answer：First, calculate the total area of the circular spinner using the formula for the area of a circle, ( A = pi r^2 ). With a radius of 10 cm, the total area is:[A = pi times 10^2 = 100pi text{ cm}^2]The probability of winning is given as ( frac{3}{7} ), which represents the ratio of the area of the WIN sector to the total area of the spinner. Let ( A_{text{win}} ) be the area of the WIN sector. Set up the equation:[frac{3}{7} = frac{A_{text{win}}}{100pi}]To find ( A_{text{win}} ), multiply both sides of the equation by ( 100pi ):[A_{text{win}} = frac{3}{7} times 100pi = frac{300pi}{7}]Therefore, the area of the WIN sector is ( frac{300pi}{7} ) square centimeters.

answer：Okay, so I have this problem about an arithmetic sequence. Let me try to understand what it's asking. First, it says that the arithmetic sequence {a_n} has a common difference of 5. That means each term is 5 more than the previous one. So, if the first term is a_1, then the second term a_2 is a_1 + 5, the third term a_3 is a_1 + 10, and so on. Got that.Next, it mentions that the sum of the first n terms is S_n. I remember that the formula for the sum of the first n terms of an arithmetic sequence is S_n = n/2 * (2a_1 + (n-1)d), where d is the common difference. So, in this case, d is 5, so the formula becomes S_n = n/2 * (2a_1 + 5(n-1)). I'll keep that in mind.Now, the problem states that a_1, a_2, a_5 form a geometric sequence. Hmm, okay. A geometric sequence is one where each term is a multiple of the previous term by a common ratio. So, if a_1, a_2, a_5 are in geometric sequence, then the ratio between a_2 and a_1 should be the same as the ratio between a_5 and a_2. That gives me the equation a_2 / a_1 = a_5 / a_2.Let me write that down:a_2 / a_1 = a_5 / a_2Cross-multiplying, I get:(a_2)^2 = a_1 * a_5Since this is an arithmetic sequence, I can express a_2 and a_5 in terms of a_1 and the common difference d=5.a_2 = a_1 + 5a_5 = a_1 + 4*5 = a_1 + 20So, substituting these into the equation:(a_1 + 5)^2 = a_1 * (a_1 + 20)Let me expand the left side:(a_1 + 5)^2 = a_1^2 + 10a_1 + 25So, the equation becomes:a_1^2 + 10a_1 + 25 = a_1^2 + 20a_1Hmm, let's subtract a_1^2 from both sides:10a_1 + 25 = 20a_1Now, subtract 10a_1 from both sides:25 = 10a_1Divide both sides by 10:a_1 = 25 / 10 = 2.5So, the first term a_1 is 2.5. That seems a bit unusual because it's not an integer, but arithmetic sequences can have any real numbers as terms, so that's fine.Now, I need to find S_6, the sum of the first 6 terms. Using the formula I wrote earlier:S_n = n/2 * (2a_1 + (n-1)d)Plugging in n=6, a_1=2.5, and d=5:S_6 = 6/2 * (2*2.5 + (6-1)*5)Simplify step by step:First, 6/2 is 3.Then, inside the parentheses:2*2.5 = 5(6-1)*5 = 5*5 = 25So, adding those together: 5 + 25 = 30Now, multiply by 3:3 * 30 = 90So, S_6 is 90. Let me check the options given:A: 80B: 85C: 90D: 95So, 90 is option C. That should be the answer.Wait, let me double-check my calculations to make sure I didn't make a mistake.First, a_1 = 2.5, d=5.So, the terms are:a_1 = 2.5a_2 = 2.5 + 5 = 7.5a_3 = 7.5 + 5 = 12.5a_4 = 12.5 + 5 = 17.5a_5 = 17.5 + 5 = 22.5a_6 = 22.5 + 5 = 27.5Now, let's compute the sum S_6:2.5 + 7.5 + 12.5 + 17.5 + 22.5 + 27.5Let me add them step by step:2.5 + 7.5 = 1010 + 12.5 = 22.522.5 + 17.5 = 4040 + 22.5 = 62.562.5 + 27.5 = 90Yes, that adds up to 90. So, my calculation seems correct.Just to be thorough, let me check if a_1, a_2, a_5 form a geometric sequence.a_1 = 2.5a_2 = 7.5a_5 = 22.5So, the ratios:a_2 / a_1 = 7.5 / 2.5 = 3a_5 / a_2 = 22.5 / 7.5 = 3Yes, both ratios are 3, so it is indeed a geometric sequence with common ratio 3.Therefore, everything checks out. The sum S_6 is 90, which is option C.**Final Answer**boxed{C}

answer：First, I recognize that the problem involves finding the hypotenuse of a right triangle with legs measuring 15 inches and 36 inches.To solve this, I'll use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (( h )) is equal to the sum of the squares of the other two sides (( a ) and ( b )).Applying the theorem:[h^2 = 15^2 + 36^2]Calculating the squares:[h^2 = 225 + 1296 = 1521]To find ( h ), I'll take the square root of 1521:[h = sqrt{1521} = 39]Therefore, the length of the hypotenuse is 39 inches.

answer：Okay, so I have this problem where I'm given a set of numbers: {4, 7, 11, 12}. I need to add a fifth element, n, which isn't equal to any of the existing numbers. The condition is that the median of the new set should be equal to its mean. I have to find all possible values of n and then sum them up. The options given are A) 15, B) 21, C) 22, D) 24, E) 30.First, let me recall what median and mean are. The median is the middle number when the numbers are arranged in order, and the mean is the average, which is the sum of the numbers divided by how many there are.Since we're adding a fifth element, the set will have five numbers. The median of five numbers is the third one when they're sorted. So, depending on where n falls in the order, the median could be different.Let me think about the original set: {4, 7, 11, 12}. If I add n, the new set will be five numbers. Let's consider different scenarios based on where n is placed when the set is sorted.Case 1: n is less than or equal to 7. Then, the sorted set would be {n, 4, 7, 11, 12} or {4, n, 7, 11, 12} depending on where n is. Wait, actually, if n is less than 4, it would be first. If it's between 4 and 7, it would be second. But in both cases, the median would still be 7 because the third number is 7.Case 2: n is between 7 and 11. Then, the sorted set would be {4, 7, n, 11, 12}, so the median is n.Case 3: n is greater than or equal to 11. Then, the sorted set would be {4, 7, 11, 12, n} or {4, 7, 11, n, 12} if n is between 11 and 12. But regardless, the median would be 11.So, there are three cases: median is 7, median is n, or median is 11. In each case, the median equals the mean, so I can set up equations for each case.Let me calculate the mean in each case.First, the sum of the original set is 4 + 7 + 11 + 12 = 34. Adding n, the total sum becomes 34 + n. The mean is (34 + n)/5.Case 1: Median is 7. So, mean = 7.Set up the equation: (34 + n)/5 = 7.Multiply both sides by 5: 34 + n = 35.Subtract 34: n = 1.So, n = 1 is a possible value. But wait, n has to be different from the existing numbers, which are 4, 7, 11, 12. 1 isn't among them, so that's okay.Case 2: Median is n. So, mean = n.Set up the equation: (34 + n)/5 = n.Multiply both sides by 5: 34 + n = 5n.Subtract n: 34 = 4n.Divide by 4: n = 34/4 = 8.5.Hmm, 8.5 is not an integer, but the problem doesn't specify that n has to be an integer. Wait, actually, looking back, the original set has integers, and n is just a number not equal to any of them. So, 8.5 is a possible value. But wait, let me check if n is between 7 and 11, which it is (8.5 is between 7 and 11), so that's valid.But wait, the problem says "the set {4,7,11,12} is augmented by a fifth element n, not equal to any of the other four." It doesn't specify that n has to be an integer, so 8.5 is acceptable. But in the answer choices, all are integers. So, maybe I need to reconsider.Wait, perhaps I made a mistake. Let me check the problem again. It says "the median of the resulting set is equal to its mean." It doesn't specify that n has to be an integer, so 8.5 is a valid value. But since the answer choices are all integers, maybe I need to see if 8.5 is among them or if I made a mistake.Wait, 8.5 isn't among the options, so maybe I need to reconsider. Alternatively, perhaps n must be an integer because the original set consists of integers, but the problem doesn't explicitly say that. Hmm.Alternatively, maybe I made a mistake in setting up the equation. Let me check.In Case 2, the median is n, so the mean is n. So, (34 + n)/5 = n. Multiply both sides by 5: 34 + n = 5n. Subtract n: 34 = 4n. So, n = 8.5. That seems correct.But since 8.5 isn't an option, maybe I need to consider that n must be an integer. Let me assume that n must be an integer. Then, 8.5 is not acceptable, so this case doesn't give a valid integer solution.So, only n = 1 and n = 21 are valid. Wait, where did 21 come from? Let me check Case 3.Case 3: Median is 11. So, mean = 11.Set up the equation: (34 + n)/5 = 11.Multiply both sides by 5: 34 + n = 55.Subtract 34: n = 21.So, n = 21 is another possible value. 21 isn't in the original set, so that's okay.So, the possible values of n are 1 and 21. Their sum is 1 + 21 = 22.Looking at the options, C) 22 is the answer.Wait, but earlier I thought about n = 8.5, but since it's not an integer and not among the options, I can ignore it. So, the sum is 22.I think that's it. So, the answer is C) 22.

answer：First, I need to simplify the given expression (frac{2+3i}{i}). To do this, I'll multiply both the numerator and the denominator by the complex conjugate of the denominator to eliminate the imaginary unit in the denominator.So, I'll multiply by (frac{-i}{-i}):[frac{2+3i}{i} times frac{-i}{-i} = frac{(2+3i)(-i)}{i times -i}]Next, I'll expand the numerator:[(2+3i)(-i) = 2 times -i + 3i times -i = -2i - 3i^2]Since (i^2 = -1), this simplifies to:[-2i - 3(-1) = -2i + 3]Now, the denominator is:[i times -i = -i^2 = -(-1) = 1]Putting it all together, the simplified expression is:[frac{-2i + 3}{1} = 3 - 2i]Comparing this with the standard form (a + bi), I can see that (a = 3) and (b = -2).Finally, to find (ab):[ab = 3 times (-2) = -6]

answer：First, I need to determine how much work each person can complete in one day. If B can finish the job in 6 days, then B's work rate is 1/6 of the job per day. Similarly, if A can complete the job in 10 days, A's work rate is 1/10 of the job per day.Next, I'll add their work rates together to find the combined work rate when they work together. Adding 1/6 and 1/10 requires finding a common denominator, which is 30. This gives me 5/30 + 3/30 = 8/30, which simplifies to 4/15 of the job per day.Finally, to find out how many days it will take for them to complete the job together, I'll take the reciprocal of the combined work rate. The reciprocal of 4/15 is 15/4, which equals 3.75 days.