe the length is the arc length, and the width is the chord length? Because the chord length is the straight line, which would be the width of the mural on the curved wall. So, chord length is 20√2 meters, as I calculated earlier. So, scaling that down, the width would be (20√2)/50 = (2√2)/5 meters, which is approximately 0.5657 meters or 56.57 cm.But now I'm confused because I don't know whether the width of the mural is the chord length or the radius. The problem doesn't specify. Hmm. Maybe I should assume that the mural is a flat rectangle on the curved wall, so the width is the same as the radius? Or perhaps the height is the same as the radius? Wait, if the wall is part of a circular garden with radius 20 meters, the height of the wall would be the same as the radius? Or is the height separate?Wait, no, the radius is 20 meters, so the wall is 20 meters from the center. The height of the wall isn't specified, so maybe the mural is just a flat rectangle with length equal to the arc length and height equal to the radius? Or perhaps the height is arbitrary? Hmm.Wait, maybe the problem is only concerned with the arc length for the length of the canvas, and the width is not specified because it's a curved wall. But the sketch is rectangular, so perhaps the width is the same as the diameter? No, that doesn't make sense.Wait, perhaps I'm overcomplicating. The problem says the mural spans an arc of 90 degrees on the circle with radius 20 meters. So, the length of the mural is the arc length, which is 10π meters. The width of the mural, since it's on a circular wall, would be the same as the radius? Or perhaps the width is the same as the chord length? I think I need to make an assumption here.Alternatively, maybe the mural is a flat rectangle on the curved wall, so the width is the same as the radius. So, the dimensions of the mural are 10π meters by 20 meters. Then, scaling down by 1:50, the sketch would be (10π)/50 by 20/50, which is π/5 meters by 0.4 meters. So, approximately 0.628 meters by 0.4 meters.Alternatively, if the width is the chord length, which is 20√2 meters, then scaling down would give (20√2)/50 = (2√2)/5 meters, which is approximately 0.5657 meters. But I think the more logical assumption is that the width is the radius, because the mural is on the outer circumference, so the height would be the radius. So, I'll go with that.So, the dimensions of the sketch would be π/5 meters by 0.4 meters. To express this in centimeters, π/5 meters is approximately 62.8 cm, and 0.4 meters is 40 cm. So, the sketch canvas is approximately 62.8 cm by 40 cm.Wait, but the problem says "determine the dimensions of a rectangular canvas." It doesn't specify whether it's length and width or something else. So, maybe it's just the length of the arc, which is 10π meters, and the height is the radius, 20 meters, so scaling down both by 1:50. So, length becomes 10π/50 = π/5 meters, and height becomes 20/50 = 0.4 meters. So, the canvas is π/5 meters by 0.4 meters.I think that's the way to go. So, part 1 is done.Now, moving on to part 2. The color intensity function is given by I(x, y) = 100 sin(πx/L) cos(πy/W), where L and W are the dimensions of the mural. From part 1, L is the arc length, which is 10π meters, and W is the width, which I assumed is 20 meters. So, L = 10π, W = 20.So, plugging these into the function: I(x, y) = 100 sin(πx/(10π)) cos(πy/20). Simplify that: sin(πx/(10π)) is sin(x/10), and cos(πy/20) is cos(πy/20). So, I(x, y) = 100 sin(x/10) cos(πy/20).We need to find the maximum color intensity and the coordinates where this maximum occurs.First, let's recall that the maximum value of sin and cos functions is 1. So, the maximum value of I(x, y) would be when both sin(x/10) and cos(πy/20) are equal to 1. So, the maximum intensity would be 100 * 1 * 1 = 100.Now, we need to find the coordinates (x, y) where this occurs.For sin(x/10) = 1, we have x/10 = π/2 + 2πk, where k is an integer. So, x = 10*(π/2 + 2πk) = 5π + 20πk. But since the mural's length is 10π meters, x must be between 0 and 10π. So, let's find k such that x is within this range.When k=0: x = 5π ≈ 15.707 meters.When k=1: x = 5π + 20π = 25π ≈ 78.539 meters, which is way beyond 10π, so that's outside the mural.Similarly, negative k would give negative x, which is also outside the range. So, the only solution within 0 to 10π is x = 5π.Similarly, for cos(πy/20) = 1, we have πy/20 = 2πn, where n is an integer. So, y = 20n. Since the width of the mural is 20 meters, y must be between 0 and 20. So, n=0 gives y=0, n=1 gives y=20, which is the edge. So, y=0 and y=20 are both within the range.But wait, cos(πy/20) = 1 when πy/20 is 0, 2π, 4π, etc. So, y=0, 40, 80,... but since y is only up to 20, the only solutions are y=0 and y=20.But wait, at y=0 and y=20, cos(πy/20) = cos(0) = 1 and cos(π*20/20) = cos(π) = -1. Wait, hold on. cos(πy/20) at y=20 is cos(π) = -1, not 1. So, that's a mistake.Wait, let's correct that. The general solution for cos(θ) = 1 is θ = 2πn, where n is integer. So, πy/20 = 2πn => y = 40n.But since y is between 0 and 20, the only possible n is 0, giving y=0. So, y=0 is the only point where cos(πy/20) = 1.Wait, but at y=20, cos(π*20/20) = cos(π) = -1, so that's the minimum. So, the maximum occurs only at y=0.But wait, let me double-check. If y=0, cos(0) = 1. If y=20, cos(π) = -1. So, yes, only y=0 gives cos(πy/20) = 1.Therefore, the maximum intensity occurs at x=5π and y=0.So, the coordinates are (5π, 0). But wait, let me confirm the dimensions. The mural is 10π meters in length (x-axis) and 20 meters in width (y-axis). So, x ranges from 0 to 10π, and y ranges from 0 to 20.So, at x=5π, which is halfway along the length, and y=0, which is the bottom edge, the intensity is maximum.Therefore, the maximum color intensity is 100, occurring at (5π, 0).Wait, but let me think again. The function is I(x, y) = 100 sin(x/10) cos(πy/20). So, the maximum of sin(x/10) is 1 at x=5π, and the maximum of cos(πy/20) is 1 at y=0. So, their product is 1, so I=100.Alternatively, could there be a point where sin(x/10) and cos(πy/20) are both 1? Yes, at x=5π and y=0, as we found.But wait, is there another point where sin(x/10) is 1 and cos(πy/20) is 1? Well, sin(x/10)=1 only at x=5π + 20πk, but within 0 to 10π, only x=5π. Similarly, cos(πy/20)=1 only at y=0 + 40n, but within 0 to 20, only y=0. So, yes, only one point.Alternatively, could the maximum occur elsewhere? For example, if sin(x/10) is 1 and cos(πy/20) is 1, that's the maximum. If either is less than 1, the product would be less than 100. So, yes, the maximum is indeed 100 at (5π, 0).Wait, but let me think about the function again. It's a product of sine and cosine functions. So, the maximum of the product occurs when both factors are at their maximum. So, yes, that's correct.Alternatively, could the maximum be higher if one function is higher and the other is lower? For example, sin(x/10) could be higher than cos(πy/20), but since both are bounded by 1, their product can't exceed 100.So, I think that's solid.Therefore, the maximum color intensity is 100, occurring at the point (5π, 0) meters on the mural.Wait, but let me just confirm the coordinates. If x is measured along the arc from one end to the other, starting at 0 and going to 10π, and y is measured from the bottom (y=0) to the top (y=20), then yes, the point (5π, 0) is halfway along the arc at the bottom edge.So, that seems correct.So, summarizing:1. The arc length is 10π meters. The dimensions of the mural are 10π meters by 20 meters. Scaling down by 1:50, the sketch dimensions are π/5 meters by 0.4 meters.2. The maximum color intensity is 100, occurring at the point (5π, 0) meters on the mural.I think that's it.</think>

🧠:<think>Okay, so I have two problems to solve here. The first one is about projectile motion in soccer, and the second one is about modeling the number of goals scored in different leagues. Let me tackle them one by one.Starting with the first problem: A South African midfielder kicks a ball from (0,0) with an initial velocity of 20 m/s at a 45-degree angle. I need to find the maximum height H reached by the ball. The equation given is ( y = -frac{g}{2v^2}x^2 + tan(theta)x ). Hmm, okay. I remember that in projectile motion, the maximum height can be found using a specific formula, but let me see if I can derive it from the given equation or use the standard formula.Wait, the equation provided is the trajectory equation. It's a quadratic in terms of x, which makes sense because projectile motion follows a parabolic path. The standard form of a projectile's trajectory is ( y = x tan(theta) - frac{g x^2}{2 v^2 cos^2(theta)} ). Comparing that with the given equation, it looks similar. So, in the given equation, the coefficient of ( x^2 ) is ( -frac{g}{2v^2} ), but in the standard form, it's ( -frac{g}{2 v^2 cos^2(theta)} ). Hmm, so maybe the given equation is simplified or assumes something about the angle?Wait, if theta is 45 degrees, then ( cos(45) ) is ( frac{sqrt{2}}{2} ), so ( cos^2(45) ) is ( frac{1}{2} ). Therefore, the standard equation would have ( -frac{g}{2 v^2 times frac{1}{2}} = -frac{g}{v^2} ). But in the given equation, it's ( -frac{g}{2 v^2} ). So, that suggests that maybe the given equation is not the standard one, or perhaps it's using a different coordinate system or something.Wait, maybe I'm overcomplicating. Let me think. The maximum height in projectile motion is given by ( H = frac{v^2 sin^2(theta)}{2g} ). Yeah, that's the formula I remember. So, maybe I can use that instead of dealing with the trajectory equation.Given that, let me compute H. The initial velocity v is 20 m/s, theta is 45 degrees, and g is 9.8 m/s².First, compute ( sin(45^circ) ). Since 45 degrees is a common angle, ( sin(45) = frac{sqrt{2}}{2} approx 0.7071 ).So, ( sin^2(45) = (frac{sqrt{2}}{2})^2 = frac{2}{4} = frac{1}{2} ).Therefore, ( H = frac{20^2 times frac{1}{2}}{2 times 9.8} ).Calculating numerator: 20 squared is 400. 400 times 1/2 is 200.Denominator: 2 times 9.8 is 19.6.So, H = 200 / 19.6.Let me compute that. 200 divided by 19.6. Hmm, 19.6 times 10 is 196, so 200 - 196 is 4. So, 10 + (4/19.6). 4 divided by 19.6 is approximately 0.2041. So, total H is approximately 10.2041 meters.Wait, let me verify that division more accurately. 19.6 goes into 200 how many times?19.6 * 10 = 196200 - 196 = 4So, 4 / 19.6 = 0.20408163265...So, H ≈ 10.2041 meters.Alternatively, maybe I can write it as a fraction. 200 / 19.6 is equal to 2000 / 196, which simplifies to 500 / 49. Because both numerator and denominator can be divided by 4: 2000 ÷ 4 = 500, 196 ÷ 4 = 49. So, 500/49 is approximately 10.2041.So, the maximum height is approximately 10.2041 meters. I can write it as 500/49 m or approximately 10.20 m.Alternatively, maybe I can use the trajectory equation given to find the maximum height. Let me try that as a cross-check.The trajectory equation is ( y = -frac{g}{2v^2}x^2 + tan(theta)x ). To find the maximum height, I can take the derivative of y with respect to x, set it to zero, and solve for x, then plug back into y.So, dy/dx = - (g / v^2) x + tan(theta). Setting this equal to zero:- (g / v^2) x + tan(theta) = 0So, x = (v^2 / g) tan(theta)Then, plugging back into y:y = - (g / 2v^2) * (v^2 / g tan(theta))^2 + tan(theta) * (v^2 / g tan(theta))Simplify:First term: - (g / 2v^2) * (v^4 / g² tan²(theta)) = - (g / 2v^2) * (v^4 tan²(theta) / g²) = - (v² tan²(theta) / (2g))Second term: tan(theta) * (v² / g tan(theta)) = v² / gSo, total y = - (v² tan²(theta) / (2g)) + (v² / g) = (v² / g)(1 - tan²(theta)/2)Wait, let me compute that:= (v² / g) [1 - (tan²(theta)/2)]Hmm, let's compute that.Given theta is 45 degrees, so tan(theta) is 1. So, tan²(theta) is 1.Therefore, y = (v² / g)(1 - 1/2) = (v² / g)(1/2) = v² / (2g)Wait, that's different from the earlier formula. Wait, hold on.Wait, earlier I had H = (v² sin²(theta)) / (2g). But here, I have y = (v² / (2g))(1 - tan²(theta)/2). But with theta = 45 degrees, tan(theta) = 1, so 1 - 1/2 = 1/2, so y = v² / (4g). Wait, that's conflicting with the earlier result.Wait, something's wrong here. Because using the derivative method, I get y = v² / (4g), but using the standard formula, I get H = v² sin²(theta) / (2g) = (400 * 0.5) / (2*9.8) = 200 / 19.6 ≈ 10.2041.But according to the derivative method, y = v² / (4g) = 400 / (4*9.8) = 100 / 9.8 ≈ 10.2041. Wait, that's the same result. Wait, so both methods give the same result.Wait, hold on. Let me re-examine.When I took the derivative, I got x = (v² / g) tan(theta). Then, plugging back into y:First term: - (g / 2v²) * x² = - (g / 2v²) * (v^4 / g² tan²(theta)) = - (v² tan²(theta) / (2g))Second term: tan(theta) * x = tan(theta) * (v² / g tan(theta)) = v² / gTherefore, y = - (v² tan²(theta) / (2g)) + (v² / g) = (v² / g)(1 - tan²(theta)/2)But for theta = 45 degrees, tan(theta) = 1, so:y = (v² / g)(1 - 1/2) = (v² / g)(1/2) = v² / (2g)Wait, but earlier, when I thought it was conflicting, I thought it was v² / (4g). Wait, no, let me compute:Wait, 1 - tan²(theta)/2 when tan(theta)=1 is 1 - 1/2 = 1/2. So, y = (v² / g)(1/2) = v² / (2g). But wait, that's different from the standard formula.Wait, standard formula is H = (v² sin²(theta)) / (2g). For theta=45, sin(theta)=sqrt(2)/2, so sin²(theta)=1/2. Therefore, H = (v² * 1/2) / (2g) = v² / (4g). Wait, so which one is correct?Wait, hold on. There's a discrepancy here. Using the derivative method, I get y = v² / (2g). Using the standard formula, I get H = v² / (4g). Which one is correct?Wait, let me compute both:Given v=20 m/s, g=9.8 m/s².Using standard formula: H = (20² * sin²(45)) / (2*9.8) = (400 * 0.5) / 19.6 = 200 / 19.6 ≈ 10.2041 m.Using derivative method: y = v² / (2g) = 400 / (2*9.8) = 400 / 19.6 ≈ 20.4082 m.Wait, that's conflicting. So, which one is correct?Wait, perhaps I made a mistake in the derivative method.Wait, let's go back. The trajectory equation is given as y = - (g / 2v²) x² + tan(theta) x.Taking derivative: dy/dx = - (g / v²) x + tan(theta). Setting to zero: x = (v² tan(theta)) / g.Then, plugging back into y:y = - (g / 2v²) * (v^4 tan²(theta) / g²) + tan(theta) * (v² tan(theta) / g)Simplify term by term:First term: - (g / 2v²) * (v^4 tan²(theta) / g²) = - (v² tan²(theta) / (2g))Second term: tan(theta) * (v² tan(theta) / g) = (v² tan²(theta) / g)So, total y = (- v² tan²(theta) / (2g)) + (v² tan²(theta) / g) = (v² tan²(theta) / (2g)).So, y = (v² tan²(theta)) / (2g)Ah, okay, so I made a mistake in the previous calculation. So, y = (v² tan²(theta)) / (2g). For theta=45 degrees, tan(theta)=1, so y = v² / (2g). Which is 400 / (2*9.8) = 200 / 9.8 ≈ 20.4082 m.But wait, that conflicts with the standard formula.Wait, what's the standard formula? H = (v² sin²(theta)) / (2g). For theta=45, sin(theta)=sqrt(2)/2, so sin²(theta)=1/2, so H = (400 * 1/2) / (2*9.8) = 200 / 19.6 ≈ 10.2041 m.So, which one is correct?Wait, perhaps the given trajectory equation is not the standard one. Let me check.The standard trajectory equation is y = x tan(theta) - (g x²) / (2 v² cos²(theta)). So, comparing with the given equation: y = - (g / 2v²) x² + tan(theta) x. So, the given equation is missing the cos²(theta) term in the denominator. Therefore, the given equation is not the standard one. So, perhaps in the given equation, they have simplified it by assuming cos(theta)=1, which would be the case for theta=0, but that's not the case here.Wait, that can't be. So, perhaps the given equation is incorrect, or perhaps it's using a different coordinate system.Alternatively, maybe the given equation is correct, but in that case, the maximum height would be different.Wait, so if I use the given equation, the maximum height is y = (v² tan²(theta)) / (2g). For theta=45, that's 400 / (2*9.8) ≈ 20.4082 m.But according to the standard formula, it's 10.2041 m.So, which one is correct? Hmm.Wait, perhaps the given equation is incorrect. Because in the standard equation, the coefficient of x² is -g/(2 v² cos²(theta)). So, if the given equation is y = -g/(2v²) x² + tan(theta) x, then that would be equivalent to assuming cos(theta)=1, which is only true for theta=0. So, perhaps the given equation is only valid for theta=0, which is not the case here.Therefore, perhaps the given equation is incorrect, and the standard formula should be used.Alternatively, perhaps the given equation is correct, but in that case, the maximum height would be different.Wait, maybe I can check by plugging in the maximum height into the given equation.Wait, if I take x = (v² tan(theta)) / g, which is the x-coordinate at maximum height, and plug into the given equation:y = - (g / 2v²) * (v^4 tan²(theta) / g²) + tan(theta) * (v² tan(theta) / g)Simplify:First term: - (g / 2v²) * (v^4 tan²(theta) / g²) = - (v² tan²(theta) / (2g))Second term: tan(theta) * (v² tan(theta) / g) = (v² tan²(theta) / g)So, total y = (- v² tan²(theta) / (2g)) + (v² tan²(theta) / g) = (v² tan²(theta) / (2g))So, that's consistent with what I got earlier. So, according to the given equation, the maximum height is (v² tan²(theta)) / (2g). For theta=45, that's 400 / (2*9.8) ≈ 20.4082 m.But according to the standard formula, it's (v² sin²(theta)) / (2g) ≈ 10.2041 m.So, which one is correct?Wait, perhaps the given equation is incorrect because it's missing the cos²(theta) term in the denominator. So, perhaps the correct maximum height is 10.2041 m, and the given equation is wrong.Alternatively, maybe the given equation is correct, but in that case, the maximum height would be 20.4082 m.Wait, let me think about the physics. The standard formula for maximum height is H = (v² sin²(theta)) / (2g). That's a well-known formula. So, for theta=45, sin(theta)=sqrt(2)/2, so H = (400 * 0.5) / (2*9.8) = 200 / 19.6 ≈ 10.2041 m.Therefore, I think the standard formula is correct, and the given equation might be incorrect or perhaps it's a different formulation.Alternatively, maybe the given equation is correct, but in that case, the maximum height is different.Wait, perhaps the given equation is derived under a different assumption. Let me think.Wait, the standard trajectory equation is y = x tan(theta) - (g x²) / (2 v² cos²(theta)). So, if the given equation is y = - (g / 2v²) x² + tan(theta) x, then that would be equivalent to assuming cos(theta)=1, which is only true for theta=0. So, that equation is only valid for a horizontal kick, which is not the case here.Therefore, the given equation is incorrect for theta=45 degrees. Therefore, I should use the standard formula.Therefore, the maximum height is approximately 10.2041 meters.But wait, let me confirm by plugging in the numbers into the standard formula.H = (v² sin²(theta)) / (2g) = (20² * sin²(45)) / (2*9.8) = (400 * 0.5) / 19.6 = 200 / 19.6 ≈ 10.2041 m.Yes, that's correct.So, despite the given equation, which seems to be incorrect, the standard formula gives the correct maximum height.Therefore, the answer is approximately 10.2041 meters, which can be written as 500/49 meters exactly.So, 500 divided by 49 is approximately 10.2041.Therefore, the maximum height H is 500/49 meters, or approximately 10.20 meters.Okay, that's the first problem.Now, moving on to the second problem: The midfielder participates in 3 different leagues, playing n1, n2, n3 matches respectively. The number of goals he scores is modeled by G(n) = a n² + b n + c. Given that he scores 10 goals in 5 matches, 24 goals in 8 matches, and 54 goals in 10 matches, determine the constants a, b, and c.So, we have three equations:1. G(5) = 10: a*(5)^2 + b*(5) + c = 10 => 25a + 5b + c = 102. G(8) = 24: a*(8)^2 + b*(8) + c = 24 => 64a + 8b + c = 243. G(10) = 54: a*(10)^2 + b*(10) + c = 54 => 100a + 10b + c = 54So, we have a system of three equations:1. 25a + 5b + c = 102. 64a + 8b + c = 243. 100a + 10b + c = 54We need to solve for a, b, c.Let me write them down:Equation 1: 25a + 5b + c = 10Equation 2: 64a + 8b + c = 24Equation 3: 100a + 10b + c = 54Let me subtract Equation 1 from Equation 2 to eliminate c:Equation 2 - Equation 1: (64a - 25a) + (8b - 5b) + (c - c) = 24 - 10So, 39a + 3b = 14 --> Let's call this Equation 4.Similarly, subtract Equation 2 from Equation 3:Equation 3 - Equation 2: (100a - 64a) + (10b - 8b) + (c - c) = 54 - 24So, 36a + 2b = 30 --> Let's call this Equation 5.Now, we have:Equation 4: 39a + 3b = 14Equation 5: 36a + 2b = 30Now, let's solve Equations 4 and 5 for a and b.First, let's simplify Equation 4 and Equation 5.Equation 4: 39a + 3b = 14We can divide Equation 4 by 3:13a + b = 14/3 --> Let's call this Equation 6.Equation 5: 36a + 2b = 30We can divide Equation 5 by 2:18a + b = 15 --> Let's call this Equation 7.Now, we have:Equation 6: 13a + b = 14/3Equation 7: 18a + b = 15Subtract Equation 6 from Equation 7:(18a - 13a) + (b - b) = 15 - 14/3So, 5a = (45/3 - 14/3) = 31/3Therefore, a = (31/3) / 5 = 31/15 ≈ 2.0667Wait, 31 divided by 15 is approximately 2.0667.Now, plug a = 31/15 into Equation 6:13*(31/15) + b = 14/3Compute 13*(31/15):13*31 = 403So, 403/15 + b = 14/3Convert 14/3 to fifteenths: 14/3 = 70/15So, 403/15 + b = 70/15Therefore, b = 70/15 - 403/15 = (70 - 403)/15 = (-333)/15 = -22.2Wait, -333 divided by 15 is -22.2.So, b = -22.2Wait, let me express that as a fraction:-333/15 simplifies to -111/5, because both numerator and denominator can be divided by 3: 333 ÷ 3 = 111, 15 ÷ 3 = 5. So, -111/5.So, b = -111/5.Now, we can find c using Equation 1: 25a + 5b + c = 10Plug in a = 31/15 and b = -111/5:25*(31/15) + 5*(-111/5) + c = 10Compute each term:25*(31/15) = (25/15)*31 = (5/3)*31 = 155/3 ≈ 51.66675*(-111/5) = -111So, 155/3 - 111 + c = 10Convert 111 to thirds: 111 = 333/3So, 155/3 - 333/3 = (155 - 333)/3 = (-178)/3Therefore, (-178)/3 + c = 10So, c = 10 + 178/3Convert 10 to thirds: 10 = 30/3So, c = 30/3 + 178/3 = 208/3 ≈ 69.3333So, c = 208/3.Therefore, the constants are:a = 31/15 ≈ 2.0667b = -111/5 = -22.2c = 208/3 ≈ 69.3333Let me verify these values with the original equations.First, Equation 1: 25a + 5b + c25*(31/15) + 5*(-111/5) + 208/3Compute each term:25*(31/15) = (25/15)*31 = (5/3)*31 = 155/3 ≈ 51.66675*(-111/5) = -111208/3 ≈ 69.3333So, total: 155/3 - 111 + 208/3Convert all to thirds:155/3 - 333/3 + 208/3 = (155 - 333 + 208)/3 = (155 + 208 - 333)/3 = (363 - 333)/3 = 30/3 = 10. Correct.Equation 2: 64a + 8b + c64*(31/15) + 8*(-111/5) + 208/3Compute each term:64*(31/15) = (64/15)*31 ≈ (4.2667)*31 ≈ 132.2667But let's compute exactly:64*31 = 19841984/15 ≈ 132.26678*(-111/5) = -888/5 = -177.6208/3 ≈ 69.3333So, total: 1984/15 - 888/5 + 208/3Convert all to fifteenths:1984/15 - (888/5)*(3/3) = -2664/15 + (208/3)*(5/5) = 1040/15So, total: 1984/15 - 2664/15 + 1040/15 = (1984 - 2664 + 1040)/15 = (1984 + 1040 - 2664)/15 = (3024 - 2664)/15 = 360/15 = 24. Correct.Equation 3: 100a + 10b + c100*(31/15) + 10*(-111/5) + 208/3Compute each term:100*(31/15) = (100/15)*31 = (20/3)*31 ≈ 206.666710*(-111/5) = -222208/3 ≈ 69.3333So, total: 206.6667 - 222 + 69.3333 ≈ (206.6667 + 69.3333) - 222 = 276 - 222 = 54. Correct.Therefore, the constants are:a = 31/15b = -111/5c = 208/3Alternatively, in decimal form:a ≈ 2.0667b ≈ -22.2c ≈ 69.3333But since the problem doesn't specify the form, fractions are probably better.So, a = 31/15, b = -111/5, c = 208/3.I think that's the solution.</think>