answer：First, I need to determine the number of ways to choose the president. Since there are 15 boys and 15 girls, there are a total of 30 possible choices for the president.Next, the vice-president must be of the same gender as the president. If a boy is chosen as president, there are 14 remaining boys to choose from for vice-president. Similarly, if a girl is chosen as president, there are 14 remaining girls for vice-president.Finally, the treasurer must be of the opposite gender to the president. If a boy is president, there are 15 girls available for treasurer, and if a girl is president, there are 15 boys available.To calculate the total number of ways, I multiply the number of choices for each position:- For a boy president: 15 (president) × 14 (vice-president) × 15 (treasurer) = 3150 ways- For a girl president: 15 (president) × 14 (vice-president) × 15 (treasurer) = 3150 waysAdding both possibilities together gives the total number of ways to choose the officers under the given conditions.

answer：Okay, so I need to find the product of the two smallest one-digit primes and the smallest two-digit prime. Hmm, let's break this down step by step.First, what are one-digit primes? Well, prime numbers are numbers greater than 1 that have no positive divisors other than 1 and themselves. So, the one-digit primes are 2, 3, 5, and 7. Among these, the two smallest are 2 and 3. Got that.Next, what's the smallest two-digit prime? Two-digit numbers start from 10, but 10 isn't prime because it's divisible by 2 and 5. The next one is 11. Is 11 a prime? Yes, because its only divisors are 1 and 11. So, 11 is the smallest two-digit prime.Now, I need to find the product of these three numbers: 2, 3, and 11. Let me calculate that. First, multiply 2 and 3. 2 times 3 is 6. Then, multiply that result by 11. 6 times 11 is 66. So, the product should be 66.Wait, let me double-check to make sure I didn't make any mistakes. 2 is the smallest one-digit prime, then 3, and 11 is indeed the smallest two-digit prime. Multiplying them together: 2 times 3 is 6, and 6 times 11 is 66. Yep, that seems right.I think I've got it. The product is 66.

answer：Okay, so I have this problem where I'm given two functions, p(x) and q(x). p(x) is 4x + 5, and q(x) is 3x - 4. Then, r(x) is defined as p(q(x)), which means I need to compose these two functions. After finding r(x), I have to determine its inverse function, r⁻¹(x). Alright, let's start by understanding what function composition means. When we say r(x) = p(q(x)), it means that I substitute q(x) into p(x) wherever there's an x. So, essentially, I'm plugging the entire expression of q(x) into p(x). Let me write down p(x) and q(x) again for clarity:- p(x) = 4x + 5- q(x) = 3x - 4So, to find r(x) = p(q(x)), I replace every x in p(x) with q(x). That would look like:r(x) = p(q(x)) = 4*(q(x)) + 5Now, substituting q(x) into that:r(x) = 4*(3x - 4) + 5Okay, now I need to simplify this expression. Let's do the multiplication first:4*(3x) is 12x, and 4*(-4) is -16. So, expanding that:r(x) = 12x - 16 + 5Now, combining the constant terms (-16 and +5):-16 + 5 is -11. So, r(x) simplifies to:r(x) = 12x - 11Alright, so now I have r(x) = 12x - 11. The next step is to find its inverse function, r⁻¹(x). To find the inverse of a function, I remember that the process involves switching the roles of x and y and then solving for y. So, let's start by writing r(x) as an equation:y = 12x - 11Now, to find the inverse, I need to swap x and y:x = 12y - 11Now, I have to solve this equation for y. Let's do that step by step. First, I can add 11 to both sides to isolate the term with y:x + 11 = 12yNow, to solve for y, I need to divide both sides by 12:(x + 11)/12 = ySo, y = (x + 11)/12Therefore, the inverse function r⁻¹(x) is:r⁻¹(x) = (x + 11)/12Wait, let me double-check my steps to make sure I didn't make any mistakes. Starting from r(x) = 12x - 11, swapping x and y gives x = 12y - 11. Adding 11 to both sides gives x + 11 = 12y. Dividing both sides by 12 gives y = (x + 11)/12. That seems correct.Just to be thorough, let me verify if this inverse function works correctly. Let's pick a value for x, say x = 1. Calculating r(1):r(1) = 12*1 - 11 = 12 - 11 = 1Now, applying the inverse function to this result:r⁻¹(1) = (1 + 11)/12 = 12/12 = 1That works. Let's try another value, say x = 2.Calculating r(2):r(2) = 12*2 - 11 = 24 - 11 = 13Now, applying the inverse function:r⁻¹(13) = (13 + 11)/12 = 24/12 = 2Perfect, that also works. It seems like the inverse function is correctly derived.So, in summary, the steps were:1. Compose p(q(x)) to get r(x) = 12x - 11.2. Set y = 12x - 11.3. Swap x and y to get x = 12y - 11.4. Solve for y to find the inverse function y = (x + 11)/12.5. Verify the inverse function with sample values to ensure correctness.I think that covers everything. The inverse function is straightforward once you follow the steps methodically. It's important to check your work by plugging in values to confirm that applying the function and then its inverse brings you back to the original input. This helps catch any algebraic errors that might have occurred during the process.Another thing to note is that since r(x) is a linear function, its inverse will also be linear, which makes sense because we ended up with another linear function for the inverse. If the original function had been nonlinear, finding the inverse might have been more complicated or even impossible without additional constraints.In this case, since r(x) is a straight line with a slope of 12, the inverse function has a slope of 1/12, which is the reciprocal of the original slope. This is a characteristic of inverse functions for linear equations, where the slopes are reciprocals of each other. Also, the y-intercept of the original function was -11, and in the inverse function, this becomes part of the numerator, shifted by the reciprocal scaling. This maintains the symmetry required for inverse functions about the line y = x.Thinking about the graphical interpretation, the original function r(x) = 12x - 11 is a straight line that increases quite steeply, while its inverse r⁻¹(x) = (x + 11)/12 is a straight line that increases much more gradually, reflecting the reciprocal relationship in their slopes. If I were to graph both functions, they should be symmetrical with respect to the line y = x, which is a good way to visualize why they are inverses of each other.It's also worth noting that not all functions have inverses. For a function to have an inverse, it must be bijective, meaning it's both injective (one-to-one) and surjective (onto). In the case of linear functions with non-zero slopes, they are bijective over the real numbers, so they do have inverses. If the original function had been something like a quadratic, which isn't one-to-one over all real numbers, we would have had to restrict its domain to find an inverse. However, since we're dealing with linear functions here, we don't have to worry about that.In conclusion, by carefully following the steps of function composition and inverse function derivation, I was able to find that the inverse of r(x) = 12x - 11 is r⁻¹(x) = (x + 11)/12. The verification step using sample inputs confirmed that the inverse function behaves as expected, reinforcing the correctness of the solution.

answer：First, I need to determine the total weight of tomatoes Tommy has. He has 3 crates, and each crate can hold 20 kilograms of tomatoes. Next, I'll calculate the total weight by multiplying the number of crates by the capacity of each crate:3 crates * 20 kg/crate = 60 kg of tomatoes.However, 3 kilograms of the tomatoes were rotten, so Tommy can't sell those. I'll subtract the rotten tomatoes from the total weight:60 kg - 3 kg = 57 kg of sellable tomatoes.Tommy plans to sell the tomatoes for 6 per kilogram. To find out how much money he will make from selling the tomatoes, I'll multiply the sellable weight by the selling price:57 kg * 6/kg = 342.Tommy initially spent 330 to buy the crates. To calculate his profit, I'll subtract the total cost from the total revenue:342 (revenue) - 330 (cost) = 12 profit.Therefore, Tommy's profit from selling the tomatoes is 12.

answer：Okay, so I have this multiple-choice question here with four statements labeled A, B, C, and D. I need to figure out which one is correct. Let me go through each statement one by one and think about them carefully.Starting with **Statement A**: "The diameter perpendicular to the chord bisects the chord." Hmm, I remember something about diameters and chords in circles. I think if a diameter is perpendicular to a chord, it should indeed bisect the chord. That means it cuts the chord exactly in half. I think this is a standard property of circles. So, I believe Statement A is correct.Moving on to **Statement B**: "A line passing through one end of a radius and perpendicular to the radius is a tangent to the circle." Okay, so if I imagine a circle with a radius, and at the endpoint of that radius, there's a line that's perpendicular to the radius. I recall that a tangent to a circle is a line that touches the circle at exactly one point and is perpendicular to the radius at that point. So, if the line is passing through the endpoint of the radius and is perpendicular to it, that should make it a tangent. Therefore, Statement B seems correct too.Next is **Statement C**: "Three points in a plane determine a circle." I think this is related to the concept of a circle passing through three points. I believe that if you have three non-collinear points (meaning they don't all lie on the same straight line), there is exactly one circle that passes through all three. However, if the three points are collinear, they don't determine a circle because you can't have a circle passing through three points on a straight line. The statement doesn't specify whether the points are non-collinear, so I'm a bit unsure. Maybe it's assuming they are non-collinear? I think in general, three points can determine a circle, but only if they are not all on the same line. Since the statement doesn't specify, I'm not entirely sure if it's always true. I'll keep this in mind and move on.Finally, **Statement D**: "The distance from the circumcenter of a triangle to each of its vertices is equal." I remember that the circumcenter is the center of the circumscribed circle around a triangle, and by definition, it is equidistant from all three vertices of the triangle. This distance is called the circumradius. So, yes, the distance from the circumcenter to each vertex is equal. Therefore, Statement D is correct.Wait, so Statements A, B, and D all seem correct based on my understanding. But the question asks which of the following statements is correct, implying only one answer. Maybe I made a mistake somewhere. Let me double-check.For **Statement A**, I'm pretty confident that a diameter perpendicular to a chord does bisect the chord. That's a standard theorem in geometry.For **Statement B**, I think it's also correct because a tangent is defined as a line perpendicular to the radius at the point of contact. So, if a line passes through the endpoint of a radius and is perpendicular to it, it should be a tangent.For **Statement C**, I'm still a bit uncertain. The statement says "three points in a plane determine a circle." If the three points are non-collinear, then yes, they determine a unique circle. But if they are collinear, they don't determine a circle. Since the statement doesn't specify, it's a bit ambiguous. Maybe in general, three points can determine a circle, but only under certain conditions. So, perhaps Statement C is not always correct.For **Statement D**, I'm certain it's correct because the circumcenter is defined as the point equidistant from all three vertices of the triangle.So, if I have to choose the most correct statement, I think Statement D is definitely correct. Statements A and B are also correct, but maybe the question expects only one answer. Alternatively, perhaps there's a nuance I'm missing in Statements A or B.Let me think about Statement B again. It says, "A line passing through one end of a radius and perpendicular to the radius is a tangent to the circle." I think this is correct because, by definition, a tangent is perpendicular to the radius at the point of contact. So, if the line passes through the endpoint of the radius and is perpendicular to it, it must be a tangent.Wait, but does it have to pass through the endpoint? If it's passing through the endpoint and is perpendicular, then yes, it's a tangent. If it's passing through some other point, it wouldn't necessarily be a tangent. So, Statement B is correct.Similarly, Statement A is correct because a diameter perpendicular to a chord will bisect the chord.Statement C is tricky because it depends on whether the three points are collinear or not. If they are non-collinear, then yes, they determine a circle. If they are collinear, they don't. Since the statement doesn't specify, it's not universally true. So, Statement C is not always correct.Therefore, Statements A, B, and D are correct, while Statement C is not necessarily correct. But the question asks which of the following statements is correct, and it's possible that multiple are correct. However, in the original problem, it might be expecting only one answer. Maybe I need to check the definitions again.Alternatively, perhaps Statement C is considered correct because in general, three points can determine a circle, assuming they are non-collinear. Maybe the question assumes that the three points are non-collinear. In that case, Statement C would be correct.But I'm not entirely sure. Given that, I think the safest choice is Statement D because it's a well-defined property of the circumcenter. Statements A and B are also correct, but if I have to choose one, I'll go with D.

answer：First, I need to understand the relationship between the radius, height, and volume of a cylinder. The volume of a cylinder is given by the formula ( V = pi r^2 h ), where ( r ) is the radius and ( h ) is the height.The problem states that the radius is increased by some factor, let's call it ( x ), and the height is doubled. This results in the new volume being 18 times the original volume. So, I'll set up the equation for the new volume:[V_{text{new}} = pi (x r)^2 (2 h)]Simplifying this, I get:[V_{text{new}} = 2 x^2 pi r^2 h]According to the problem, this new volume is 18 times the original volume:[2 x^2 pi r^2 h = 18 pi r^2 h]I can cancel out the common terms ( pi r^2 h ) from both sides:[2 x^2 = 18]Solving for ( x^2 ):[x^2 = frac{18}{2} = 9]Taking the square root of both sides gives:[x = sqrt{9} = 3]Therefore, the new radius is 3 times larger than the original radius.