🧠:<think>Alright, so I have this problem about exponential growth in student enrollments at a school in Jamshedpur. It's divided into two parts. Let me try to work through each part step by step. I'm a bit nervous because exponential growth models can sometimes trip me up, but I'll take it slow.Starting with part 1: I need to find the constant growth rate ( k ) using the given information. The model is ( N(t) = N_0 e^{kt} ). First, let's note down the given data:- The school was established in 1923, so that's our starting point, ( t = 0 ).- The initial enrollment ( N_0 ) is 150 students.- By the year 1973, the enrollment had grown to 2400 students.So, I need to find ( k ). To do that, I can plug in the values we know into the exponential growth formula.Let me figure out the time ( t ) between 1923 and 1973. That's 1973 minus 1923, which is 50 years. So, ( t = 50 ) years.The formula becomes:( 2400 = 150 e^{k times 50} )Okay, so I need to solve for ( k ). Let's write that equation again:( 2400 = 150 e^{50k} )First, I can divide both sides by 150 to simplify. Let's do that:( frac{2400}{150} = e^{50k} )Calculating the left side: 2400 divided by 150. Hmm, 150 times 16 is 2400 because 150*10=1500, 150*6=900, so 1500+900=2400. So, 2400/150 is 16.So now we have:( 16 = e^{50k} )To solve for ( k ), I need to take the natural logarithm of both sides because the base is ( e ). Remember, ( ln(e^{x}) = x ).Taking natural logs:( ln(16) = ln(e^{50k}) )Simplifying the right side:( ln(16) = 50k )Now, solve for ( k ):( k = frac{ln(16)}{50} )So, that's the value of ( k ). The problem says to express ( k ) in terms of natural logarithms, so I think that's the answer for part 1.Wait, let me double-check my steps to make sure I didn't make a mistake.1. Calculated time between 1923 and 1973: 50 years. That seems right.2. Plugged into the formula: 2400 = 150 e^{50k}. Correct.3. Divided both sides by 150: 16 = e^{50k}. Yes, because 2400/150 is 16.4. Took natural logs: ln(16) = 50k. That's correct.5. Solved for k: k = ln(16)/50. Yep, that seems right.I think that's solid. So, part 1 is done. Now, moving on to part 2.Part 2 asks to calculate the projected number of students in the year 2023 using the growth rate ( k ) found in part 1.First, let's figure out how many years have passed since the school was established in 1923 until 2023.2023 minus 1923 is 100 years. So, ( t = 100 ) years.We can use the same exponential growth formula:( N(t) = N_0 e^{kt} )We know ( N_0 = 150 ), ( k = frac{ln(16)}{50} ), and ( t = 100 ).Plugging in the values:( N(100) = 150 e^{left( frac{ln(16)}{50} times 100 right)} )Let me simplify the exponent first:( frac{ln(16)}{50} times 100 = 2 ln(16) )So, the equation becomes:( N(100) = 150 e^{2 ln(16)} )Hmm, ( e^{2 ln(16)} ). I remember that ( e^{ln(a)} = a ), so maybe I can simplify this further.Wait, ( e^{2 ln(16)} ) can be rewritten as ( (e^{ln(16)})^2 ) because ( a times ln(b) = ln(b^a) ), but actually, exponent rules: ( e^{k ln(a)} = a^k ). So, yes, ( e^{2 ln(16)} = 16^2 ).Calculating ( 16^2 ): 16*16 is 256.So, substituting back:( N(100) = 150 times 256 )Now, let's compute 150 multiplied by 256.Breaking it down:150 * 200 = 30,000150 * 56 = 8,400Adding them together: 30,000 + 8,400 = 38,400So, ( N(100) = 38,400 ) students.Wait, let me verify that multiplication again to be sure.150 * 256:First, 100 * 256 = 25,60050 * 256 = 12,800Adding them: 25,600 + 12,800 = 38,400. Yep, same result.So, the projected number of students in 2023 is 38,400.Just to recap the steps for part 2:1. Calculated time since establishment: 100 years.2. Plugged into the formula with known ( N_0 ) and ( k ).3. Simplified the exponent: 2 ln(16) becomes ln(16^2) which is ln(256), but since it's exponentiated, it's 16^2 = 256.4. Multiplied 150 by 256 to get 38,400.Everything seems to check out. I don't see any errors in my calculations.So, summarizing:1. The growth rate ( k ) is ( frac{ln(16)}{50} ).2. The projected enrollment in 2023 is 38,400 students.I think that's all. It was a bit of work, but breaking it down step by step helped me manage it without getting lost.Final Answer1. The constant growth rate ( k ) is boxed{dfrac{ln 16}{50}}.2. The projected number of students in 2023 is boxed{38400}.</think>

🧠:<think>Okay, so I have this problem where I need to plan a world tour. It involves two main tasks: calculating the total geodesic distance using the Haversine formula and optimizing the number of cultural exchange activities within a 30-day limit. Let me try to break this down step by step.First, for the geodesic distance calculation. I know that the Haversine formula is used to find the distance between two points on a sphere given their latitudes and longitudes. The Earth's radius is given as 6,371 km. The order of the countries to visit is Japan, India, Italy, Egypt, South Africa, Brazil, and Australia. I need to calculate the distance between each consecutive pair of capitals and sum them up.Let me list out the coordinates again to make sure I have them right:- Tokyo, Japan: (35.6895° N, 139.6917° E)- New Delhi, India: (28.6139° N, 77.2090° E)- Rome, Italy: (41.9028° N, 12.4964° E)- Cairo, Egypt: (30.0444° N, 31.2357° E)- Pretoria, South Africa: (25.7461° S, 28.1881° E)- Brasília, Brazil: (15.7939° S, 47.8828° W)- Canberra, Australia: (35.2809° S, 149.1300° E)I remember that the Haversine formula involves converting degrees to radians, then applying the formula which includes the sine and cosine of the differences in coordinates. The formula is:a = sin²(Δφ/2) + cos φ1 * cos φ2 * sin²(Δλ/2)c = 2 * atan2(√a, √(1−a))d = R * cWhere φ is latitude, λ is longitude, R is Earth's radius, and d is the distance.I think I should write down each pair of cities and compute the distance step by step.Let me start by converting all the coordinates from degrees to radians because the trigonometric functions in the formula require radians.For Tokyo (35.6895° N, 139.6917° E):φ1 = 35.6895° * π/180 ≈ 0.622 radiansλ1 = 139.6917° * π/180 ≈ 2.438 radiansNext, New Delhi (28.6139° N, 77.2090° E):φ2 = 28.6139° * π/180 ≈ 0.499 radiansλ2 = 77.2090° * π/180 ≈ 1.347 radiansSo, the difference in latitudes, Δφ = φ2 - φ1 ≈ 0.499 - 0.622 ≈ -0.123 radiansDifference in longitudes, Δλ = λ2 - λ1 ≈ 1.347 - 2.438 ≈ -1.091 radiansNow, compute a:sin²(Δφ/2) = sin²(-0.123/2) ≈ sin²(-0.0615) ≈ (sin(-0.0615))² ≈ ( -0.0614 )² ≈ 0.00377cos φ1 * cos φ2 = cos(0.622) * cos(0.499) ≈ 0.814 * 0.878 ≈ 0.714sin²(Δλ/2) = sin²(-1.091/2) ≈ sin²(-0.5455) ≈ (sin(-0.5455))² ≈ ( -0.519 )² ≈ 0.269So, a ≈ 0.00377 + 0.714 * 0.269 ≈ 0.00377 + 0.192 ≈ 0.1958Then, c = 2 * atan2(√a, √(1−a)) ≈ 2 * atan2(√0.1958, √0.8042) ≈ 2 * atan2(0.4425, 0.897) ≈ 2 * 0.475 ≈ 0.95 radiansDistance d = R * c ≈ 6371 km * 0.95 ≈ 6052 kmWait, that seems a bit high. Let me check my calculations again.Wait, maybe I made a mistake in calculating a. Let me recalculate:sin²(Δφ/2) = sin²(-0.123/2) = sin²(-0.0615) ≈ (sin(-0.0615))² ≈ ( -0.0614 )² ≈ 0.00377cos φ1 = cos(0.622) ≈ 0.814cos φ2 = cos(0.499) ≈ 0.878So, cos φ1 * cos φ2 ≈ 0.814 * 0.878 ≈ 0.714sin²(Δλ/2) = sin²(-1.091/2) = sin²(-0.5455) ≈ (sin(-0.5455))² ≈ ( -0.519 )² ≈ 0.269So, a = 0.00377 + 0.714 * 0.269 ≈ 0.00377 + 0.192 ≈ 0.1958Then, √a ≈ 0.4425, √(1 - a) ≈ √0.8042 ≈ 0.897atan2(0.4425, 0.897) ≈ 0.475 radiansSo, c ≈ 2 * 0.475 ≈ 0.95 radiansDistance d ≈ 6371 * 0.95 ≈ 6052 kmHmm, that seems correct. But I thought the distance from Japan to India might be less. Maybe I should verify with an online calculator or something. But for now, I'll proceed.Next, from New Delhi to Rome.New Delhi: φ1 = 28.6139° N ≈ 0.499 radians, λ1 = 77.2090° E ≈ 1.347 radiansRome: φ2 = 41.9028° N ≈ 0.731 radians, λ2 = 12.4964° E ≈ 0.218 radiansΔφ = 0.731 - 0.499 ≈ 0.232 radiansΔλ = 0.218 - 1.347 ≈ -1.129 radiansCompute a:sin²(Δφ/2) = sin²(0.232/2) ≈ sin²(0.116) ≈ (0.1157)² ≈ 0.0134cos φ1 * cos φ2 = cos(0.499) * cos(0.731) ≈ 0.878 * 0.743 ≈ 0.653sin²(Δλ/2) = sin²(-1.129/2) ≈ sin²(-0.5645) ≈ (sin(-0.5645))² ≈ ( -0.536 )² ≈ 0.287So, a ≈ 0.0134 + 0.653 * 0.287 ≈ 0.0134 + 0.187 ≈ 0.2004c = 2 * atan2(√0.2004, √(1 - 0.2004)) ≈ 2 * atan2(0.4477, 0.897) ≈ 2 * 0.475 ≈ 0.95 radiansDistance d ≈ 6371 * 0.95 ≈ 6052 km again? That seems similar. Maybe it's correct.Wait, but the actual distance from India to Italy is probably less. Maybe I made a mistake in the longitude difference. Let me check:New Delhi is at 77.2090° E, Rome is at 12.4964° E. So Δλ = 12.4964 - 77.2090 ≈ -64.7126°, which is -1.129 radians. That's correct.Hmm, maybe the distance is indeed around 6000 km. Let me proceed.Next, Rome to Cairo.Rome: φ1 = 41.9028° N ≈ 0.731 radians, λ1 = 12.4964° E ≈ 0.218 radiansCairo: φ2 = 30.0444° N ≈ 0.524 radians, λ2 = 31.2357° E ≈ 0.545 radiansΔφ = 0.524 - 0.731 ≈ -0.207 radiansΔλ = 0.545 - 0.218 ≈ 0.327 radiansCompute a:sin²(Δφ/2) = sin²(-0.207/2) ≈ sin²(-0.1035) ≈ ( -0.1033 )² ≈ 0.0107cos φ1 * cos φ2 = cos(0.731) * cos(0.524) ≈ 0.743 * 0.853 ≈ 0.635sin²(Δλ/2) = sin²(0.327/2) ≈ sin²(0.1635) ≈ (0.163)² ≈ 0.0266So, a ≈ 0.0107 + 0.635 * 0.0266 ≈ 0.0107 + 0.0168 ≈ 0.0275c = 2 * atan2(√0.0275, √(1 - 0.0275)) ≈ 2 * atan2(0.1658, 0.986) ≈ 2 * 0.166 ≈ 0.332 radiansDistance d ≈ 6371 * 0.332 ≈ 2115 kmThat seems more reasonable.Next, Cairo to Pretoria.Cairo: φ1 = 30.0444° N ≈ 0.524 radians, λ1 = 31.2357° E ≈ 0.545 radiansPretoria: φ2 = 25.7461° S ≈ -0.449 radians, λ2 = 28.1881° E ≈ 0.492 radiansΔφ = -0.449 - 0.524 ≈ -0.973 radiansΔλ = 0.492 - 0.545 ≈ -0.053 radiansCompute a:sin²(Δφ/2) = sin²(-0.973/2) ≈ sin²(-0.4865) ≈ ( -0.467 )² ≈ 0.218cos φ1 * cos φ2 = cos(0.524) * cos(-0.449) ≈ 0.853 * 0.900 ≈ 0.768sin²(Δλ/2) = sin²(-0.053/2) ≈ sin²(-0.0265) ≈ ( -0.0265 )² ≈ 0.0007So, a ≈ 0.218 + 0.768 * 0.0007 ≈ 0.218 + 0.0005 ≈ 0.2185c = 2 * atan2(√0.2185, √(1 - 0.2185)) ≈ 2 * atan2(0.467, 0.876) ≈ 2 * 0.513 ≈ 1.026 radiansDistance d ≈ 6371 * 1.026 ≈ 6534 kmHmm, that seems quite long. Let me check the coordinates again. Cairo is in the northern hemisphere, Pretoria is in the southern. So the distance should be significant. Maybe it's correct.Next, Pretoria to Brasília.Pretoria: φ1 = -25.7461° ≈ -0.449 radians, λ1 = 28.1881° E ≈ 0.492 radiansBrasília: φ2 = -15.7939° ≈ -0.275 radians, λ2 = -47.8828° W ≈ -0.836 radiansΔφ = -0.275 - (-0.449) ≈ 0.174 radiansΔλ = -0.836 - 0.492 ≈ -1.328 radiansCompute a:sin²(Δφ/2) = sin²(0.174/2) ≈ sin²(0.087) ≈ (0.0868)² ≈ 0.0075cos φ1 * cos φ2 = cos(-0.449) * cos(-0.275) ≈ 0.900 * 0.963 ≈ 0.867sin²(Δλ/2) = sin²(-1.328/2) ≈ sin²(-0.664) ≈ ( -0.616 )² ≈ 0.379So, a ≈ 0.0075 + 0.867 * 0.379 ≈ 0.0075 + 0.328 ≈ 0.3355c = 2 * atan2(√0.3355, √(1 - 0.3355)) ≈ 2 * atan2(0.579, 0.832) ≈ 2 * 0.644 ≈ 1.288 radiansDistance d ≈ 6371 * 1.288 ≈ 8193 kmThat seems quite long, but considering the distance from South Africa to Brazil across the Atlantic, maybe it's correct.Finally, Brasília to Canberra.Brasília: φ1 = -15.7939° ≈ -0.275 radians, λ1 = -47.8828° ≈ -0.836 radiansCanberra: φ2 = -35.2809° ≈ -0.615 radians, λ2 = 149.1300° E ≈ 2.603 radiansΔφ = -0.615 - (-0.275) ≈ -0.34 radiansΔλ = 2.603 - (-0.836) ≈ 3.439 radiansBut wait, longitude difference can be more than π, so we might need to take the shorter path. Since 3.439 radians is more than π (≈3.1416), the shorter path would be 2π - 3.439 ≈ 2.844 radians. But let me check:Δλ = 149.1300° E - (-47.8828° W) = 149.13 + 47.88 ≈ 197.01°, which is more than 180°, so the shorter path is 360 - 197.01 ≈ 162.99°, which is 162.99 * π/180 ≈ 2.839 radians.So, Δλ ≈ 2.839 radians.Compute a:sin²(Δφ/2) = sin²(-0.34/2) ≈ sin²(-0.17) ≈ ( -0.169 )² ≈ 0.0286cos φ1 * cos φ2 = cos(-0.275) * cos(-0.615) ≈ 0.963 * 0.816 ≈ 0.786sin²(Δλ/2) = sin²(2.839/2) ≈ sin²(1.4195) ≈ (sin(1.4195))² ≈ (0.989)² ≈ 0.978So, a ≈ 0.0286 + 0.786 * 0.978 ≈ 0.0286 + 0.768 ≈ 0.7966c = 2 * atan2(√0.7966, √(1 - 0.7966)) ≈ 2 * atan2(0.8925, 0.436) ≈ 2 * 1.105 ≈ 2.21 radiansDistance d ≈ 6371 * 2.21 ≈ 14080 kmWait, that seems extremely long. The distance from Brazil to Australia is indeed very long, but 14,000 km? Let me check the calculation.Wait, the Δλ was converted correctly to the shorter path, which is 2.839 radians. The formula should handle that. Let me recalculate a:sin²(Δφ/2) ≈ 0.0286cos φ1 * cos φ2 ≈ 0.963 * 0.816 ≈ 0.786sin²(Δλ/2) ≈ sin²(1.4195) ≈ (0.989)² ≈ 0.978So, a ≈ 0.0286 + 0.786 * 0.978 ≈ 0.0286 + 0.768 ≈ 0.7966Yes, that's correct.c ≈ 2 * atan2(0.8925, 0.436) ≈ 2 * 1.105 ≈ 2.21 radiansDistance ≈ 6371 * 2.21 ≈ 14080 kmOkay, I'll go with that.Now, summing up all the distances:Japan to India: ~6052 kmIndia to Italy: ~6052 kmItaly to Egypt: ~2115 kmEgypt to South Africa: ~6534 kmSouth Africa to Brazil: ~8193 kmBrazil to Australia: ~14080 kmTotal distance ≈ 6052 + 6052 + 2115 + 6534 + 8193 + 14080 ≈ Let's compute step by step:6052 + 6052 = 1210412104 + 2115 = 1421914219 + 6534 = 2075320753 + 8193 = 2894628946 + 14080 = 43026 kmSo, approximately 43,026 km total distance.Now, for the cultural exchange optimization. I have 30 days and need to maximize the number of activities. Each country offers a certain number of activities per day:- Japan: 5- India: 4- Italy: 6- Egypt: 3- South Africa: 2- Brazil: 4- Australia: 5I need to decide how many days to spend in each country, with the total days ≤30, to maximize the total activities.This is a knapsack problem where each country is an item with a "weight" (days) and a "value" (activities per day * days). But since the activities are per day, it's more like maximizing the sum of (days * activities per day) with the constraint that the total days ≤30.But since each country can be visited multiple times? Wait, no, the problem says "exactly 30 days" and "spend full days in each country." So, it's more like assigning days to each country such that the sum is 30, and the total activities are maximized.This is an integer allocation problem. To maximize the total activities, I should allocate as many days as possible to the country with the highest activities per day, then the next, etc.Looking at the activities per day:Italy: 6 (highest)Japan: 5Australia: 5Brazil: 4India: 4Egypt: 3South Africa: 2So, priority order: Italy, Japan, Australia, Brazil, India, Egypt, South Africa.I should allocate days starting from the highest.First, allocate as many days as possible to Italy.But how many days can I allocate? Since I have 30 days, and I can choose to spend all 30 in Italy, but that would give 30*6=180 activities. But maybe I can get more by combining with other high countries.Wait, no, because Italy has the highest rate, so allocating all days to Italy would give the maximum. But let me check if that's the case.Wait, but the problem says "exactly 30 days" and "spend full days in each country." So, I can choose to spend all 30 days in Italy, but I might have to visit all countries? Wait, no, the problem doesn't specify that I have to visit all countries. It just says I have a list of countries I want to visit, but the optimization is about maximizing activities within 30 days, so I can choose to spend all days in the highest activity country.But wait, the first part of the problem was about visiting all countries in a specific order, but the second part is separate: "you have exactly 30 days for your world tour, and you want to maximize the number of cultural exchange activities you can participate in." So, it's a separate optimization, not necessarily visiting all countries. So, I can choose to spend all 30 days in Italy, giving 180 activities.But let me check the problem statement again: "you want to visit some of the most culturally rich countries on each continent." So, maybe I have to visit at least one country per continent? Wait, the list includes countries from each continent: Japan (Asia), India (Asia), Italy (Europe), Egypt (Africa), South Africa (Africa), Brazil (South America), Australia (Australia). So, if I have to visit at least one country per continent, then I need to spend at least one day in each of the seven countries. But the problem doesn't specify that. It just says "you have a list of 7 countries you want to visit" but the optimization is about maximizing activities within 30 days, so I think I can choose to spend all days in the highest activity country, unless the problem implies that I must visit all countries.Wait, the problem says: "you have exactly 30 days for your world tour, and you want to maximize the number of cultural exchange activities you can participate in. Each country offers a different number of activities per day, but your schedule constraints allow you only to spend full days in each country."So, it's about allocating days to countries, possibly multiple days in the same country, but you can choose which countries to visit. So, to maximize, you should spend as many days as possible in the country with the highest activities per day, which is Italy at 6 per day.But wait, if you can only spend full days in each country, but you can choose to spend multiple days in the same country. So, the optimal is to spend all 30 days in Italy, giving 30*6=180 activities.But let me check if the problem requires visiting all countries. It says "you have a list of 7 countries you want to visit," but the optimization is about maximizing activities within 30 days. So, unless specified otherwise, I think you can choose to spend all days in the highest activity country.But maybe the problem expects you to visit all countries, spending at least one day in each. Let me check the problem statement again:"you have exactly 30 days for your world tour, and you want to maximize the number of cultural exchange activities you can participate in. Each country offers a different number of activities per day, but your schedule constraints allow you only to spend full days in each country."It doesn't say you have to visit all countries, just that you have a list of 7 countries you want to visit, but the optimization is about maximizing activities within 30 days. So, I think you can choose to spend all days in Italy.But to be safe, maybe the problem expects you to visit all countries, so you have to spend at least one day in each. Let me assume that for a moment.If I have to spend at least one day in each country, that's 7 days, leaving 23 days to allocate. Then, I should allocate the remaining days to the country with the highest activities per day, which is Italy.So, days in Italy: 1 + 23 = 24 daysOther countries: 1 day each.Total activities: 24*6 + 1*5 + 1*5 + 1*4 + 1*4 + 1*3 + 1*2 = 144 + 5 +5 +4 +4 +3 +2 = 144 + 23 = 167But if I don't have to visit all countries, then 30 days in Italy gives 180, which is higher.But the problem says "you have a list of 7 countries you want to visit," so maybe you have to visit all of them, meaning at least one day each. So, the optimal is to spend 24 days in Italy and 1 day each in the others, giving 167 activities.But let me check the problem statement again:"you have exactly 30 days for your world tour, and you want to maximize the number of cultural exchange activities you can participate in. Each country offers a different number of activities per day, but your schedule constraints allow you only to spend full days in each country."It doesn't specify that you have to visit all countries, just that you have a list of 7 countries you want to visit. So, perhaps you can choose to visit only the highest activity countries. But since the list includes all seven, maybe you have to visit all. It's a bit ambiguous.But in the first part, you are visiting all countries in a specific order, but the second part is a separate optimization. So, perhaps in the second part, you can choose which countries to visit, not necessarily all.Therefore, the optimal is to spend all 30 days in Italy, giving 180 activities.But let me think again. If you have to visit all countries, then the maximum is 167. If not, it's 180.Given the problem statement, I think it's safer to assume that you can choose to visit any subset of the countries, so the maximum is 180.But let me check the exact wording:"you have exactly 30 days for your world tour, and you want to maximize the number of cultural exchange activities you can participate in. Each country offers a different number of activities per day, but your schedule constraints allow you only to spend full days in each country."It doesn't say you have to visit all countries, so the optimal is to spend all days in Italy.But wait, the first part was about visiting all countries in a specific order, but the second part is a separate optimization. So, perhaps the second part is about the same tour, meaning you have to visit all countries, but you can choose how many days to spend in each, with total days ≤30.Wait, the problem says:"Imagine you are planning a world tour that includes visiting some of the most culturally rich countries on each continent. You have a list of 7 countries you want to visit: Japan, India, Italy, Egypt, South Africa, Brazil, and Australia."So, you want to visit these 7 countries, but the second part is about optimizing the days spent in each to maximize activities within 30 days.Therefore, you have to visit all 7 countries, spending at least one day in each, and the rest of the days allocated to maximize activities.So, total days = 30Minimum days spent: 7 (1 in each country)Remaining days: 23To maximize, allocate all remaining days to the country with the highest activities per day, which is Italy.So, days in Italy: 1 + 23 = 24Other countries: 1 day each.Total activities: 24*6 + 1*5 + 1*5 + 1*4 + 1*4 + 1*3 + 1*2 = 144 + 5 +5 +4 +4 +3 +2 = 144 + 23 = 167Alternatively, if you can spend more days in other high countries, but since Italy is the highest, it's optimal to spend as many days as possible there.So, the optimal allocation is 24 days in Italy, 1 day each in the others, total activities 167.But wait, let me check if allocating some days to other high countries might yield more. For example, if I take one day from Italy and give it to Japan or Australia, which have 5 activities per day, which is less than Italy's 6. So, 6 >5, so it's better to keep all extra days in Italy.Similarly, Brazil and India have 4, which is less than 6, so no point.Therefore, the optimal is 24 days in Italy, 1 day each in the others, total activities 167.But let me check the math again:24*6 = 1441*5 (Japan) =51*5 (Australia)=51*4 (Brazil)=41*4 (India)=41*3 (Egypt)=31*2 (South Africa)=2Total: 144+5+5+4+4+3+2= 144+23=167Yes.Alternatively, if you don't have to visit all countries, you could get 180, but since the problem says "you have a list of 7 countries you want to visit," I think you have to visit all, so 167 is the answer.But let me think again. The problem says "you have a list of 7 countries you want to visit," but the optimization is about maximizing activities within 30 days. It doesn't say you have to visit all of them. So, perhaps you can choose to visit only the highest activity countries.But given that the first part was about visiting all in order, maybe the second part is also about the same tour, meaning you have to visit all 7 countries, so you have to spend at least one day in each, and the rest allocated to maximize.Therefore, the optimal is 24 days in Italy, 1 day each in the others, total activities 167.But let me check if the problem allows skipping some countries. It says "you have a list of 7 countries you want to visit," but the optimization is about maximizing activities, so perhaps you can choose to visit only some of them.But I think the problem expects you to visit all 7 countries, as it's a world tour including each continent. So, I'll proceed with the 167 answer.Wait, but let me think again. If you don't have to visit all countries, then the maximum is 30*6=180. But if you have to visit all, it's 167. Since the problem says "you have a list of 7 countries you want to visit," but the optimization is about maximizing activities, it's possible that you can choose to visit only some of them. However, the first part was about visiting all in a specific order, but the second part is a separate optimization. So, perhaps in the second part, you can choose to visit only the highest activity countries.But I'm not sure. The problem is a bit ambiguous. To be safe, I'll provide both possibilities, but I think the intended answer is to visit all countries, so 167.But let me check the problem statement again:"Imagine you are planning a world tour that includes visiting some of the most culturally rich countries on each continent. You have a list of 7 countries you want to visit: Japan, India, Italy, Egypt, South Africa, Brazil, and Australia."So, you want to visit these 7 countries, but the optimization is about how many days to spend in each to maximize activities within 30 days. Therefore, you have to spend at least one day in each, because you want to visit them. So, the optimal is 24 days in Italy, 1 day each in the others, total activities 167.Therefore, the answers are:1. Total geodesic distance: approximately 43,026 km2. Optimal days: 24 in Italy, 1 each in the others, total activities 167</think>

🧠:<think>Okay, so I have this problem about a young couple designing their sustainable, minimalist home. They want to maximize the volume of their house while keeping the surface area under 300 square meters and having exactly 100 square meters of windows and skylights combined. The house is a rectangular box with a flat roof. First, let me parse the problem step by step. The house is a rectangular box, so it has length, width, and height. They mentioned that the length is twice its width, and the height is equal to the width. So, let me denote the width as 'w'. Then, the length would be '2w', and the height would also be 'w'. So, in terms of variables, we have:- Width = w- Length = 2w- Height = wSo, the volume of the house would be length × width × height, which is 2w × w × w = 2w³. That's the volume we need to maximize.Now, the constraints are on the surface area and the total area of windows and skylights. Let me tackle the surface area first. The total surface area of the house, including the roof and walls but excluding the skylights and windows, should not exceed 300 square meters.Since it's a rectangular box, the surface area (SA) is calculated as 2(lw + lh + wh). But since the roof is flat, I think the surface area includes the roof as well. So, the formula remains the same: 2(lw + lh + wh). But wait, the problem says to exclude the skylights and windows. So, actually, the surface area given (300 m²) is the area of the walls, roof, and floor, but without the windows and skylights. Hmm, that complicates things a bit.Wait, no, actually, the problem says: "the total surface area of the house (including the roof and walls but excluding the skylights and windows) does not exceed 300 square meters." So, that means the surface area of the house, which includes the roof and walls, but subtracting the areas where the skylights and windows are. So, the surface area is 300 m², which is the area of the walls, roof, and floor, minus the windows and skylights.But hold on, the floor isn't mentioned. The house is a rectangular box, so it has a floor as well. But in the surface area, do we include the floor? Typically, when people talk about the surface area of a house, they might not include the floor because it's on the ground. But the problem says "including the roof and walls," so maybe the floor is excluded. Let me check the problem statement again.It says: "the total surface area of the house (including the roof and walls but excluding the skylights and windows) does not exceed 300 square meters." So, it includes the roof and walls, but excludes skylights and windows. So, the floor is not included in the surface area. So, the surface area is just the walls and the roof.So, the surface area (SA) is the sum of the areas of the four walls and the roof. Let me calculate that.The four walls consist of two walls with area length × height and two walls with area width × height. So, the total area of the walls is 2*(length*height) + 2*(width*height). The roof is length × width.So, SA = 2*(l*h) + 2*(w*h) + l*w.But since we are excluding the windows and skylights, the actual surface area is 300 m², which is the area of the walls and roof minus the areas of the windows and skylights.Wait, no. The problem says the surface area (excluding windows and skylights) is 300 m². So, the surface area without the windows and skylights is 300 m². So, the total surface area including windows and skylights would be 300 m² plus the area of windows and skylights.But the total area of windows and skylights is exactly 100 m². So, the total surface area including everything would be 300 + 100 = 400 m². But I'm not sure if that's necessary right now.Wait, perhaps I need to model the surface area without windows and skylights as 300 m². So, the walls and roof have a total area of 300 m², and then the windows and skylights add 100 m² to the total. But actually, the windows and skylights are openings, so they don't contribute to the surface area. So, the surface area is 300 m², which is the area of the walls and roof, minus the windows and skylights. So, the actual walls and roof have an area of 300 + 100 = 400 m²? Hmm, maybe.Wait, no. Let me think again. The surface area is the area of the walls and roof, but without the windows and skylights. So, if the walls and roof had no windows or skylights, their area would be 400 m², but since we have 100 m² of windows and skylights, the actual surface area is 400 - 100 = 300 m². So, the surface area without the windows and skylights is 300 m².Therefore, the total area of the walls and roof is 300 + 100 = 400 m². So, the walls and roof, if they had no windows or skylights, would be 400 m². So, we can model the surface area as 400 m², and then subtract the 100 m² for the windows and skylights to get 300 m².But perhaps it's better to model it as the surface area of the walls and roof is 300 m², which already excludes the windows and skylights. So, in that case, the surface area is 300 m², and the windows and skylights are 100 m². So, the total area of the walls, roof, windows, and skylights would be 300 + 100 = 400 m².But I think the problem is saying that the surface area, excluding the windows and skylights, is 300 m². So, the walls and roof have an area of 300 m², and the windows and skylights are 100 m². So, the total area (including windows and skylights) is 300 + 100 = 400 m². But I'm not sure if that's necessary.Wait, perhaps I should just model the surface area as 300 m², which is the area of the walls and roof, excluding the windows and skylights. So, the surface area is 300 m², and the windows and skylights add 100 m² to the total, but since they are openings, they don't contribute to the surface area. So, the surface area remains 300 m², and the total area of windows and skylights is 100 m².So, perhaps the surface area is 300 m², and the windows and skylights are 100 m², which are separate. So, the surface area is 300 m², and the windows and skylights are 100 m², so the total area (surface area + windows + skylights) is 400 m². But I think that's not necessary for the problem.Wait, the problem says: "the total surface area of the house (including the roof and walls but excluding the skylights and windows) does not exceed 300 square meters." So, the surface area is 300 m², which includes the roof and walls, but excludes the skylights and windows. So, the skylights and windows are not part of the surface area. So, the surface area is 300 m², and the windows and skylights are 100 m², which are separate.So, in that case, the surface area is 300 m², which is the area of the walls and roof, and the windows and skylights are 100 m², which are the openings.So, the surface area (walls and roof) is 300 m², and the windows and skylights are 100 m².So, now, let's model this.Given that the house is a rectangular box with length = 2w, width = w, height = w.So, the surface area (walls and roof) is:- Two walls of length × height: 2*(2w * w) = 4w²- Two walls of width × height: 2*(w * w) = 2w²- Roof: length × width = 2w * w = 2w²So, total surface area (walls and roof) is 4w² + 2w² + 2w² = 8w².But according to the problem, this surface area is 300 m². So, 8w² = 300.Wait, but hold on. The surface area is 300 m², which is the area of the walls and roof, excluding the skylights and windows. So, if the total surface area (including skylights and windows) would be 8w², but since we exclude the skylights and windows, the surface area is 300 m².But actually, the skylights are on the roof, and the windows are on the walls. So, the surface area is 8w² minus the area of the skylights and windows, which is 100 m². So, 8w² - 100 = 300.So, 8w² = 400, so w² = 50, so w = sqrt(50) = 5*sqrt(2) ≈ 7.071 m.But wait, let me make sure. If the total surface area (walls and roof) is 8w², and we have 100 m² of windows and skylights, then the surface area excluding windows and skylights is 8w² - 100 = 300. So, 8w² = 400, so w² = 50, so w = sqrt(50). So, that's approximately 7.071 meters.But hold on, is that correct? Let me think again.The surface area of the walls and roof is 8w². The windows and skylights are 100 m². So, the surface area excluding windows and skylights is 8w² - 100 = 300. So, 8w² = 400, so w² = 50, so w = sqrt(50). So, that seems correct.But let me check if the surface area is correctly calculated.The walls: two walls are length × height = 2w × w = 2w² each, so two of them is 4w².The other two walls are width × height = w × w = w² each, so two of them is 2w².The roof is length × width = 2w × w = 2w².So, total surface area (walls and roof) is 4w² + 2w² + 2w² = 8w². That's correct.So, 8w² - 100 = 300 => 8w² = 400 => w² = 50 => w = sqrt(50). So, w = 5*sqrt(2) ≈ 7.071 m.So, the dimensions would be:- Width: w = 5√2 m- Length: 2w = 10√2 m- Height: w = 5√2 mSo, that's the first part.But wait, the problem says to maximize the volume. So, is this the maximum volume? Or is this just a constraint?Wait, in the problem, it says: "maximize the volume of the house while ensuring that the total surface area... does not exceed 300 square meters." So, we have a constraint on the surface area (300 m²) and another constraint on the total area of windows and skylights (100 m²). So, we need to find the dimensions that maximize the volume under these constraints.So, perhaps I need to set up an optimization problem with constraints.Let me denote:- Let w = width- Then, length = 2w- Height = wSo, volume V = length × width × height = 2w × w × w = 2w³.We need to maximize V = 2w³, subject to:1. Surface area constraint: 8w² - 100 ≤ 300 => 8w² ≤ 400 => w² ≤ 50 => w ≤ sqrt(50)But wait, the surface area is exactly 300 m², right? Because it says "does not exceed 300 square meters." So, it's ≤ 300, but to maximize the volume, we probably need to set it to 300.So, 8w² - 100 = 300 => 8w² = 400 => w² = 50 => w = sqrt(50). So, that's the maximum possible w under the surface area constraint.But is that the only constraint? Also, the total area of windows and skylights is exactly 100 m².So, let's think about the windows and skylights.The walls have windows that cover 40% of the surface area on each side. So, each wall has windows covering 40% of its area.Wait, the problem says: "the walls will have windows that cover 40% of the surface area on each side." So, each wall has windows covering 40% of its area.So, for each wall, the window area is 0.4 × (area of the wall).So, let's calculate the window area on each wall.First, the walls:- There are two walls with area length × height = 2w × w = 2w² each. So, each of these walls has windows covering 40% of 2w², which is 0.4 × 2w² = 0.8w² per wall. Since there are two such walls, total window area on these walls is 2 × 0.8w² = 1.6w².- Then, there are two walls with area width × height = w × w = w² each. So, each of these walls has windows covering 40% of w², which is 0.4 × w² = 0.4w² per wall. Since there are two such walls, total window area on these walls is 2 × 0.4w² = 0.8w².So, total window area on all walls is 1.6w² + 0.8w² = 2.4w².Additionally, there are skylights on the roof. The total area of all windows and skylights is exactly 100 m². So, the skylights must account for the remaining area.So, total windows and skylights = 2.4w² + skylights = 100.So, skylights = 100 - 2.4w².But the skylights are on the roof, which has an area of length × width = 2w × w = 2w².So, the skylights cannot exceed the area of the roof. So, skylights ≤ 2w².So, 100 - 2.4w² ≤ 2w² => 100 ≤ 4.4w² => w² ≥ 100 / 4.4 ≈ 22.727.So, w² ≥ approximately 22.727, so w ≥ sqrt(22.727) ≈ 4.768 m.But from the surface area constraint, we have w² = 50, so w = sqrt(50) ≈ 7.071 m.So, let's compute the skylights area when w² = 50.Skylights = 100 - 2.4w² = 100 - 2.4*50 = 100 - 120 = -20.Wait, that can't be. We can't have negative skylights. That suggests that when w² = 50, the skylights would be negative, which is impossible. So, that means our initial assumption that the surface area is exactly 300 m² might not hold because the skylights can't be negative.So, perhaps we need to adjust the surface area to ensure that the skylights are non-negative.So, skylights = 100 - 2.4w² ≥ 0 => 2.4w² ≤ 100 => w² ≤ 100 / 2.4 ≈ 41.6667.So, w² ≤ 41.6667, so w ≤ sqrt(41.6667) ≈ 6.455 m.But from the surface area constraint, we had w² = 50, which is higher than 41.6667. So, that's a conflict.So, this suggests that if we set the surface area to 300 m², the skylights would have to be negative, which is impossible. Therefore, we need to adjust the surface area to a lower value so that the skylights are non-negative.So, perhaps the surface area is not 300 m², but less, so that when we subtract the windows and skylights, we get a feasible surface area.Wait, no. The surface area is 300 m², which is the area of the walls and roof excluding the windows and skylights. So, the surface area is fixed at 300 m², and the windows and skylights are 100 m².But when we calculated, with w² = 50, the skylights would be negative, which is impossible. So, perhaps the surface area cannot be 300 m² if we have 100 m² of windows and skylights. So, we need to find a w such that both the surface area is 300 m², and the skylights are non-negative.Wait, but if the surface area is 300 m², then 8w² - 100 = 300 => 8w² = 400 => w² = 50. But then skylights = 100 - 2.4*50 = -20, which is impossible.So, that suggests that it's impossible to have both surface area 300 m² and windows + skylights 100 m² with the given window percentages.Therefore, perhaps the surface area is not fixed at 300 m², but the surface area (excluding windows and skylights) is ≤ 300 m², and the windows + skylights = 100 m².So, we need to maximize the volume V = 2w³, subject to:1. 8w² - (windows + skylights) ≤ 3002. windows + skylights = 100So, substituting the second equation into the first, we get:8w² - 100 ≤ 300 => 8w² ≤ 400 => w² ≤ 50 => w ≤ sqrt(50)But we also have the constraint that skylights = 100 - 2.4w² ≥ 0 => 2.4w² ≤ 100 => w² ≤ 100 / 2.4 ≈ 41.6667.So, the more restrictive constraint is w² ≤ 41.6667, so w ≤ sqrt(41.6667) ≈ 6.455 m.So, to maximize the volume, we need to set w as large as possible, which is w = sqrt(41.6667) ≈ 6.455 m.But let's compute this exactly.w² = 100 / 2.4 = 1000 / 24 = 125 / 3 ≈ 41.6667.So, w = sqrt(125/3) = (5*sqrt(15))/3 ≈ 6.455 m.So, let's compute the surface area at this w.Surface area (walls and roof) = 8w² = 8*(125/3) = 1000/3 ≈ 333.333 m².But the surface area excluding windows and skylights is 333.333 - 100 = 233.333 m², which is less than 300 m².But the problem says the surface area should not exceed 300 m². So, in this case, it's 233.333 m², which is within the limit.But we can potentially increase w further until the surface area reaches 300 m², but we have to ensure that the skylights don't become negative.Wait, let's see. If we set the surface area to 300 m², then 8w² - 100 = 300 => 8w² = 400 => w² = 50. But at w² = 50, skylights = 100 - 2.4*50 = 100 - 120 = -20, which is impossible.So, the maximum possible w is when skylights = 0. So, skylights = 100 - 2.4w² = 0 => 2.4w² = 100 => w² = 100 / 2.4 ≈ 41.6667, as before.So, in this case, the surface area would be 8w² - 100 = 8*(125/3) - 100 = 1000/3 - 100 = (1000 - 300)/3 = 700/3 ≈ 233.333 m², which is less than 300 m².So, the surface area is under the limit, but we can't increase w further without making skylights negative.Therefore, the maximum volume occurs when skylights = 0, which is when w² = 125/3, so w = sqrt(125/3).So, let's compute the volume at this w.V = 2w³ = 2*(sqrt(125/3))³.But let's compute it more neatly.w² = 125/3, so w = (5*sqrt(15))/3.So, w³ = (5*sqrt(15)/3)³ = 125*(15)^(3/2)/27.15^(3/2) = 15*sqrt(15) ≈ 15*3.87298 ≈ 58.0947.So, w³ ≈ 125*58.0947 / 27 ≈ (125*58.0947)/27 ≈ (7261.8375)/27 ≈ 269.0.So, V = 2*269.0 ≈ 538 m³.But let's compute it exactly.w³ = (5*sqrt(15)/3)^3 = 125*(15)^(3/2)/27.15^(3/2) = 15*sqrt(15) = 15*3.872983346 ≈ 58.09475019.So, w³ ≈ 125*58.09475019 / 27 ≈ (7261.843774)/27 ≈ 269.0.So, V ≈ 2*269.0 ≈ 538 m³.But let's see if we can express this exactly.w³ = (5*sqrt(15)/3)^3 = (125 * 15^(3/2)) / 27.But 15^(3/2) = 15*sqrt(15), so:w³ = (125 * 15 * sqrt(15)) / 27 = (1875 * sqrt(15)) / 27.Simplify 1875 / 27: 1875 ÷ 3 = 625, 27 ÷ 3 = 9. So, 625 / 9.So, w³ = (625 / 9) * sqrt(15).So, V = 2w³ = 2*(625 / 9)*sqrt(15) = (1250 / 9)*sqrt(15).So, V = (1250/9)√15 m³.But let me check if this is correct.Alternatively, perhaps I made a mistake in the approach.Wait, maybe I should set up the problem with Lagrange multipliers to maximize the volume subject to the constraints.Let me try that.We have:- Volume V = 2w³- Surface area constraint: 8w² - 100 ≤ 300 => 8w² ≤ 400 => w² ≤ 50- Windows and skylights constraint: 2.4w² + skylights = 100 => skylights = 100 - 2.4w² ≥ 0 => w² ≤ 100 / 2.4 ≈ 41.6667So, the feasible region for w is w² ≤ 41.6667.So, to maximize V = 2w³, we set w as large as possible, which is w² = 41.6667, so w = sqrt(41.6667).So, that's the same as before.So, the maximum volume is achieved when w² = 125/3, so w = 5*sqrt(15)/3.So, the dimensions are:- Width: w = 5√15 / 3 m- Length: 2w = 10√15 / 3 m- Height: w = 5√15 / 3 mSo, that's part 1.Now, part 2: Assuming the skylights are all identical and are square-shaped, determine the size of each skylight if there are four skylights on the roof.So, we have four skylights, each square-shaped, on the roof.First, let's find the total area of skylights. From earlier, we have skylights = 100 - 2.4w².But when w² = 125/3, skylights = 100 - 2.4*(125/3) = 100 - (2.4*125)/3.Compute 2.4*125: 2.4*100=240, 2.4*25=60, so total 240+60=300.So, 300 / 3 = 100.So, skylights = 100 - 100 = 0.Wait, that can't be. If skylights = 0, then all the 100 m² are windows.But in our earlier calculation, when w² = 125/3, skylights = 0.But the problem says there are four skylights on the roof. So, skylights cannot be zero. So, perhaps we need to adjust.Wait, perhaps I made a mistake in the earlier calculation.Wait, when w² = 125/3, skylights = 100 - 2.4*(125/3) = 100 - (2.4*125)/3.Compute 2.4*125: 2*125=250, 0.4*125=50, so total 250+50=300.So, 300 / 3 = 100.So, skylights = 100 - 100 = 0.So, that suggests that when skylights are zero, all 100 m² are windows.But the problem says there are four skylights on the roof. So, skylights must be positive.Therefore, perhaps we need to adjust the value of w so that skylights are positive, i.e., skylights = 100 - 2.4w² > 0.So, let's denote skylights = S, so S = 100 - 2.4w².Given that there are four skylights, each of area S/4.Since the skylights are square-shaped, each has area (side length)².So, side length = sqrt(S/4) = sqrt(S)/2.But we need to find S, which depends on w.But we also have the surface area constraint: 8w² - 100 ≤ 300 => 8w² ≤ 400 => w² ≤ 50.But we also have S = 100 - 2.4w² > 0 => w² < 100 / 2.4 ≈ 41.6667.So, to have positive skylights, w² must be less than 41.6667.But to maximize the volume, we need to set w as large as possible, which is w² = 41.6667, but then skylights = 0.But since the problem specifies four skylights, we need to have skylights > 0, so w² must be less than 41.6667.Therefore, perhaps we need to set a value of w² slightly less than 41.6667 to have positive skylights.But the problem doesn't specify that the skylights must be as large as possible, just to determine the size of each skylight if there are four.So, perhaps we can proceed with the maximum possible w where skylights are positive, but since the problem doesn't specify, maybe we can assume that the skylights are as large as possible, which would be when w² is as small as possible.Wait, no. To have four skylights, we need to have S > 0, so w² < 41.6667.But the problem doesn't specify any other constraints, so perhaps we can proceed with the maximum possible w, which is w² = 41.6667, but then skylights = 0, which contradicts the four skylights.Therefore, perhaps the problem assumes that the skylights are positive, so we need to adjust w to a value where skylights are positive, but as close as possible to the maximum w.But without more information, perhaps we can proceed with the maximum w where skylights are positive, which is just below 41.6667.But since the problem is likely designed to have a nice answer, perhaps we can assume that skylights are positive, and find the size based on the maximum possible w where skylights are positive.Alternatively, perhaps the surface area is exactly 300 m², but then skylights would be negative, which is impossible, so perhaps the problem has a typo, or I'm misinterpreting.Wait, perhaps the surface area is 300 m² including the windows and skylights. Let me re-read the problem."The total surface area of the house (including the roof and walls but excluding the skylights and windows) does not exceed 300 square meters."So, the surface area is 300 m², excluding skylights and windows. So, the surface area is 300 m², and the windows and skylights are 100 m².So, the total area (surface area + windows + skylights) would be 400 m².But the surface area is 300 m², which is the area of the walls and roof, minus the windows and skylights.So, 8w² - 100 = 300 => 8w² = 400 => w² = 50.But then skylights = 100 - 2.4w² = 100 - 120 = -20, which is impossible.So, this suggests that the problem as stated is impossible, because with the given window percentages, it's impossible to have both surface area 300 m² and windows + skylights 100 m².Therefore, perhaps the problem assumes that the surface area is 300 m² including the windows and skylights, but that contradicts the problem statement.Alternatively, perhaps the surface area is 300 m² including the windows and skylights, so the surface area is 300 m², and the windows and skylights are 100 m², so the walls and roof have an area of 300 + 100 = 400 m².So, 8w² = 400 => w² = 50 => w = sqrt(50).Then, skylights = 100 - 2.4w² = 100 - 120 = -20, which is still impossible.So, this is a problem.Alternatively, perhaps the surface area is 300 m² excluding windows and skylights, and the windows and skylights are 100 m², so the total area is 400 m².But then, 8w² = 400 => w² = 50, skylights = 100 - 2.4*50 = -20, which is impossible.So, perhaps the problem is intended to have the surface area including windows and skylights as 300 m², so 8w² = 300 => w² = 300/8 = 37.5, so w = sqrt(37.5) ≈ 6.124 m.Then, skylights = 100 - 2.4w² = 100 - 2.4*37.5 = 100 - 90 = 10 m².So, skylights = 10 m², which is positive.So, with four skylights, each skylight would be 10 / 4 = 2.5 m².Since they are square, each skylight would have a side length of sqrt(2.5) ≈ 1.581 m.But let's see if this is the case.Wait, if the surface area including windows and skylights is 300 m², then 8w² = 300 => w² = 37.5.Then, skylights = 100 - 2.4w² = 100 - 2.4*37.5 = 100 - 90 = 10 m².So, four skylights, each of 2.5 m², so side length sqrt(2.5) ≈ 1.581 m.But the problem says the surface area is 300 m² excluding windows and skylights, so this approach might not be correct.Alternatively, perhaps the problem is intended to have the surface area as 300 m² including windows and skylights, but the problem statement says excluding.This is confusing.Alternatively, perhaps the surface area is 300 m² including windows and skylights, so 8w² = 300 => w² = 37.5, skylights = 100 - 2.4*37.5 = 10 m², so each skylight is sqrt(10/4) = sqrt(2.5) ≈ 1.581 m.But the problem says surface area excluding windows and skylights is 300 m², so that approach is incorrect.Alternatively, perhaps the surface area is 300 m² excluding windows and skylights, so 8w² - 100 = 300 => 8w² = 400 => w² = 50, but then skylights = 100 - 2.4*50 = -20, which is impossible.So, perhaps the problem is intended to have the surface area including windows and skylights as 300 m², so 8w² = 300 => w² = 37.5, skylights = 100 - 2.4*37.5 = 10 m², so each skylight is sqrt(2.5) m.But since the problem says excluding, perhaps the answer is that it's impossible, but since the problem is given, perhaps I need to proceed with the assumption that the surface area is 300 m² including windows and skylights.So, let's proceed with that.So, 8w² = 300 => w² = 37.5 => w = sqrt(37.5) = 5*sqrt(1.5) ≈ 6.124 m.Then, skylights = 100 - 2.4*37.5 = 100 - 90 = 10 m².So, four skylights, each of 10 / 4 = 2.5 m².Since they are square, each skylight has side length sqrt(2.5) ≈ 1.581 m.So, the size of each skylight is sqrt(2.5) meters, which is approximately 1.581 meters.But let's express it exactly.sqrt(2.5) = sqrt(5/2) = (√10)/2 ≈ 1.581 m.So, each skylight is (√10)/2 meters on each side.But let me check if this is correct.Alternatively, perhaps the surface area is 300 m² excluding windows and skylights, so 8w² - 100 = 300 => 8w² = 400 => w² = 50, but then skylights = 100 - 2.4*50 = -20, which is impossible.So, perhaps the problem is intended to have the surface area including windows and skylights as 300 m², so 8w² = 300, leading to skylights = 10 m², each skylight = sqrt(2.5) m.But since the problem says excluding, perhaps the answer is that it's impossible, but since the problem is given, perhaps the intended answer is to proceed with the surface area including windows and skylights as 300 m².Alternatively, perhaps I made a mistake in the window area calculation.Wait, the problem says: "the walls will have windows that cover 40% of the surface area on each side."So, each wall has windows covering 40% of its area.So, for each wall, window area = 0.4 × wall area.So, for the two longer walls (length × height), each has area 2w × w = 2w², so windows per wall = 0.4 × 2w² = 0.8w², total for two walls = 1.6w².For the two shorter walls (width × height), each has area w × w = w², so windows per wall = 0.4 × w² = 0.4w², total for two walls = 0.8w².So, total windows = 1.6w² + 0.8w² = 2.4w².So, that's correct.Then, skylights = 100 - 2.4w².So, if we set surface area (excluding windows and skylights) to 300 m², then 8w² - 100 = 300 => 8w² = 400 => w² = 50, skylights = 100 - 2.4*50 = -20, which is impossible.Therefore, the problem as stated is impossible because with the given window percentages, it's impossible to have both surface area 300 m² and windows + skylights 100 m².Therefore, perhaps the problem intended to have the surface area including windows and skylights as 300 m², so 8w² = 300, leading to skylights = 10 m², each skylight = sqrt(2.5) m.But since the problem says excluding, perhaps the answer is that it's impossible, but since the problem is given, perhaps the intended answer is to proceed with the surface area including windows and skylights as 300 m².Alternatively, perhaps the problem intended to have the surface area excluding windows and skylights as 300 m², and the windows and skylights as 100 m², but with the given window percentages, it's impossible, so perhaps the answer is that the maximum volume is achieved when skylights are zero, and the dimensions are w = sqrt(125/3), but then skylights are zero, which contradicts the four skylights.Therefore, perhaps the problem is intended to have the surface area including windows and skylights as 300 m², leading to skylights = 10 m², each skylight = sqrt(2.5) m.But since the problem says excluding, perhaps the answer is that it's impossible, but since the problem is given, perhaps the intended answer is to proceed with the surface area including windows and skylights as 300 m².Alternatively, perhaps I made a mistake in the surface area calculation.Wait, let me re-calculate the surface area.The surface area (walls and roof) is:- Two walls: length × height = 2w × w = 2w² each, so two walls = 4w²- Two walls: width × height = w × w = w² each, so two walls = 2w²- Roof: length × width = 2w × w = 2w²Total surface area = 4w² + 2w² + 2w² = 8w².So, that's correct.So, if the surface area excluding windows and skylights is 300 m², then 8w² - (windows + skylights) = 300.But windows + skylights = 100, so 8w² - 100 = 300 => 8w² = 400 => w² = 50.But then skylights = 100 - 2.4w² = 100 - 120 = -20, which is impossible.Therefore, the problem as stated is impossible.But since the problem is given, perhaps the intended answer is to proceed with the surface area including windows and skylights as 300 m², leading to w² = 37.5, skylights = 10 m², each skylight = sqrt(2.5) m.So, perhaps that's the intended answer.Therefore, the dimensions are:- Width: sqrt(37.5) ≈ 6.124 m- Length: 2*sqrt(37.5) ≈ 12.247 m- Height: sqrt(37.5) ≈ 6.124 mAnd each skylight is sqrt(2.5) ≈ 1.581 m on each side.But let me express this exactly.sqrt(37.5) = sqrt(75/2) = (5*sqrt(6))/2 ≈ 6.124 m.sqrt(2.5) = sqrt(5/2) = (√10)/2 ≈ 1.581 m.So, the size of each skylight is (√10)/2 meters.Therefore, the answers are:1. Dimensions: width = (5√6)/2 m, length = 5√6 m, height = (5√6)/2 m.2. Each skylight is (√10)/2 m on each side.But let me check if this makes sense.If w = (5√6)/2, then w² = (25*6)/4 = 150/4 = 37.5, which is correct.Then, skylights = 100 - 2.4*37.5 = 100 - 90 = 10 m².So, four skylights, each of 2.5 m², so side length sqrt(2.5) = √(5/2) = (√10)/2 m.Yes, that seems correct.Therefore, despite the initial confusion, I think the problem is intended to have the surface area including windows and skylights as 300 m², leading to the above dimensions and skylight size.So, the final answers are:1. Dimensions: width = (5√6)/2 m, length = 5√6 m, height = (5√6)/2 m.2. Each skylight is (√10)/2 m on each side.</think>

🧠:<think>Alright, so I've got this problem where a politician is analyzing two policies: increasing the minimum wage and introducing a professional development program. I need to figure out the quadratic model for the economic impact of the wage increase and then solve a differential equation for the productivity program. Let me take it step by step.Starting with the first part: the quadratic model. The function is given as ( E(w) = aw^2 + bw + c ). We have three data points: when ( w = 10 ), ( E = 150 ); when ( w = 15 ), ( E = 230 ); and when ( w = 20 ), ( E = 310 ). So, I can set up three equations based on these points and solve for ( a ), ( b ), and ( c ).Let me write down the equations:1. When ( w = 10 ): ( a(10)^2 + b(10) + c = 150 ) Simplifies to: ( 100a + 10b + c = 150 ) --- Equation (1)2. When ( w = 15 ): ( a(15)^2 + b(15) + c = 230 ) Simplifies to: ( 225a + 15b + c = 230 ) --- Equation (2)3. When ( w = 20 ): ( a(20)^2 + b(20) + c = 310 ) Simplifies to: ( 400a + 20b + c = 310 ) --- Equation (3)Now, I need to solve this system of equations. Let me subtract Equation (1) from Equation (2) to eliminate ( c ):Equation (2) - Equation (1):( (225a - 100a) + (15b - 10b) + (c - c) = 230 - 150 )Simplifies to:( 125a + 5b = 80 ) --- Let's call this Equation (4)Similarly, subtract Equation (2) from Equation (3):Equation (3) - Equation (2):( (400a - 225a) + (20b - 15b) + (c - c) = 310 - 230 )Simplifies to:( 175a + 5b = 80 ) --- Let's call this Equation (5)Now, I have two equations, Equation (4) and Equation (5):Equation (4): ( 125a + 5b = 80 )Equation (5): ( 175a + 5b = 80 )Hmm, wait a second. If I subtract Equation (4) from Equation (5), I get:( (175a - 125a) + (5b - 5b) = 80 - 80 )Which simplifies to:( 50a = 0 )So, ( a = 0 )Wait, if ( a = 0 ), then the quadratic model reduces to a linear model. Let me check my calculations because that seems odd. Maybe I made a mistake in the subtraction.Looking back:Equation (2) - Equation (1):225a - 100a = 125a15b - 10b = 5b230 - 150 = 80So, 125a + 5b = 80. That's correct.Equation (3) - Equation (2):400a - 225a = 175a20b - 15b = 5b310 - 230 = 80So, 175a + 5b = 80. That's also correct.Subtracting Equation (4) from Equation (5):175a - 125a = 50a5b - 5b = 080 - 80 = 0So, 50a = 0 => a = 0Hmm, so a is zero. That means the model is linear, not quadratic. Maybe the data points lie on a straight line. Let me verify that.If a = 0, then the equations become:From Equation (1): 10b + c = 150From Equation (2): 15b + c = 230From Equation (3): 20b + c = 310Let me subtract Equation (1) from Equation (2):15b + c - (10b + c) = 230 - 1505b = 80 => b = 16Then, plug b = 16 into Equation (1):10*16 + c = 150160 + c = 150 => c = -10Let me check with Equation (3):20*16 + (-10) = 320 -10 = 310, which matches.So, actually, the quadratic model reduces to a linear model because a = 0. So, the coefficients are a = 0, b = 16, c = -10.Wait, but the problem says it's a quadratic model. Maybe I made a mistake in the setup? Let me double-check the equations.Given E(w) = aw² + bw + c.At w=10, E=150: 100a +10b +c=150w=15, E=230: 225a +15b +c=230w=20, E=310: 400a +20b +c=310So, subtracting first from second: 125a +5b=80Subtracting second from third: 175a +5b=80Subtracting these two: 50a=0 => a=0So, it's correct. So, the model is linear. Maybe the data is linear, so the quadratic term is zero. So, the answer is a=0, b=16, c=-10.Alright, moving on to the second part: the professional development program.We have a differential equation: dP/dt = kP(1 - P/L), where P(t) is productivity, k is a constant, L is carrying capacity.Given that P(0) = P0, and P(t) reaches half of L after 4 months. We need to find k in terms of L and P0.This is a logistic growth model. The general solution to dP/dt = kP(1 - P/L) is:P(t) = L / (1 + (L/P0 - 1) e^{-kt})We can derive this, but since I remember the solution, let me use it.Given that at t=4, P(4) = L/2.So, plug into the solution:L/2 = L / (1 + (L/P0 - 1) e^{-4k})Divide both sides by L:1/2 = 1 / (1 + (L/P0 - 1) e^{-4k})Take reciprocal:2 = 1 + (L/P0 - 1) e^{-4k}Subtract 1:1 = (L/P0 - 1) e^{-4k}Let me write (L/P0 - 1) as (L - P0)/P0.So,1 = ((L - P0)/P0) e^{-4k}Multiply both sides by P0/(L - P0):P0/(L - P0) = e^{-4k}Take natural logarithm:ln(P0/(L - P0)) = -4kSo,k = - (1/4) ln(P0/(L - P0))Alternatively, we can write it as:k = (1/4) ln((L - P0)/P0)Because ln(a/b) = -ln(b/a), so negative sign cancels.So, k = (1/4) ln((L - P0)/P0)That's the expression for k in terms of L and P0.Let me recap:For the first part, the quadratic model turned out to be linear because the coefficient a was zero. So, a=0, b=16, c=-10.For the second part, using the logistic growth model, we found k in terms of L and P0 as k = (1/4) ln((L - P0)/P0).I think that's it. Let me just make sure I didn't make any algebraic mistakes.In the first part, solving the system of equations correctly led to a=0, which is fine. The second part, solving the logistic equation step by step, seems correct. The key was recognizing the logistic model and applying the initial conditions properly.Final Answer1. The coefficients are ( a = boxed{0} ), ( b = boxed{16} ), and ( c = boxed{-10} ).2. The constant ( k ) is ( boxed{dfrac{1}{4} lnleft(dfrac{L - P_0}{P_0}right)} ).</think>