🧠:<think>First, I need to determine the total number of safety nets required for the 50-floor skyscraper. Since safety nets are installed every 5 floors, I divide the total number of floors by 5 to find the number of nets.Next, I'll calculate the total area of all the safety nets by multiplying the number of nets by the area of each net, which is 200 square meters.For the second part, I need to find the halfway point of the building's height. Each floor is 3.5 meters high, so the total height of the building is 50 multiplied by 3.5 meters. Half of this total height will give me the height for the additional safety net.Finally, I'll check if this halfway height aligns with the existing safety net spacing to ensure consistency.</think>

🧠:<think>Okay, so I have this problem where an attorney is dealing with immigration cases, and I need to model how the number of cases she handles changes over time. The problem gives me two functions: one for the rate of new immigration applications, I(t), and another for the rate at which cases are resolved, R(t). First, I need to formulate a differential equation for C(t), the number of cases at time t. Hmm, I remember that in such problems, the rate of change of the number of cases is equal to the rate at which new cases are added minus the rate at which cases are resolved. So, that should translate to dC/dt = I(t) - R(t). Let me write that down:dC/dt = I(t) - R(t)Given that I(t) is 1000e^{0.05t} and R(t) is 500 + 20t, substituting these into the equation gives:dC/dt = 1000e^{0.05t} - (500 + 20t)Simplifying that, it's:dC/dt = 1000e^{0.05t} - 500 - 20tOkay, so that's the differential equation. Now, moving on to the second part, solving this differential equation with the initial condition C(0) = 500. To solve dC/dt = 1000e^{0.05t} - 500 - 20t, I need to integrate both sides with respect to t. Let me set that up:C(t) = ∫ [1000e^{0.05t} - 500 - 20t] dt + C₀Where C₀ is the constant of integration, which I'll determine using the initial condition.Breaking this integral into three separate integrals:C(t) = ∫ 1000e^{0.05t} dt - ∫ 500 dt - ∫ 20t dt + C₀Let me compute each integral one by one.First integral: ∫ 1000e^{0.05t} dtI know that the integral of e^{kt} dt is (1/k)e^{kt} + C. So here, k is 0.05. Therefore,∫ 1000e^{0.05t} dt = 1000 * (1/0.05)e^{0.05t} + C = 1000 * 20e^{0.05t} + C = 20000e^{0.05t} + CSecond integral: ∫ 500 dtThat's straightforward. The integral of a constant is the constant times t. So,∫ 500 dt = 500t + CThird integral: ∫ 20t dtAgain, integrating t with respect to t. The integral of t is (1/2)t², so:∫ 20t dt = 20*(1/2)t² + C = 10t² + CPutting it all together:C(t) = 20000e^{0.05t} - 500t - 10t² + C₀Now, apply the initial condition C(0) = 500. Let's plug t = 0 into the equation:C(0) = 20000e^{0} - 500*0 - 10*(0)² + C₀ = 500Simplify:20000*1 - 0 - 0 + C₀ = 500So,20000 + C₀ = 500Solving for C₀:C₀ = 500 - 20000 = -19500Therefore, the function C(t) is:C(t) = 20000e^{0.05t} - 500t - 10t² - 19500Let me double-check my calculations to make sure I didn't make a mistake.First, the integral of 1000e^{0.05t} is indeed 20000e^{0.05t} because 1000 divided by 0.05 is 20000. Then, the integral of 500 is 500t, and the integral of 20t is 10t². So, subtracting those gives the expression before the constant. Plugging in t=0, e^{0}=1, so 20000*1 is 20000, and the rest are zeros, so 20000 + C₀ = 500, so C₀ is -19500. That seems correct.Just to be thorough, let me compute C(t) again step by step.Starting with dC/dt = 1000e^{0.05t} - 500 - 20t.Integrate term by term:∫1000e^{0.05t} dt = 1000*(1/0.05)e^{0.05t} = 20000e^{0.05t}∫-500 dt = -500t∫-20t dt = -10t²So, combining:C(t) = 20000e^{0.05t} - 500t - 10t² + C₀Apply C(0) = 500:C(0) = 20000e^{0} - 0 - 0 + C₀ = 20000 + C₀ = 500Thus, C₀ = 500 - 20000 = -19500Yes, that seems consistent.So, the final expression for C(t) is:C(t) = 20000e^{0.05t} - 500t - 10t² - 19500I think that's the correct solution. Let me just check if the units make sense. The initial condition is in cases, and the functions I(t) and R(t) are rates, so their units are cases per year. The differential equation is in terms of dC/dt, which is cases per year, so integrating over time gives cases, which matches the units of C(t). So, the units are consistent.Also, looking at the behavior as t increases, the exponential term 20000e^{0.05t} will dominate because exponential growth outpaces the linear and quadratic terms. So, over time, the number of cases will grow exponentially, which makes sense because the rate of new applications is growing exponentially while the resolution rate is only linear. Therefore, the number of cases should increase without bound, which matches our solution.I don't see any mistakes in my reasoning, so I think I'm confident with this solution.Final AnswerThe function representing the number of cases over time is boxed{20000e^{0.05t} - 10t^2 - 500t - 19500}.</think>

🧠:<think>Alright, so Alex is expanding their product line with two new products, A and B. They want to figure out the total revenue after 12 months, considering the sales growth of each product and the boost from hiring a sales representative. Let me break this down step by step.First, let's understand the sales growth for each product without the sales representative. Product A has a linear growth model. It starts at 100 units in the first month and increases by 20 units each month. So, in month 1, it's 100 units, month 2 it's 120, month 3 it's 140, and so on. Since it's linear, the sales each month can be represented by the formula:[ S_A(t) = 100 + 20t ]where ( t ) is the number of months after launch. Product B follows a quadratic growth model. The formula given is:[ S_B(t) = 50 + 10t + 5t^2 ]So, starting at 50 units in the first month, it increases quadratically. That means the growth rate itself is increasing each month, which will lead to much higher sales as time goes on compared to Product A.Now, Alex is hiring a sales representative who is expected to boost the sales of both products by a factor of 1.5. That means each month's sales for both products will be multiplied by 1.5. So, the adjusted sales formulas become:For Product A:[ S'_A(t) = 1.5 times (100 + 20t) ]For Product B:[ S'_B(t) = 1.5 times (50 + 10t + 5t^2) ]Our goal is to calculate the total sales of both products over 12 months and then determine the total revenue based on their respective prices: 30 per unit for Product A and 45 per unit for Product B.To find the total sales, we need to sum the monthly sales for each product over 12 months. That means we'll calculate the sales for each month from t=1 to t=12 for both products, add them up, and then multiply by their respective prices to get the revenue.Let me start by calculating the total sales for Product A without the sales representative first. Since it's linear, the total sales over 12 months can be calculated using the formula for the sum of an arithmetic series. The formula for the sum ( S ) of the first ( n ) terms of an arithmetic series is:[ S = frac{n}{2} times (2a + (n - 1)d) ]where:- ( n ) is the number of terms (12 months),- ( a ) is the first term (100 units),- ( d ) is the common difference (20 units).Plugging in the numbers:[ S_A = frac{12}{2} times (2 times 100 + (12 - 1) times 20) ][ S_A = 6 times (200 + 220) ][ S_A = 6 times 420 ][ S_A = 2520 ]So, without the sales representative, Product A would sell 2520 units over 12 months. But with the 1.5 multiplier, the total sales become:[ S'_A = 1.5 times 2520 = 3780 ]Now, let's move on to Product B. Since it's a quadratic growth, the total sales over 12 months can't be calculated with a simple arithmetic series formula. Instead, we need to compute the sales for each month individually and then sum them up.The formula for Product B's sales each month is:[ S_B(t) = 50 + 10t + 5t^2 ]So, for each month from t=1 to t=12, we'll calculate ( S_B(t) ) and then sum them all. Let me create a table for clarity.| Month (t) | S_B(t) Calculation | S_B(t) Units ||-----------|--------------------|--------------|| 1 | 50 + 10(1) + 5(1)^2 | 50 + 10 + 5 = 65 || 2 | 50 + 10(2) + 5(4) | 50 + 20 + 20 = 90 || 3 | 50 + 10(3) + 5(9) | 50 + 30 + 45 = 125 || 4 | 50 + 10(4) + 5(16) | 50 + 40 + 80 = 170 || 5 | 50 + 10(5) + 5(25) | 50 + 50 + 125 = 225 || 6 | 50 + 10(6) + 5(36) | 50 + 60 + 180 = 290 || 7 | 50 + 10(7) + 5(49) | 50 + 70 + 245 = 365 || 8 | 50 + 10(8) + 5(64) | 50 + 80 + 320 = 450 || 9 | 50 + 10(9) + 5(81) | 50 + 90 + 405 = 545 || 10 | 50 + 10(10) + 5(100)|50 + 100 + 500 = 650 || 11 | 50 + 10(11) + 5(121)|50 + 110 + 605 = 765 || 12 | 50 + 10(12) + 5(144)|50 + 120 + 720 = 890 |Now, let's sum up all these S_B(t) values:65 + 90 = 155155 + 125 = 280280 + 170 = 450450 + 225 = 675675 + 290 = 965965 + 365 = 13301330 + 450 = 17801780 + 545 = 23252325 + 650 = 29752975 + 765 = 37403740 + 890 = 4630So, without the sales representative, Product B would sell 4630 units over 12 months. Applying the 1.5 multiplier:[ S'_B = 1.5 times 4630 = 6945 ]Wait, hold on. Let me double-check that multiplication. 4630 * 1.5. 4630 * 1 = 46304630 * 0.5 = 2315Adding them together: 4630 + 2315 = 6945. Yeah, that's correct.Now, we have the total adjusted sales for both products:- Product A: 3780 units- Product B: 6945 unitsNext, we need to calculate the revenue for each product. Revenue is calculated as:[ text{Revenue} = text{Total Sales} times text{Price per Unit} ]For Product A:[ text{Revenue}_A = 3780 times 30 ]Let me compute that:3780 * 30. Well, 3000 * 30 = 90,000780 * 30 = 23,400Adding them together: 90,000 + 23,400 = 113,400So, Revenue_A = 113,400For Product B:[ text{Revenue}_B = 6945 times 45 ]Hmm, 6945 * 45. Let's break this down.First, 6945 * 40 = 277,800Then, 6945 * 5 = 34,725Adding them together: 277,800 + 34,725 = 312,525So, Revenue_B = 312,525Now, to find the total revenue from both products, we add Revenue_A and Revenue_B:Total Revenue = 113,400 + 312,525 = 425,925Wait, let me verify that addition:113,400 + 312,525Adding the thousands: 113 + 312 = 425Adding the hundreds: 400 + 525 = 925So, yes, 425,925.Therefore, the total revenue generated from both products over 12 months is 425,925.But hold on a second, let me cross-verify the total sales numbers because sometimes when dealing with series, especially quadratic, it's easy to make a mistake in summation.Earlier, I summed up the monthly sales for Product B as 4630 units without the sales rep. Let me recount the monthly sales:Month 1: 65Month 2: 90 (Total: 65+90=155)Month 3: 125 (Total: 155+125=280)Month 4: 170 (Total: 280+170=450)Month 5: 225 (Total: 450+225=675)Month 6: 290 (Total: 675+290=965)Month 7: 365 (Total: 965+365=1330)Month 8: 450 (Total: 1330+450=1780)Month 9: 545 (Total: 1780+545=2325)Month 10: 650 (Total: 2325+650=2975)Month 11: 765 (Total: 2975+765=3740)Month 12: 890 (Total: 3740+890=4630)Yes, that adds up correctly. So, 4630 units without the rep, and 6945 with the 1.5 multiplier.Similarly, for Product A, the total without the rep was 2520, which with 1.5 becomes 3780. That seems correct.Calculating the revenues again:Product A: 3780 * 30 = 113,400Product B: 6945 * 45 = 312,525Total Revenue: 113,400 + 312,525 = 425,925Yes, that seems consistent.Alternatively, another way to approach this is to calculate the monthly sales with the multiplier and then sum them up. Maybe that would be a good way to cross-verify.Let's try that for Product A:Each month, the sales are 1.5*(100 + 20t). So, for each t from 1 to 12, compute 1.5*(100 + 20t) and sum them.Alternatively, since it's linear, the total with the multiplier is just 1.5 times the original total, which we already did. So, 1.5*2520=3780.Same for Product B: 1.5*4630=6945.So, the total sales are correct.Therefore, the total revenue is indeed 425,925.Final AnswerThe total revenue generated from the sales of both products over 12 months is boxed{425925} dollars.</think>

🧠:<think>Okay, so I'm trying to figure out the total salary and bonus for this athlete. Let me break it down step by step.First, the athlete has a fixed salary component of 1,200,000 per year. That part is straightforward. Now, the performance-based salary is a bit more complicated because it's a combination of linear and exponential growth. The formula given is S(x) = 50,000x + 10,000e^{0.05x}, where x is the average points scored per game. Alright, so for part 1, the athlete averages 25 points per game. I need to plug x = 25 into the function S(x). Let me write that out:S(25) = 50,000 * 25 + 10,000 * e^{0.05 * 25}First, calculating the linear part: 50,000 multiplied by 25. Let me do that:50,000 * 25 = 1,250,000Okay, that's the linear component. Now, the exponential part: 10,000 multiplied by e raised to the power of 0.05 times 25. Let me compute the exponent first:0.05 * 25 = 1.25So, e^{1.25}. I remember that e is approximately 2.71828. So, e^1.25 is... Hmm, I might need to calculate this. Let me recall that e^1 is about 2.718, e^1.2 is roughly 3.32, and e^1.25 is a bit more. Maybe around 3.49? Wait, let me check that more accurately.Alternatively, I can use the Taylor series expansion for e^x, but that might take too long. Maybe I can use a calculator approximation. Let me think: e^1.25 is e^(5/4) which is the fourth root of e^5. e^5 is approximately 148.413, so the fourth root of that is roughly 3.49. Yeah, that seems right.So, e^{1.25} ≈ 3.49Therefore, the exponential part is:10,000 * 3.49 = 34,900Adding that to the linear part:1,250,000 + 34,900 = 1,284,900So, the performance-based salary is 1,284,900. Now, adding the fixed salary:Total salary = Fixed salary + Performance-based salaryTotal salary = 1,200,000 + 1,284,900 = 2,484,900Wait, that seems high. Let me double-check my calculations. Maybe I made a mistake with the exponent. Let me recalculate e^{1.25} more accurately.Using a calculator, e^1.25 is approximately 3.49034285. So, 10,000 multiplied by that is 34,903.4285. So, approximately 34,903.43.So, the performance-based salary is 1,250,000 + 34,903.43 = 1,284,903.43Adding the fixed salary: 1,200,000 + 1,284,903.43 = 2,484,903.43Rounding to the nearest dollar, that's 2,484,903.Okay, that seems correct.Now, moving on to part 2. The bonus clause is given if the athlete scores more than 30 points per game. The bonus amount B(y) = 200,000y, where y is the number of points exceeding 30. If the athlete averages 32 points per game, then y is 32 - 30 = 2.So, plugging into the bonus formula:B(2) = 200,000 * 2 = 400,000Therefore, the total bonus is 400,000.But wait, is the bonus per game or for the entire season? The problem says "the bonus is given if the athlete scores more than 30 points per game." It also mentions "y is the number of points scored per game exceeding 30." So, if the athlete averages 32 points per game, then each game they score 2 points over 30, so y = 2.But does that mean the bonus is 200,000 * 2 per game? Or is it 200,000 multiplied by the total points over 30 for the season?Wait, the problem says "the bonus amount B is modeled by B(y) = 200,000y, where y is the number of points scored per game exceeding 30." So, y is per game. But does that mean per game bonus or total?Wait, the wording is a bit ambiguous. It says "the bonus is given if the athlete scores more than 30 points per game." So, if they score more than 30 per game, they get a bonus. The bonus amount is 200,000 multiplied by y, where y is the number of points exceeding 30 per game.So, if they score 32 per game, y = 2, so the bonus is 200,000 * 2 = 400,000. But is that per game or for the entire season?Wait, the problem doesn't specify whether it's per game or total. It just says "the bonus amount B is modeled by B(y) = 200,000y, where y is the number of points scored per game exceeding 30." So, if y is per game, then the bonus is per game? Or is it total?Wait, the function B(y) is given, and y is defined as the number of points per game exceeding 30. So, if the athlete averages 32 points per game, then y = 2. So, B(y) = 200,000 * 2 = 400,000. So, that would be the total bonus for the season, assuming that the bonus is calculated based on the average points per game.Alternatively, if it's per game, then we would need to know the number of games to calculate the total bonus. But the problem doesn't specify the number of games in the season. It just gives the average points per game.Therefore, I think the bonus is calculated based on the average points per game exceeding 30, so y = 2, and the total bonus is 200,000 * 2 = 400,000.So, the total bonus is 400,000.Wait, but let me think again. If the bonus is given if they score more than 30 points per game, does that mean for each game they score over 30, they get a bonus? Or is it a one-time bonus if they exceed 30 on average?The wording says "the bonus is given if the athlete scores more than 30 points per game." So, it's conditional on scoring more than 30 per game. Then, the bonus amount is based on how many points they exceed 30 per game. So, if they average 32, then y = 2, so the bonus is 200,000 * 2 = 400,000.I think that's the correct interpretation.So, to recap:1. Total salary: Fixed + Performance-based = 1,200,000 + 1,284,903.43 ≈ 2,484,9032. Total bonus: 400,000But wait, the problem says "calculate the total bonus if the athlete scores an average of 32 points per game over a season." So, the bonus is 400,000.But do we need to add this bonus to the total salary? The question is separate: part 1 is the total salary, part 2 is the total bonus. So, they are separate components.So, the answers are:1. Total salary: 2,484,9032. Total bonus: 400,000I think that's it. Let me just make sure I didn't make any calculation errors.For part 1:S(25) = 50,000*25 + 10,000*e^{0.05*25}50,000*25 = 1,250,0000.05*25 = 1.25e^1.25 ≈ 3.4903410,000*3.49034 ≈ 34,903.4Total performance-based: 1,250,000 + 34,903.4 ≈ 1,284,903.4Total salary: 1,200,000 + 1,284,903.4 ≈ 2,484,903.4, which is 2,484,903 when rounded.Part 2:y = 32 - 30 = 2B(y) = 200,000*2 = 400,000Yes, that seems correct.</think>