🧠:<think>To determine the common ratio of the geometric progression, I start by noting the given information: the first book has 150 pages, and the fourth book has 600 pages. In a geometric progression, each term is the previous term multiplied by the common ratio ( r ). Therefore, the fourth term can be expressed as ( 150 times r^{3} = 600 ). Solving for ( r ), I divide both sides by 150 to get ( r^{3} = 4 ), and then take the cube root of both sides to find ( r = sqrt[3]{4} ).Next, to find the minimum number of books needed to reach at least 10,000 pages, I use the formula for the sum of the first ( n ) terms of a geometric series: ( S_{n} = a_{1} times frac{r^{n} - 1}{r - 1} ). Plugging in the known values, the equation becomes ( 150 times frac{(sqrt[3]{4})^{n} - 1}{sqrt[3]{4} - 1} geq 10,000 ). Simplifying further, I divide both sides by 150 to get ( frac{(sqrt[3]{4})^{n} - 1}{sqrt[3]{4} - 1} geq frac{200}{3} ). To solve for ( n ), I approximate ( sqrt[3]{4} ) as 1.5874 and set up the inequality ( frac{1.5874^{n} - 1}{0.5874} geq 66.6667 ). Multiplying both sides by 0.5874 gives ( 1.5874^{n} - 1 geq 39.1667 ), and adding 1 to both sides results in ( 1.5874^{n} geq 40.1667 ). Taking the natural logarithm of both sides, I find ( n times ln(1.5874) geq ln(40.1667) ), which simplifies to ( n geq frac{ln(40.1667)}{ln(1.5874)} approx 10.3 ). Since ( n ) must be an integer, the reader needs to collect at least 11 books to reach the desired total of 10,000 pages.</think>

🧠:<think>Okay, so I have this problem about a restaurant owner who is investing in two projects, P and E. The total budget is 500,000. There are two sub-problems here. Let me tackle them one by one.Starting with Sub-problem 1: The owner wants the investment in Project P to be directly proportional to the square of the investment in Project E. The constant of proportionality is given as k = 0.005. I need to find out how much is invested in each project.Alright, let me parse that. Directly proportional means that P = k * E^2. So, if I let P be the amount invested in preservation and E be the amount in economic growth, then P = 0.005 * E^2.Also, the total investment is P + E = 500,000. So, I can set up two equations:1. P = 0.005 * E^22. P + E = 500,000I can substitute the first equation into the second one. So, replacing P in the second equation with 0.005 * E^2 gives:0.005 * E^2 + E = 500,000Hmm, that's a quadratic equation in terms of E. Let me write it as:0.005E^2 + E - 500,000 = 0To make it easier, maybe multiply all terms by 1000 to eliminate the decimal:5E^2 + 1000E - 500,000,000 = 0Wait, let me check that multiplication:0.005 * 1000 = 5, so 0.005E^2 becomes 5E^2E * 1000 = 1000E500,000 * 1000 = 500,000,000So, the equation becomes:5E^2 + 1000E - 500,000,000 = 0Hmm, that seems a bit messy, but manageable. Alternatively, maybe I can keep it as is and use the quadratic formula.Let me write the original equation again:0.005E^2 + E - 500,000 = 0Let me denote this as a quadratic equation in the form aE^2 + bE + c = 0, where:a = 0.005b = 1c = -500,000Using the quadratic formula, E = [-b ± sqrt(b^2 - 4ac)] / (2a)Plugging in the values:E = [-1 ± sqrt(1^2 - 4 * 0.005 * (-500,000))] / (2 * 0.005)First, compute the discriminant:D = 1 - 4 * 0.005 * (-500,000)Calculate 4 * 0.005 = 0.02Then, 0.02 * (-500,000) = -10,000So, D = 1 - (-10,000) = 1 + 10,000 = 10,001So, sqrt(D) = sqrt(10,001). Hmm, sqrt(10,000) is 100, so sqrt(10,001) is just a bit more than 100. Let me approximate it.Let me compute 100^2 = 10,000100.005^2 = (100 + 0.005)^2 = 100^2 + 2*100*0.005 + (0.005)^2 = 10,000 + 1 + 0.000025 = 10,001.000025So, sqrt(10,001) ≈ 100.005So, sqrt(D) ≈ 100.005Therefore, E = [-1 ± 100.005] / (0.01)Wait, because 2a = 2 * 0.005 = 0.01So, E = (-1 + 100.005)/0.01 or E = (-1 - 100.005)/0.01Compute the first solution:(-1 + 100.005) = 99.005Divide by 0.01: 99.005 / 0.01 = 9900.5Second solution:(-1 - 100.005) = -101.005Divide by 0.01: -101.005 / 0.01 = -10,100.5But investment can't be negative, so we discard the negative solution.So, E ≈ 9900.5Therefore, E ≈ 9,900.50Wait, that seems low. Let me check my calculations again.Wait, if E is approximately 9,900.50, then P = 0.005 * E^2Compute E^2: (9900.5)^2Let me compute 9900^2 = (10,000 - 100)^2 = 100,000,000 - 2,000,000 + 10,000 = 98,010,000Then, 9900.5^2 is approximately (9900 + 0.5)^2 = 9900^2 + 2*9900*0.5 + 0.5^2 = 98,010,000 + 9,900 + 0.25 = 98,019,900.25So, P = 0.005 * 98,019,900.25 ≈ 0.005 * 98,019,900 ≈ 490,099.5So, P ≈ 490,099.50Then, P + E ≈ 490,099.50 + 9,900.50 = 500,000, which matches the total budget.So, seems correct.But just to make sure, let me check the quadratic equation again.Original equation: 0.005E^2 + E - 500,000 = 0Multiply all terms by 1000 to eliminate decimals:5E^2 + 1000E - 500,000,000 = 0Wait, 0.005 * 1000 = 5, yes.E * 1000 = 1000E500,000 * 1000 = 500,000,000So, 5E^2 + 1000E - 500,000,000 = 0Divide all terms by 5 to simplify:E^2 + 200E - 100,000,000 = 0Now, quadratic equation: E = [-200 ± sqrt(200^2 - 4*1*(-100,000,000))]/2Compute discriminant:D = 40,000 + 400,000,000 = 400,040,000sqrt(D) = sqrt(400,040,000). Hmm, sqrt(400,000,000) is 20,000. So, sqrt(400,040,000) is a bit more.Compute 20,000^2 = 400,000,000Compute 20,001^2 = 400,040,001So, sqrt(400,040,000) is approximately 20,001 - (1 / (2*20,001)) ≈ 20,001 - 0.000025 ≈ 20,000.999975So, sqrt(D) ≈ 20,001Therefore, E = [-200 ± 20,001]/2Compute the positive solution:(-200 + 20,001)/2 = (19,801)/2 = 9,900.5Which is the same as before. So, E ≈ 9,900.5, which is approximately 9,900.50So, that seems consistent.Therefore, the amount invested in Project E is approximately 9,900.50, and in Project P is approximately 490,099.50.Wait, but that seems like a huge imbalance. Project P is getting almost all the money, and Project E is getting just a small fraction. Is that correct?Let me think. The investment in P is directly proportional to the square of E. So, P = k * E^2. So, as E increases, P increases quadratically. So, if E is small, P is even smaller, but if E is large, P becomes very large.But in this case, with k = 0.005, which is a small constant, so even if E is, say, 10,000, then P would be 0.005*(10,000)^2 = 0.005*100,000,000 = 500,000. But since E is 9,900.5, P is just slightly less.Wait, actually, if E were 10,000, P would be 0.005*(10,000)^2 = 500,000, which would make the total investment 10,000 + 500,000 = 510,000, which is over the budget.So, in our case, E is slightly less than 10,000, so that P is slightly less than 500,000, so that the total is 500,000.So, that makes sense.So, the conclusion is that E is approximately 9,900.50 and P is approximately 490,099.50.Wait, but let me check if E is 9,900.50, then E^2 is approximately 98,019,900.25, and 0.005 times that is 490,099.50, which is correct.So, yes, that seems accurate.So, Sub-problem 1 is solved.Moving on to Sub-problem 2: After the initial investments, both projects generate revenue. The revenue functions are given as:R_P(t) = 10,000 ln(t + 1)R_E(t) = 20,000 e^{0.1t}We need to calculate the total expected revenue after 5 years, so t = 5.So, total revenue R_total(5) = R_P(5) + R_E(5)Compute each separately.First, R_P(5) = 10,000 ln(5 + 1) = 10,000 ln(6)Compute ln(6). I remember that ln(6) is approximately 1.7918So, R_P(5) ≈ 10,000 * 1.7918 ≈ 17,918Next, R_E(5) = 20,000 e^{0.1*5} = 20,000 e^{0.5}Compute e^{0.5}. I know that e^0.5 is approximately 1.6487So, R_E(5) ≈ 20,000 * 1.6487 ≈ 32,974Therefore, total revenue R_total(5) ≈ 17,918 + 32,974 ≈ 50,892So, approximately 50,892 total revenue after 5 years.Wait, let me compute more accurately.First, ln(6):ln(6) = ln(2*3) = ln(2) + ln(3) ≈ 0.6931 + 1.0986 ≈ 1.7917So, R_P(5) = 10,000 * 1.7917 ≈ 17,917Similarly, e^{0.5}:e^{0.5} ≈ 1.64872So, R_E(5) = 20,000 * 1.64872 ≈ 32,974.4So, total revenue ≈ 17,917 + 32,974.4 ≈ 50,891.4So, approximately 50,891.40Rounding to the nearest dollar, that's 50,891.Alternatively, if we use more precise values:Compute ln(6):Using calculator: ln(6) ≈ 1.791759So, R_P(5) = 10,000 * 1.791759 ≈ 17,917.59Compute e^{0.5}:e^{0.5} ≈ 1.6487212707So, R_E(5) = 20,000 * 1.6487212707 ≈ 32,974.4254Total revenue ≈ 17,917.59 + 32,974.4254 ≈ 50,891.0154So, approximately 50,891.02So, depending on rounding, it's about 50,891.Therefore, the total expected revenue after 5 years is approximately 50,891.Wait, but let me make sure I didn't make a mistake in the calculations.Double-checking R_P(5):t = 5, so R_P(5) = 10,000 ln(6). Yes, ln(6) ≈ 1.7918, so 10,000 * 1.7918 ≈ 17,918.R_E(5) = 20,000 e^{0.5} ≈ 20,000 * 1.6487 ≈ 32,974.Total ≈ 17,918 + 32,974 ≈ 50,892.Yes, that seems correct.So, summarizing:Sub-problem 1: E ≈ 9,900.50 and P ≈ 490,099.50Sub-problem 2: Total revenue after 5 years ≈ 50,891I think that's it.</think>

🧠:<think>First, I'll address Sub-problem 1 by calculating the new budget for each department based on the given percentage changes. For the Emergency department, I'll increase the initial budget by 30%. Similarly, I'll increase the ICU budget by 50%, the General Wards by 20%, and decrease the Outpatient Services by 10%. After determining each department's new budget, I'll sum them up to find the total budget required.Next, for Sub-problem 2, I'll calculate the expected increase in patient admissions, which is a 25% increase from the current 1,200 patients per day. This will give the new patient load. I'll then determine the total patient-handling capacity of the current staff by multiplying the number of doctors and nurses by their respective patient capacities. By comparing the new patient load to the current capacity, I'll identify the additional staff needed to maintain the same patient-to-staff ratio. This will involve calculating the required number of doctors and nurses and then determining how many more are needed beyond the current staffing levels.</think>

🧠:<think>First, I need to calculate the total area of the wall to determine the amount of paint required. The wall is 150 meters long and 4 meters high, so the area is 150 multiplied by 4, which equals 600 square meters.For the first sub-problem, there are two layers of paint: a base layer costing 2 per square meter and a detailed artwork layer costing 3 per square meter. I'll calculate the cost for each layer separately and then sum them up. The base layer cost is 600 square meters multiplied by 2, totaling 1,200. The detailed artwork layer cost is 600 square meters multiplied by 3, totaling 1,800. Adding both costs gives a total paint cost of 3,000.Moving on to the second sub-problem, Mrs. Thompson's design divides the mural into squares and triangles. 60% of the area will be squares, and 40% will be triangles. Calculating these percentages of the total area: 60% of 600 square meters is 360 square meters for squares, and 40% is 240 square meters for triangles.Each square panel is 2 meters by 2 meters, giving an area of 4 square meters per panel. To find the number of square panels needed, I divide 360 square meters by 4, resulting in 90 square panels.For the triangular panels, each is an equilateral triangle with a side length of 2 meters. The area of an equilateral triangle is calculated using the formula (√3/4) multiplied by the side length squared. This gives approximately 1.732 square meters per triangle. Dividing the total triangular area of 240 square meters by 1.732 yields approximately 138.56 triangles. Since we can't have a fraction of a panel, we'll need 139 triangular panels to cover the area.In summary, the total cost for the paint is 3,000, and the mural will consist of 90 square panels and 139 triangular panels.</think>