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question:The edges of a cuboid are 2 cm, some cm, and 8 cm. The volume of the cuboid is 80 cm³. What is the length of the missing edge?
answer:The volume of a cuboid is given by the formula: Volume = length × width × height We are given that the volume of the cuboid is 80 cm³, and two of the edges are 2 cm and 8 cm. Let's call the missing edge "x" cm. So, we have: 80 cm³ = 2 cm × x cm × 8 cm To find the missing edge "x", we need to divide both sides of the equation by the product of the two known edges: x = 80 cm³ / (2 cm × 8 cm) x = 80 cm³ / 16 cm² x = 5 cm Therefore, the length of the missing edge is boxed{5} cm.
question:On an indoor circular track of circumference 100 feet, Mia starts at point S, runs in a counterclockwise direction, and then stops when she has run exactly two miles (10560 feet). On which quarter of the circle, A, B, C, or D, did Mia stop? [asy] import graph; draw(Circle((0,0),40)); draw((50,0)(50,0)); draw((0,50)(0,50)); dot((40,0)); label(S,(40,0),SE); label(A,(35,35),E); label(B,(35,35),W); label(C,(35,35),W); label(D,(35,35),E); [/asy]
answer:First, calculate how many complete laps Mia ran on the track: - The circumference of the track is 100 feet. - One complete lap is 10560 feet divided by 100 feet per lap, which equals 105.6 laps. Mia completed 105 full laps and a part of another lap. To determine how far into the additional lap she ran: - (105 times 100 = 10500) feet in complete laps. - (10560 - 10500 = 60) feet into the partial lap. The track is divided into four quarters, each being 25 feet long (100 feet / 4). Mia runs 60 feet into the new lap: - The first 25 feet take her to the end of quarter A. - The next 25 feet take her through quarter B. - The final 10 feet take her into quarter C. Thus, Mia stops in quarter boxed{C}.
question:Find the product of the nonreal roots of the equation x^4 4x^3 + 6x^2 4x = 2010.
answer:We start by noticing a similarity with (x-1)^4 in the left-hand side of the equation. Thus, we try adding 1 to both sides: [x^4 - 4x^3 + 6x^2 - 4x + 1 = 2011.] This transforms our equation to: [(x-1)^4 = 2011.] We then proceed to solve for x: [x-1 = sqrt[4]{2011}, isqrt[4]{2011}, -sqrt[4]{2011}, -isqrt[4]{2011}.] Since we are interested in the nonreal roots, they are: [x = 1 + isqrt[4]{2011}, , x = 1 - isqrt[4]{2011}.] To find their product: [P = (1 + isqrt[4]{2011})(1 - isqrt[4]{2011}) = 1^2 - (isqrt[4]{2011})^2 = 1 + sqrt{2011}.] Thus, the product of the nonreal roots is: [boxed{1 + sqrt{2011}}.]
question:A function f(x) defined on mathbb{R} is called an H function if for any x_1 neq x_2, it satisfies x_1 f(x_1) + x_2 f(x_2) > x_1 f(x_2) + x_2 f(x_1). Determine the number of H functions in the following list: 1. y = x^2 + x + 1; 2. y = 3x 2(sin x cos x); 3. y = e^x + 1; 4. f(x) = begin{cases} |ln x|, &x neq 0 0, &x = 0 end{cases} A: 4 B: 3 C: 2 D: 1
answer:Given any distinct real numbers x_1 and x_2, if the inequality x_1 f(x_1) + x_2 f(x_2) > x_1 f(x_2) + x_2 f(x_1) always holds, then function f(x) is defined as an "H function," which is equivalent to being a strictly increasing function on mathbb{R}. 1. y = -x^2 + x + 1: The function's axis of symmetry is x = frac{1}{2}. It is not monotonic on its entire domain, so it does not satisfy the condition. 2. y = 3x - 2(sin x - cos x): The derivative is y' = 3 - 2(cos x + sin x) = 3 - 2 sqrt{2} sin left(x + frac{pi}{4}right) > 0. The function is strictly increasing, so it satisfies the condition. 3. y = e^x + 1: The function is strictly increasing, so it satisfies the condition. 4. f(x) = begin{cases} |ln x|, &x neq 0 0, &x = 0 end{cases}: The function is strictly increasing for x > 0 and strictly decreasing for x < 0. It does not satisfy the condition. Thus, the "H functions" are 2 and 3. The answer is: boxed{text{C}}.