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question:a northwestern town, it rained 4 inches per day during the first 15 days of November. For the remainder of the month, the average daily rainfall was some multiple of the amount observed during the first 15 days. The total amount of rainfall for this town in November was 180 inches. What is the ratio of the average daily rainfall for the remainder of the month to the first 15 days?
answer:Let's first calculate the total rainfall for the first 15 days of November: Rainfall for the first 15 days = 4 inches/day * 15 days = 60 inches Now, we know the total rainfall for the entire month of November was 180 inches. So, the rainfall for the remainder of the month (which is 30 - 15 = 15 days) is: Total rainfall for the remainder of the month = 180 inches (total) - 60 inches (first 15 days) = 120 inches Now, let's find the average daily rainfall for the remainder of the month: Average daily rainfall for the remainder of the month = Total rainfall for the remainder of the month / Number of days in the remainder of the month Average daily rainfall for the remainder of the month = 120 inches / 15 days = 8 inches/day Now we have the average daily rainfall for the first 15 days (4 inches/day) and the average daily rainfall for the remainder of the month (8 inches/day). The ratio of the average daily rainfall for the remainder of the month to the first 15 days is: Ratio = Average daily rainfall for the remainder of the month / Average daily rainfall for the first 15 days Ratio = 8 inches/day / 4 inches/day = 2 So, the ratio of the average daily rainfall for the remainder of the month to the first 15 days is boxed{2:1} .
question:A new dartboard is divided into six regions by three radii and two concentric circles of radii 4 and 8. The smaller inner regions have point values of 3, 5, and 5 respectively, and the larger outer regions have point values of 4, 3, and 3. The probability that a dart hits a given region is proportional to the area of that region. If a player throws two darts at the board, the score is the sum of the point values where the darts land. What is the probability that the score will be odd? A) frac{2}{9} B) frac{1}{3} C) frac{4}{9} D) frac{5}{12} E) frac{1}{2}
answer:1. **Identify the regions and their point values**: - Inner regions: 3, 5, 5 (radii = 4) - Outer regions: 4, 3, 3 (outer radius = 8, inner radius = 4) 2. **Calculate the areas**: - **Area of the inner circle**: pi times 4^2 = 16pi. - **Area of each inner region**: frac{16pi}{3}. - **Area of the outer ring (annulus)**: pi times 8^2 - pi times 4^2 = 64pi - 16pi = 48pi. - **Area of each outer region**: frac{48pi}{3} = 16pi. 3. **Calculate the probability of hitting an odd or even number**: - **Odd-numbered regions**: Two outer regions (3, 3) and two inner regions (5, 5) are labeled with odd numbers, so the total area for odd numbers is 16pi + 16pi + frac{16pi}{3} + frac{16pi}{3} = 32pi + frac{32pi}{3}. - **Probability of hitting an odd number**: frac{32pi + frac{32pi}{3}}{64pi} = frac{128pi}{192pi} = frac{2}{3}. - **Even-numbered regions**: One inner region (3) and one outer region (4), so the total area for even numbers is frac{16pi}{3} + 16pi = frac{64pi}{3}. - **Probability of hitting an even number**: frac{frac{64pi}{3}}{64pi} = frac{1}{3}. 4. **Calculate the probability of scoring an odd sum**: - **Odd then even or even then odd**: frac{2}{3} times frac{1}{3} = frac{2}{9}. - **Total probability of an odd sum**: frac{2}{9} + frac{2}{9} = frac{4}{9}. 5. **Conclusion**: The probability that the score is odd is frac{4{9}}. The final answer is boxed{C}.
question:The monotonic increasing interval of the function f(x) = log_{frac{1}{2}}(x^2 1) is ( ) A: (0, +infty) B: (infty, 0) C: (1, +infty) D: (infty, 1)
answer:Given x^2 - 1 > 0, we find x < -1 or x > 1. Therefore, the domain of the function f(x) = log_{frac{1}{2}}(x^2 - 1) is (-infty, -1) cup (1, +infty). Let t = x^2 - 1, The outer function y = log_{frac{1}{2}}t is a decreasing function within its domain, while the inner function t = x^2 - 1 is a decreasing function within (-infty, -1). Based on the monotonicity of composite functions, the monotonic increasing interval of the function f(x) = log_{frac{1}{2}}(x^2 - 1) is (-infty, -1). Therefore, the correct answer is: boxed{D}. To find the original function's domain, and knowing that the outer function is a decreasing function within its domain, it suffices to find the decreasing interval of the inner function to determine the increasing interval of the original function. This problem examines the monotonicity of composite functions and how to find monotonic intervals. Regarding the monotonicity of composite functions, it is important to first determine the function's domain and then use the relationship between the monotonicity of the composite function and that of the inner and outer functions. The judgment is based on "increasing together, decreasing separately," which is considered a medium-level question.
question:Let x, y, and z be nonzero complex numbers such that x + y + z = 20 and [(x y)^2 + (x z)^2 + (y z)^2 = xyz.]Find frac{x^3 + y^3 + z^3}{xyz}.
answer:To solve for frac{x^3 + y^3 + z^3}{xyz} given the conditions x + y + z = 20 and (x - y)^2 + (x - z)^2 + (y - z)^2 = xyz, we proceed as follows: First, we recognize a useful factorization for the sum of cubes: [x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - xz - yz).] Next, we expand the given equation (x - y)^2 + (x - z)^2 + (y - z)^2 = xyz to find a relationship between the terms x^2 + y^2 + z^2 - xy - xz - yz and xyz: begin{align*} (x - y)^2 + (x - z)^2 + (y - z)^2 &= 2x^2 + 2y^2 + 2z^2 - 2xy - 2xz - 2yz &= xyz. end{align*} This simplifies to: [x^2 + y^2 + z^2 - xy - xz - yz = frac{xyz}{2}.] Substituting x + y + z = 20 and x^2 + y^2 + z^2 - xy - xz - yz = frac{xyz}{2} into the factorization, we get: begin{align*} x^3 + y^3 + z^3 - 3xyz &= (x + y + z)left(x^2 + y^2 + z^2 - xy - xz - yzright) &= 20 cdot frac{xyz}{2} &= 10xyz. end{align*} Adding 3xyz to both sides, we find: [x^3 + y^3 + z^3 = 13xyz.] Therefore, the value of frac{x^3 + y^3 + z^3}{xyz} is: [frac{x^3 + y^3 + z^3}{xyz} = boxed{13}.]