💡First, calculate the ratio of royalties to sales for the first 20 million in sales:Royalties for first 20 million = 6 millionSales for first 20 million = 20 millionRatio for first 20 million = Royalties / Sales = 6 million / 20 million = 0.3 or 30%Next, calculate the ratio of royalties to sales for the next 108 million in sales:Royalties for next 108 million = 9 millionSales for next 108 million = 108 millionRatio for next 108 million = Royalties / Sales = 9 million / 108 million ≈ 0.0833 or 8.33%Now, calculate the decrease in the ratio of royalties to sales:Decrease in ratio = Ratio for first 20 million - Ratio for next 108 millionDecrease in ratio = 30% - 8.33% = 21.67%Therefore, the ratio of royalties to sales decreased by approximately 21.67%.

💡First, I need to simplify the given complex number ( z = frac{5}{2i - 1} ). To do this, I'll multiply both the numerator and the denominator by the complex conjugate of the denominator, which is (-2i - 1), to eliminate the imaginary unit from the denominator.[z = frac{5}{2i - 1} times frac{-2i - 1}{-2i - 1} = frac{5(-2i - 1)}{(2i - 1)(-2i - 1)}]Next, I'll simplify the denominator by expanding the product:[(2i - 1)(-2i - 1) = -4i^2 - 2i + 2i + 1 = 4 + 1 = 5]Now, I'll simplify the numerator:[5(-2i - 1) = -10i - 5]Putting it all together:[z = frac{-10i - 5}{5} = -2i - 1]The complex conjugate of ( z ), denoted as ( overline{z} ), is obtained by changing the sign of the imaginary part:[overline{z} = -1 + 2i]In the complex plane, the real part is (-1) and the imaginary part is (2). This places the point in the second quadrant, where the real part is negative and the imaginary part is positive.

💡Okay, so I have this problem: Compute (cos left( arcsin frac{8}{17} right)). Hmm, that looks a bit tricky at first glance, but maybe I can break it down step by step. Let me think about what each part means.First, (arcsin frac{8}{17}) is the angle whose sine is (frac{8}{17}). So, if I let (theta = arcsin frac{8}{17}), then (sin theta = frac{8}{17}). Got that part. Now, I need to find (cos theta). So, essentially, I need to find the cosine of an angle whose sine is (frac{8}{17}).I remember there's a Pythagorean identity that relates sine and cosine: (sin^2 theta + cos^2 theta = 1). Maybe I can use that to find (cos theta). Let me write that down:[sin^2 theta + cos^2 theta = 1]Since I know (sin theta = frac{8}{17}), I can substitute that in:[left(frac{8}{17}right)^2 + cos^2 theta = 1]Calculating (left(frac{8}{17}right)^2):[frac{64}{289} + cos^2 theta = 1]Now, subtract (frac{64}{289}) from both sides to solve for (cos^2 theta):[cos^2 theta = 1 - frac{64}{289}]Calculating (1 - frac{64}{289}):First, express 1 as (frac{289}{289}):[cos^2 theta = frac{289}{289} - frac{64}{289} = frac{225}{289}]So, (cos^2 theta = frac{225}{289}). To find (cos theta), I take the square root of both sides:[cos theta = sqrt{frac{225}{289}} = frac{sqrt{225}}{sqrt{289}} = frac{15}{17}]Wait a second, but cosine can be positive or negative, right? So, do I need to consider the sign here? Hmm, let's think about the angle (theta). Since (theta = arcsin frac{8}{17}), and the arcsine function returns angles in the range (-frac{pi}{2}) to (frac{pi}{2}), which is the fourth and first quadrants. In these quadrants, cosine is positive. So, (cos theta) must be positive. Therefore, (cos theta = frac{15}{17}).So, putting it all together, (cos left( arcsin frac{8}{17} right) = frac{15}{17}).Just to make sure I didn't make any mistakes, let me visualize this. If (sin theta = frac{8}{17}), then in a right triangle, the opposite side is 8, and the hypotenuse is 17. Using the Pythagorean theorem, the adjacent side should be (sqrt{17^2 - 8^2} = sqrt{289 - 64} = sqrt{225} = 15). Therefore, (cos theta = frac{15}{17}), which matches what I found earlier.I think that makes sense. I don't see any errors in my calculations, so I'm confident that the answer is (frac{15}{17}).

💡First, calculate the total number of ways to choose two distinct integers from 1 to 15, which is (binom{15}{2} = 105).Next, determine the number of ways to choose two odd integers. There are 8 odd integers between 1 and 15, so the number of ways to choose two odd integers is (binom{8}{2} = 28).The number of ways to choose at least one even integer is the total number of ways minus the number of ways to choose two odd integers: (105 - 28 = 77).Finally, the probability that the product of the two chosen integers is even is the number of favorable outcomes divided by the total number of outcomes: (dfrac{77}{105}).

💡First, I need to identify all the two-digit multiples of 5. These numbers start from 10 and go up to 95, increasing by 5 each time.Next, I'll list out these multiples to ensure I have all of them:10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95.Now, I'll count how many numbers there are in this list. Starting from 10 and ending at 95, with each number increasing by 5, there are a total of 18 numbers.To find the arithmetic mean, I need to calculate the sum of all these numbers and then divide by the total count. Calculating the sum:10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50 + 55 + 60 + 65 + 70 + 75 + 80 + 85 + 90 + 95 = 945.Finally, I'll divide the sum by the number of terms to find the mean:945 ÷ 18 = 52.5.Therefore, the arithmetic mean of all positive two-digit multiples of 5 is 52.5.

💡Okay, so I have this question here: "x > 1" is a ( ). The options are (A) sufficient but not necessary condition for "x geqslant 1", (B) necessary but not sufficient condition for "x geqslant 1", (C) both necessary and sufficient condition for "x geqslant 1", and (D) neither necessary nor sufficient condition for "x geqslant 1". I need to figure out which one is correct.First, let me recall what necessary and sufficient conditions mean. A necessary condition is something that must be true for another statement to be true. On the other hand, a sufficient condition is something that, if it's true, guarantees that another statement is true. If a condition is both necessary and sufficient, then it means that the two statements imply each other.So, in this case, the main statement is "x geqslant 1". The condition given is "x > 1". I need to see if "x > 1" is a necessary condition, a sufficient condition, both, or neither for "x geqslant 1".Let me think about sufficiency first. If "x > 1", does that guarantee that "x geqslant 1"? Well, yes, because if x is greater than 1, it's certainly greater than or equal to 1. So, "x > 1" is a sufficient condition for "x geqslant 1".Now, is it a necessary condition? That would mean that if "x geqslant 1", then "x > 1" must be true. But wait, if x is equal to 1, then "x geqslant 1" is true, but "x > 1" is false. So, "x > 1" isn't a necessary condition because there are cases where "x geqslant 1" is true without "x > 1" being true.Therefore, "x > 1" is sufficient but not necessary for "x geqslant 1". So, the answer should be option (A).Let me double-check to make sure I didn't make a mistake. If I consider x = 1, then "x geqslant 1" is true, but "x > 1" is false. This confirms that "x > 1" isn't necessary. And since any x > 1 will definitely satisfy x ≥ 1, it's sufficient. Yeah, I think I'm confident with that.**Final Answer**boxed{A}