answer：Alright, so I've got this math problem here: "The difference between 2 times a number and -8 is -12. Find the number." Hmm, okay, let's break this down step by step. First, I need to understand what the problem is asking. It mentions a "number," which we'll call "x" for simplicity. The problem says that if you take 2 times this number and then find the difference between that and -8, the result is -12. So, in other words, 2 times the number minus -8 equals -12.Wait, hold on. The wording says "the difference between 2 times a number and -8." Difference usually means subtraction, right? So, it's like 2x minus (-8). But subtracting a negative is the same as adding the positive, so that would make it 2x + 8. And this difference is equal to -12. So, the equation would be 2x + 8 = -12.Okay, that makes sense. Now, I need to solve for x. Let's write that equation down:2x + 8 = -12To solve for x, I'll need to isolate it on one side of the equation. First, I'll subtract 8 from both sides to get rid of the constant term on the left side.2x + 8 - 8 = -12 - 8Simplifying that, I have:2x = -20Now, to get x by itself, I'll divide both sides by 2:2x / 2 = -20 / 2Which simplifies to:x = -10Wait a second, let me double-check that. If I plug x = -10 back into the original equation, does it hold true?2*(-10) + 8 = -20 + 8 = -12Yes, that matches the given condition. So, x = -10 is the correct solution.But just to make sure I didn't make any mistakes, let me go through the steps again.Starting with the problem: The difference between 2 times a number and -8 is -12.Translating that into an equation: 2x - (-8) = -12Simplifying: 2x + 8 = -12Subtracting 8 from both sides: 2x = -20Dividing by 2: x = -10Yep, that all checks out. So, the number we're looking for is -10.I think I've got it. It's important to carefully translate the words into the correct mathematical expression, especially with the signs. Sometimes, misinterpreting "difference" can lead to errors, but in this case, it worked out.

answer：Okay, so I have this problem here about sets M and N, and I need to find the complement of their intersection. Let me try to understand what each set represents.First, set M is defined as {x | y = sqrt(3x - 1)}. Hmm, okay, so this is the set of all x values for which the expression under the square root is non-negative because you can't take the square root of a negative number. So, I need to solve the inequality 3x - 1 >= 0. Let me do that step by step.3x - 1 >= 0 Add 1 to both sides: 3x >= 1 Divide both sides by 3: x >= 1/3So, set M is all real numbers x such that x is greater than or equal to 1/3. In interval notation, that would be [1/3, ∞). Got that.Now, moving on to set N, which is {x | y = log₂(x - 2x²)}. Okay, logarithms have their own set of rules. The argument of the logarithm must be positive because you can't take the log of zero or a negative number. So, I need to solve the inequality x - 2x² > 0.Let me factor that expression: x(1 - 2x) > 0. So, this is a quadratic inequality. To solve it, I should find the critical points where the expression equals zero. Setting x(1 - 2x) = 0, we get x = 0 or 1 - 2x = 0, which gives x = 1/2.So, the critical points are x = 0 and x = 1/2. These divide the real number line into three intervals:1. (-∞, 0)2. (0, 1/2)3. (1/2, ∞)Now, I need to test each interval to see where the product x(1 - 2x) is positive.First interval: (-∞, 0). Let's pick x = -1. Plugging into x(1 - 2x): (-1)(1 - 2*(-1)) = (-1)(1 + 2) = (-1)(3) = -3, which is negative. So, this interval doesn't satisfy the inequality.Second interval: (0, 1/2). Let's pick x = 1/4. Plugging in: (1/4)(1 - 2*(1/4)) = (1/4)(1 - 1/2) = (1/4)(1/2) = 1/8, which is positive. So, this interval satisfies the inequality.Third interval: (1/2, ∞). Let's pick x = 1. Plugging in: 1*(1 - 2*1) = 1*(1 - 2) = 1*(-1) = -1, which is negative. So, this interval doesn't satisfy the inequality.Therefore, the solution to x(1 - 2x) > 0 is (0, 1/2). So, set N is the open interval from 0 to 1/2.Now, I need to find the intersection of sets M and N. Set M is [1/3, ∞) and set N is (0, 1/2). The intersection would be the overlap of these two intervals.Looking at the intervals, [1/3, ∞) starts at 1/3 and goes to infinity, while (0, 1/2) starts just above 0 and goes up to 1/2. The overlap between them is from 1/3 to 1/2 because that's where both intervals are present.So, M ∩ N is [1/3, 1/2). Now, the question asks for the complement of this intersection with respect to the real numbers. The complement of a set is all the real numbers not in that set.So, the complement of [1/3, 1/2) would be everything less than 1/3 and everything greater than or equal to 1/2. In interval notation, that's (-∞, 1/3) ∪ [1/2, ∞).Looking at the options given:A: (1/3, 1/2) B: (-∞, 1/3) ∪ [1/2, ∞) C: [0, 1/2] D: (-∞, 0] ∪ [1/2, ∞)So, option B matches the complement I found. Therefore, the correct answer should be B.Wait, let me double-check my steps to make sure I didn't make a mistake.1. For set M: sqrt(3x - 1) requires 3x - 1 >= 0 => x >= 1/3. That seems correct.2. For set N: log₂(x - 2x²) requires x - 2x² > 0 => x(1 - 2x) > 0. Factoring and solving the inequality, I found the critical points at 0 and 1/2, tested intervals, and concluded (0, 1/2). That seems right.3. Intersection of [1/3, ∞) and (0, 1/2) is [1/3, 1/2). That makes sense because 1/3 is approximately 0.333 and 1/2 is 0.5, so the overlap is between 1/3 and 1/2.4. Complement of [1/3, 1/2) is everything else on the real line, which is (-∞, 1/3) and [1/2, ∞). So, option B is correct.I think I did everything correctly. Maybe I should visualize it on a number line to confirm.Imagine a number line with points at 0, 1/3, and 1/2.- Set M covers from 1/3 onwards.- Set N covers from just above 0 to just below 1/2.- Their intersection is from 1/3 to just below 1/2.- The complement would be everything to the left of 1/3 and everything to the right of 1/2, including 1/2 since it's a closed interval on the right.Yes, that seems consistent. So, I'm confident that option B is the correct answer.**Final Answer**boxed{B}

answer：Alright, so I've got this problem here about an odd function f(x) defined on all real numbers. It says that f(x) equals f(1 - x) for any real number x. Also, when x is between 0 and 1/2, f(x) is given as 2x squared. I need to find the value of f(3) plus f(-5/2). Hmm, okay, let's break this down step by step.First, let me recall what an odd function is. An odd function satisfies the property f(-x) = -f(x) for all x in its domain. So, if I know the value of f at some positive x, I can find f at the corresponding negative x by just taking the negative of that value. That might come in handy later.Next, the problem states that f(x) = f(1 - x) for any real number x. That seems like a symmetry condition. If I plug in x into the function, it's the same as plugging in 1 - x. So, this suggests that the function is symmetric around the line x = 1/2. That is, if you reflect the graph of the function over the vertical line x = 1/2, it remains unchanged. Interesting.Also, we're given the specific form of f(x) on the interval (0, 1/2]. It's 2x squared. So, for any x in that interval, f(x) = 2x². Since the function is odd, I can use that to find f(x) on the interval [-1/2, 0). Specifically, for x in [-1/2, 0), f(x) = -f(-x) = -2(-x)² = -2x². So, that gives me f(x) on [-1/2, 1/2].But what about outside of that interval? Well, the function is defined on all real numbers, so I need to figure out how to extend f(x) beyond [-1/2, 1/2]. The key here is the given symmetry condition f(x) = f(1 - x). Let's see if we can use that to find f(x) for x outside of (0, 1/2].Let me consider x = 3. That's way outside of (0, 1/2]. How can I relate f(3) to something within the known interval? Maybe by using the symmetry condition f(x) = f(1 - x). So, f(3) = f(1 - 3) = f(-2). Hmm, okay, so f(3) is equal to f(-2). But since f is odd, f(-2) = -f(2). So, f(3) = -f(2).Now, I need to find f(2). Let's apply the symmetry condition again. f(2) = f(1 - 2) = f(-1). Again, using the odd property, f(-1) = -f(1). So, f(2) = -f(1). Therefore, f(3) = -(-f(1)) = f(1).Alright, so f(3) is equal to f(1). Now, let's find f(1). Using the symmetry condition, f(1) = f(1 - 1) = f(0). But wait, f(0) is a specific value. Since f is odd, f(-0) = -f(0), but since -0 is 0, this implies f(0) = -f(0), which can only be true if f(0) = 0. So, f(1) = f(0) = 0. Therefore, f(3) = 0.Okay, so f(3) is 0. Now, let's tackle f(-5/2). That's the same as f(-2.5). Since f is odd, f(-2.5) = -f(2.5). So, I need to find f(2.5). Let's use the symmetry condition again. f(2.5) = f(1 - 2.5) = f(-1.5). Then, using the odd property, f(-1.5) = -f(1.5). So, f(2.5) = -f(1.5). Therefore, f(-2.5) = -(-f(1.5)) = f(1.5).Now, let's find f(1.5). Again, using the symmetry condition, f(1.5) = f(1 - 1.5) = f(-0.5). Since f is odd, f(-0.5) = -f(0.5). So, f(1.5) = -f(0.5). Therefore, f(-2.5) = -f(0.5).But wait, f(0.5) is in the interval (0, 1/2]? No, 0.5 is exactly at the boundary. The interval is (0, 1/2], so 0.5 is included. So, f(0.5) = 2*(0.5)^2 = 2*(0.25) = 0.5. Therefore, f(-2.5) = -0.5.Putting it all together, f(3) is 0 and f(-2.5) is -0.5. So, f(3) + f(-5/2) = 0 + (-0.5) = -0.5.Wait, let me double-check my steps to make sure I didn't make a mistake. Starting with f(3):1. f(3) = f(1 - 3) = f(-2)2. f(-2) = -f(2) because f is odd3. f(2) = f(1 - 2) = f(-1)4. f(-1) = -f(1) because f is odd5. f(1) = f(1 - 1) = f(0) = 06. So, f(2) = -0 = 0, hence f(-2) = -0 = 0, and f(3) = 0That seems correct.Now for f(-5/2):1. f(-5/2) = -f(5/2) because f is odd2. f(5/2) = f(1 - 5/2) = f(-3/2)3. f(-3/2) = -f(3/2) because f is odd4. f(3/2) = f(1 - 3/2) = f(-1/2)5. f(-1/2) = -f(1/2) because f is odd6. f(1/2) = 2*(1/2)^2 = 0.57. So, f(-1/2) = -0.5, hence f(3/2) = -0.58. Therefore, f(-3/2) = -(-0.5) = 0.59. So, f(5/2) = 0.5, and f(-5/2) = -0.5Wait a minute, hold on. In step 9, I have f(-5/2) = -f(5/2) = -0.5. But earlier, I thought f(5/2) was 0.5, so f(-5/2) would be -0.5. That seems consistent.But let me check f(3/2):1. f(3/2) = f(1 - 3/2) = f(-1/2)2. f(-1/2) = -f(1/2) = -0.53. So, f(3/2) = -0.5Then, f(-3/2) = -f(3/2) = -(-0.5) = 0.5Then, f(5/2) = f(-3/2) = 0.5Therefore, f(-5/2) = -f(5/2) = -0.5Yes, that seems correct.So, f(3) is 0 and f(-5/2) is -0.5, so their sum is -0.5.I think that's the answer. Let me just recap:- Use the given symmetry f(x) = f(1 - x) to relate f at larger x to f at smaller x.- Use the oddness of f to relate f at negative x to f at positive x.- For x in (0, 1/2], f(x) is given, so use that to find f at those points.- Combine these properties to express f(3) and f(-5/2) in terms of f at points within the known interval.- Calculate the values step by step, ensuring each transformation is correct.I don't see any errors in my reasoning, so I think the final answer is -0.5.

answer：Okay, so I have this problem where a right circular cylinder is inscribed in a right circular cone. The cone has a diameter of 12 units and an altitude of 15 units. The cylinder's height is three times its radius, and their axes coincide. I need to find the radius of the cylinder and express it as a common fraction.First, let me visualize the problem. There's a cone with a diameter of 12, so its radius is half of that, which is 6 units. The height of the cone is 15 units. Inside this cone, there's a cylinder whose height is three times its radius. The cylinder is perfectly centered within the cone, meaning their central axes are the same.I think the key here is to use similar triangles. Since the cylinder is inscribed in the cone, the top edge of the cylinder will touch the cone's surface. If I imagine a vertical cross-section through the axis of the cone and cylinder, I can see two similar triangles: one large triangle representing the cone and a smaller triangle representing the space above the cylinder.Let me denote the radius of the cylinder as ( r ). Then, the height of the cylinder is ( 3r ) because it's given that the height is three times the radius. The height of the cone is 15 units, so the remaining height from the top of the cylinder to the apex of the cone is ( 15 - 3r ).Now, looking at the similar triangles. The large triangle (the cone) has a height of 15 and a base radius of 6. The smaller triangle above the cylinder has a height of ( 15 - 3r ) and a base radius of ( r ). Since these triangles are similar, their corresponding sides are proportional.So, I can set up a proportion:[frac{text{Height of small triangle}}{text{Height of large triangle}} = frac{text{Radius of small triangle}}{text{Radius of large triangle}}]Plugging in the values:[frac{15 - 3r}{15} = frac{r}{6}]Now, I need to solve for ( r ). Let's cross-multiply to get rid of the fractions:[6(15 - 3r) = 15r]Expanding the left side:[90 - 18r = 15r]Now, let's get all the terms involving ( r ) on one side:[90 = 15r + 18r][90 = 33r]Finally, solving for ( r ):[r = frac{90}{33}]Simplifying the fraction by dividing numerator and denominator by 3:[r = frac{30}{11}]So, the radius of the cylinder is ( frac{30}{11} ) units.Wait, let me double-check my steps to make sure I didn't make a mistake. I set up the proportion correctly based on similar triangles, cross-multiplied, expanded, and solved for ( r ). The calculations seem correct. I think I got it right.**Final Answer**The radius of the cylinder is boxed{dfrac{30}{11}}.

answer：First, I need to determine the total number of ways to select three contestants out of eight. This can be calculated using the combination formula:[{8 choose 3} = frac{8 times 7 times 6}{3 times 2 times 1} = 56]Next, I'll calculate the number of ways to select three contestants who are all female. There are five female contestants, so:[{5 choose 3} = frac{5 times 4 times 3}{3 times 2 times 1} = 10]To find the number of ways to select at least one male, I'll subtract the all-female selections from the total selections:[56 - 10 = 46]Finally, the probability of selecting at least one male contestant is:[frac{46}{56} = frac{23}{28}]So, the probability that at least one male is among the selected contestants is (boxed{frac{23}{28}}).

answer：Okay, so I have this problem about an exponential function, f(x) = a^x. They told me that a is greater than 0 and not equal to 1, which makes sense because exponential functions usually have positive bases not equal to 1. The graph of this function passes through the point (2, 1/9). Part (1) asks me to find the value of a. Hmm, okay. So, if the graph passes through (2, 1/9), that means when x is 2, f(x) is 1/9. So, plugging that into the equation, I get a^2 = 1/9. Wait, so a squared equals 1/9. To find a, I need to take the square root of both sides. But since a is positive, I don't have to worry about the negative root. So, a should be sqrt(1/9). The square root of 1 is 1, and the square root of 9 is 3, so that simplifies to 1/3. Okay, so a is 1/3. That seems straightforward.Let me double-check. If a is 1/3, then f(2) should be (1/3)^2, which is 1/9. Yep, that matches the point given. So, I think that's correct.Moving on to part (2). It asks me to compare the magnitudes of f(3) and f(4). So, I need to figure out whether f(3) is greater than f(4) or vice versa.Since I found that a is 1/3, the function is f(x) = (1/3)^x. Now, I remember that exponential functions can be either increasing or decreasing depending on the base. If the base is greater than 1, the function is increasing. If the base is between 0 and 1, the function is decreasing.In this case, the base is 1/3, which is between 0 and 1. So, the function f(x) = (1/3)^x is a decreasing function. That means as x increases, f(x) decreases.So, since 3 is less than 4, f(3) should be greater than f(4). Let me verify that by calculating both values.Calculating f(3): (1/3)^3 = 1/27.Calculating f(4): (1/3)^4 = 1/81.Comparing 1/27 and 1/81, clearly 1/27 is larger because 27 is smaller than 81, so 1 divided by a smaller number is larger. Therefore, f(3) is greater than f(4).Wait, let me think again. If the function is decreasing, then as x increases, the output decreases. So, yes, since 3 < 4, f(3) > f(4). That makes sense.I think I got both parts right. For part (1), a is 1/3, and for part (2), f(3) is greater than f(4).**Final Answer**(1) The value of ( a ) is (boxed{frac{1}{3}}).(2) The comparison result is (boxed{f(3) > f(4)}).