💡Okay, so I have this function f(x) defined piecewise. Let me write it down to make sure I understand it correctly:f(x) = - (1/2)^{|x - a|} when x < a + 1- -|x + 1| - a when x ≥ a + 1And the problem says that the maximum value of f(x) does not exceed 1. I need to find the range of the real number a. The options are given as A, B, C, D, which I can check later.First, I should analyze each piece of the function separately to find their maximum values and then ensure that both pieces don't exceed 1. Then, I can find the constraints on a.Starting with the first piece: f(x) = (1/2)^{|x - a|} for x < a + 1.I know that (1/2)^{k} is a decreasing function when k is positive. So, the exponent |x - a| is always non-negative. Therefore, the function (1/2)^{|x - a|} will have its maximum value when |x - a| is minimized, which is when x = a. At x = a, |x - a| = 0, so f(a) = (1/2)^0 = 1. So, the maximum value of the first piece is 1, which is exactly the upper limit given in the problem. So, that's good.Now, I need to check if this maximum is actually attainable. Since x < a + 1, when x = a, which is less than a + 1, so yes, x = a is within the domain of the first piece. So, the maximum value of 1 is achieved at x = a.Now, moving on to the second piece: f(x) = -|x + 1| - a for x ≥ a + 1.This is a linear function in terms of |x + 1|, but since it's negative, it's a V-shaped graph opening downward. The maximum value will occur at the vertex of this V-shape.The expression |x + 1| is minimized when x = -1, so the maximum of -|x + 1| - a occurs at x = -1. However, we need to check if x = -1 is within the domain of this piece, which is x ≥ a + 1.So, if -1 ≥ a + 1, then x = -1 is in the domain, and the maximum value is -| -1 + 1| - a = -|0| - a = -a. So, in this case, the maximum value is -a.But if -1 < a + 1, then x = -1 is not in the domain of this piece. In that case, the function f(x) = -|x + 1| - a is decreasing for x ≥ a + 1 because as x increases, |x + 1| increases, so -|x + 1| decreases. Therefore, the maximum value in this case would occur at x = a + 1.So, let's break it down:Case 1: -1 ≥ a + 1This implies that a + 1 ≤ -1, so a ≤ -2.In this case, the maximum of the second piece is at x = -1, which is -a. So, we have f(-1) = -a.We need to ensure that this maximum does not exceed 1. So:-a ≤ 1Which implies:a ≥ -1But wait, in this case, a ≤ -2. So, we have a ≤ -2 and a ≥ -1. But these two inequalities can't be true at the same time because -2 is less than -1. So, there is no solution in this case. Therefore, when a ≤ -2, the maximum value of the second piece is -a, which would have to be ≤ 1, but since a ≤ -2, -a ≥ 2, which is greater than 1. Therefore, this case is not possible because it would violate the condition that the maximum does not exceed 1.Case 2: -1 < a + 1This implies that a + 1 > -1, so a > -2.In this case, the maximum of the second piece occurs at x = a + 1. Let's compute f(a + 1):f(a + 1) = -|a + 1 + 1| - a = -|a + 2| - aSo, we have f(a + 1) = -|a + 2| - aWe need this value to be ≤ 1.So, let's write the inequality:-|a + 2| - a ≤ 1Let me solve this inequality.First, let's rewrite it:-|a + 2| - a ≤ 1Multiply both sides by -1 (remembering to reverse the inequality sign):|a + 2| + a ≥ -1But I think it's better to handle the absolute value directly.Let me consider two subcases based on the value inside the absolute value.Subcase 2a: a + 2 ≥ 0, which implies a ≥ -2In this subcase, |a + 2| = a + 2So, the inequality becomes:-(a + 2) - a ≤ 1Simplify:- a - 2 - a ≤ 1Combine like terms:-2a - 2 ≤ 1Add 2 to both sides:-2a ≤ 3Divide both sides by -2 (inequality sign reverses):a ≥ -3/2But in this subcase, a ≥ -2. So, combining these, we have a ≥ -3/2.Subcase 2b: a + 2 < 0, which implies a < -2In this subcase, |a + 2| = -(a + 2) = -a - 2So, the inequality becomes:-(-a - 2) - a ≤ 1Simplify:a + 2 - a ≤ 1Which simplifies to:2 ≤ 1But 2 ≤ 1 is false. Therefore, there is no solution in this subcase.So, combining the results from Case 2:We have a > -2 (from Case 2) and a ≥ -3/2 (from Subcase 2a). Therefore, the range of a is a ≥ -3/2.But wait, in Case 1, when a ≤ -2, we saw that the maximum of the second piece would be -a, which would have to be ≤ 1, but since a ≤ -2, -a ≥ 2, which is greater than 1. Therefore, these a's are not allowed.In Case 2, when a > -2, the maximum of the second piece is at x = a + 1, and we found that a must be ≥ -3/2.So, combining everything, the allowable range for a is a ≥ -3/2.Therefore, the range of a is [-3/2, ∞).Looking back at the options:A: [-3/2, +∞)B: (-3/2, +∞)C: [-5/4, 0)D: [-3/2, -5/4)So, the correct answer is A.Wait, but let me double-check my reasoning because sometimes when dealing with inequalities, especially with absolute values, it's easy to make a mistake.In Case 2, when a > -2, we found that a must be ≥ -3/2. So, the lower bound is -3/2, and since a can be any real number greater than or equal to -3/2, the interval is [-3/2, ∞).Yes, that seems correct. So, the answer should be A.**Final Answer**boxed{A}

💡Okay, so Melinda has four boxes and 16 textbooks, four of which are physics books. She's going to pack these textbooks into the boxes randomly. The boxes can hold 4, 5, 3, and 4 textbooks respectively. I need to find the probability that all four physics textbooks end up in the same box. Hmm, sounds like a combinatorics problem.First, let me understand the total number of ways to distribute the textbooks into the boxes. Since the order in which she packs them matters, I think I need to use combinations here. The first box can hold 4 books, so the number of ways to choose 4 books out of 16 is C(16,4). Then, the second box holds 5 books, so from the remaining 12, it's C(12,5). The third box holds 3, so from the remaining 7, it's C(7,3). And the last box will have the remaining 4 books, which is just 1 way since all are chosen already. So the total number of ways is C(16,4) * C(12,5) * C(7,3) * 1.Let me compute that. C(16,4) is 1820, C(12,5) is 792, C(7,3) is 35. Multiplying these together: 1820 * 792 = let's see, 1820 * 700 is 1,274,000, and 1820 * 92 is 167,440. So total is 1,274,000 + 167,440 = 1,441,440. Then multiply by 35: 1,441,440 * 35. Hmm, 1,441,440 * 30 is 43,243,200 and 1,441,440 * 5 is 7,207,200. So total is 43,243,200 + 7,207,200 = 50,450,400. So the total number of ways is 50,450,400.Now, for the favorable cases where all four physics books are in the same box. There are three possible boxes where this can happen: the first box (which holds 4), the second box (which holds 5), and the fourth box (which also holds 4). The third box only holds 3, so it can't contain all four physics books.Let's consider each case:1. All four physics books in the first box: Since the first box holds exactly 4 books, all of them must be the physics books. So the number of ways is 1 (choosing all four physics books) multiplied by the number of ways to distribute the remaining 12 books into the other boxes. Wait, no, actually, since we're considering the specific case where all four physics books are in the first box, the rest of the distribution is fixed. So the number of favorable ways for this case is C(12,0) * C(12,5) * C(7,3) * C(4,4). But C(12,0) is 1, so it's just 1 * C(12,5) * C(7,3) * 1. Wait, but actually, once we've chosen the four physics books for the first box, the remaining 12 books are distributed into the other boxes. So the number of ways is C(12,5) * C(7,3) * C(4,4). Which is 792 * 35 * 1 = 27,720.2. All four physics books in the second box: The second box holds 5 books, so we need to choose 1 more book from the remaining 12 to go with the four physics books. So the number of ways is C(12,1) * C(11,4) * C(7,3) * C(4,4). Wait, no, let me think again. If all four physics books are in the second box, which holds 5, then we need to choose 1 more book from the 12 non-physics books. So the number of ways is C(12,1) for choosing the extra book, then the remaining 11 books are distributed into the other boxes. So the number of ways is C(12,1) * C(11,4) * C(7,3) * C(4,4). Wait, but actually, after choosing the 1 extra book for the second box, we have 11 books left. The first box needs 4, so C(11,4), then the third box needs 3, so C(7,3), and the last box gets the remaining 4. So the number of ways is C(12,1) * C(11,4) * C(7,3) * 1. Let's compute that: C(12,1) is 12, C(11,4) is 330, C(7,3) is 35. So 12 * 330 = 3,960, then 3,960 * 35 = 138,600.3. All four physics books in the fourth box: Similar to the first case, since the fourth box holds 4 books, all four must be physics books. So the number of ways is 1 * C(12,5) * C(7,3) * 1, which is the same as the first case: 792 * 35 = 27,720.So total favorable ways are 27,720 (first box) + 138,600 (second box) + 27,720 (fourth box) = 27,720 + 138,600 = 166,320 + 27,720 = 194,040.Wait, but hold on, I think I made a mistake here. Because when I calculated the favorable ways for the second box, I considered choosing 1 extra book, but actually, the total number of ways should be calculated differently. Let me re-examine that.When all four physics books are in the second box, which holds 5 books, we need to choose 1 more book from the 12 non-physics books. So the number of ways to choose the books for the second box is C(12,1). Then, the remaining 11 books are distributed into the other boxes. The first box needs 4, so C(11,4), then the third box needs 3, so C(7,3), and the fourth box gets the remaining 4. So the number of ways is C(12,1) * C(11,4) * C(7,3). Let's compute that again: 12 * 330 * 35. 12 * 330 is 3,960, and 3,960 * 35 is indeed 138,600. So that part seems correct.Similarly, for the first and fourth boxes, since they hold exactly 4 books, the number of ways is just the number of ways to distribute the remaining books, which is C(12,5) * C(7,3). So 792 * 35 = 27,720 each. So total favorable is 27,720 + 138,600 + 27,720 = 194,040.Wait, but the total number of ways is 50,450,400, so the probability is 194,040 / 50,450,400. Let me simplify that fraction.First, let's see if both numerator and denominator can be divided by 12. 194,040 ÷ 12 = 16,170. 50,450,400 ÷ 12 = 4,204,200. So now we have 16,170 / 4,204,200. Let's divide numerator and denominator by 10: 1,617 / 420,420. Hmm, can we divide by 3? 1,617 ÷ 3 = 539. 420,420 ÷ 3 = 140,140. So now 539 / 140,140. Let's see if 539 divides into 140,140. 140,140 ÷ 539. Let me compute 539 * 260 = 140,140. Yes, because 539 * 200 = 107,800, and 539 * 60 = 32,340, so total 107,800 + 32,340 = 140,140. So 539 / 140,140 = 1 / 260. Therefore, the probability is 1 / 260.Wait, but earlier I thought the answer was 3/286, but now I'm getting 1/260. Hmm, maybe I made a mistake in my calculations. Let me double-check.Wait, when I calculated the favorable ways, I added 27,720 + 138,600 + 27,720 = 194,040. Then total ways is 50,450,400. So 194,040 / 50,450,400. Let me compute this division.Divide numerator and denominator by 12: 194,040 ÷ 12 = 16,170; 50,450,400 ÷ 12 = 4,204,200.16,170 / 4,204,200. Divide numerator and denominator by 10: 1,617 / 420,420.Divide numerator and denominator by 3: 539 / 140,140.Divide numerator and denominator by 539: 1 / 260.So the probability is 1/260. But wait, in the initial problem, the user mentioned the answer was 3/286, but I think that might be incorrect. Let me see.Alternatively, maybe I should approach this problem differently. Instead of calculating the number of ways, perhaps I can think in terms of probabilities step by step.The probability that all four physics books are in the same box can be calculated by considering each box and the probability that all four are in that box, then summing those probabilities.So, the probability that all four are in the first box: The first box holds 4 books. The number of ways to choose 4 books out of 16 is C(16,4). The number of ways to choose all four physics books is 1. So the probability is 1 / C(16,4) = 1 / 1820.Similarly, the probability that all four are in the second box: The second box holds 5 books. The number of ways to choose 5 books out of 16 is C(16,5). The number of ways to choose all four physics books plus one more book is C(12,1). So the probability is C(12,1) / C(16,5) = 12 / 4368.The probability that all four are in the fourth box: Similar to the first box, it's 1 / C(16,4) = 1 / 1820.So total probability is 1/1820 + 12/4368 + 1/1820.Let me compute these fractions:First, 1/1820 is approximately 0.000549.12/4368: Let's simplify 12/4368. Divide numerator and denominator by 12: 1 / 364 ≈ 0.002747.So total probability is approximately 0.000549 + 0.002747 + 0.000549 ≈ 0.003845.Now, let's compute the exact value:1/1820 + 12/4368 + 1/1820.First, note that 4368 = 16 choose 5 = 4368.Let me find a common denominator. The denominators are 1820 and 4368. Let's see, 1820 factors into 2^2 * 5 * 7 * 13. 4368 factors into 2^4 * 3 * 7 * 13. So the least common multiple (LCM) would be 2^4 * 3 * 5 * 7 * 13 = 16 * 3 * 5 * 7 * 13.Compute that: 16 * 3 = 48, 48 * 5 = 240, 240 * 7 = 1,680, 1,680 * 13 = 21,840.So the LCM is 21,840.Convert each fraction:1/1820 = (12)/21,84012/4368 = (60)/21,8401/1820 = (12)/21,840So total is 12 + 60 + 12 = 84 / 21,840.Simplify 84/21,840: Divide numerator and denominator by 84: 1 / 260.So the probability is 1/260, which is approximately 0.003846, matching the earlier approximation.Wait, but earlier I thought the answer was 3/286, which is approximately 0.0105, which is higher than 0.003846. So which one is correct?I think my second approach is more accurate because it directly calculates the probability by considering each box and summing the probabilities. The first approach, where I calculated the number of favorable ways as 194,040 and total ways as 50,450,400, leading to 1/260, seems consistent with the second approach.So perhaps the initial answer of 3/286 was incorrect, and the correct probability is 1/260.Wait, but let me check the total number of ways again. The total number of ways to distribute the books is indeed 16! / (4!5!3!4!) = 50,450,400. The favorable ways are 194,040, so 194,040 / 50,450,400 = 1 / 260.Alternatively, perhaps I made a mistake in the favorable ways calculation. Let me think again.When all four physics books are in the first box: The number of ways is C(12,0) * C(12,5) * C(7,3) * C(4,4) = 1 * 792 * 35 * 1 = 27,720.When all four are in the second box: C(12,1) * C(11,4) * C(7,3) * C(4,4) = 12 * 330 * 35 * 1 = 138,600.When all four are in the fourth box: Same as the first box, 27,720.Total favorable: 27,720 + 138,600 + 27,720 = 194,040.So 194,040 / 50,450,400 = 1 / 260.Therefore, the probability is 1/260, so m=1, n=260, and m+n=261.But wait, the initial problem statement said to express the probability as m/n where m and n are coprime, and then find m+n. So if the probability is 1/260, then m+n is 261.However, I recall that sometimes in such problems, the answer might be expressed differently, perhaps considering the boxes as distinguishable or not. Wait, in this case, the boxes are distinguishable because they have different capacities. So the approach I took is correct.Alternatively, maybe I should consider the probability step by step, considering the placement of each physics book.The first physics book can go into any box. The second physics book must go into the same box as the first. The probability that the second book goes into the same box as the first depends on the capacity of that box.Wait, this might be a more complicated approach, but let's try.The first physics book can go into any of the four boxes. Let's say it goes into box 1, which holds 4 books. The probability that the second physics book also goes into box 1 is 3/15, since there are 3 remaining spots in box 1 out of 15 remaining books.Similarly, if the first book is in box 2 (capacity 5), the probability that the second book is also in box 2 is 4/15.If the first book is in box 3 (capacity 3), the probability that the second book is also in box 3 is 2/15.If the first book is in box 4 (capacity 4), the probability that the second book is also in box 4 is 3/15.So the probability that the second book is in the same box as the first is:(Probability first book in box1 * 3/15) + (Probability first book in box2 * 4/15) + (Probability first book in box3 * 2/15) + (Probability first book in box4 * 3/15).Since the first book is equally likely to go into any box, but the boxes have different capacities, the probability that the first book is in a particular box is proportional to the box's capacity.Wait, no, actually, the first book is equally likely to go into any of the 16 positions, but the boxes have different numbers of positions. So the probability that the first book is in box1 is 4/16, box2 is 5/16, box3 is 3/16, box4 is 4/16.So the probability that the second book is in the same box as the first is:(4/16)*(3/15) + (5/16)*(4/15) + (3/16)*(2/15) + (4/16)*(3/15).Compute each term:(4/16)*(3/15) = (1/4)*(1/5) = 1/20.(5/16)*(4/15) = (5/16)*(4/15) = (1/4)*(1/3) = 1/12.(3/16)*(2/15) = (1/8)*(2/15) = 1/60.(4/16)*(3/15) = same as the first term, 1/20.So total probability is 1/20 + 1/12 + 1/60 + 1/20.Convert to 60 denominator:1/20 = 3/60, 1/12 = 5/60, 1/60 = 1/60, 1/20 = 3/60.Total: 3 + 5 + 1 + 3 = 12/60 = 1/5.Wait, that's the probability that the second book is in the same box as the first. But we need all four books to be in the same box. So this is just the probability for two books. To extend this to four books, we need to consider the probabilities step by step.After placing the first two books in the same box, the third book must also go into that box, and then the fourth.So the probability that all four are in the same box is the product of the probabilities at each step.But this approach might get complicated because the probabilities depend on which box the first book was placed in and how many spots are left.Alternatively, perhaps it's better to stick with the combinatorial approach, which gave us 1/260.Wait, but earlier when I calculated using the combinatorial method, I got 1/260, but when I tried the step-by-step probability approach, I got a different result. That suggests I might have made a mistake in one of the approaches.Wait, in the step-by-step approach, I only calculated the probability that the second book is in the same box as the first, which is 1/5. But to get all four books in the same box, we need the third and fourth books to also be in that box.So let's try to compute the probability step by step.Probability that all four physics books are in the same box:First, choose a box for all four physics books. The probability that all four are in box1: The number of ways to choose 4 books for box1, which includes all four physics books. So the probability is C(12,0)/C(16,4) = 1/1820.Similarly, probability all four are in box2: C(12,1)/C(16,5) = 12/4368.Probability all four are in box4: same as box1, 1/1820.So total probability is 1/1820 + 12/4368 + 1/1820 = 2/1820 + 12/4368.Simplify:2/1820 = 1/910.12/4368 = 1/364.So total probability is 1/910 + 1/364.Find a common denominator: 910 and 364. Let's see, 910 = 13*70, 364 = 4*91 = 4*13*7. So LCM is 4*13*7*5 = 1820.Convert fractions:1/910 = 2/1820.1/364 = 5/1820.So total is 2 + 5 = 7/1820 = 1/260.Ah, so that matches the combinatorial approach. So the probability is indeed 1/260.Therefore, the initial answer of 3/286 was incorrect, and the correct probability is 1/260, so m=1, n=260, and m+n=261.Wait, but I think I might have made a mistake in the step-by-step probability approach earlier when I tried to compute it as 1/5. That was only for two books, not four. So the correct approach is to sum the probabilities for each box, which gives 1/260.So, to summarize, the probability that all four physics textbooks end up in the same box is 1/260, so m+n=1+260=261.

💡Okay, so I have this problem about a torus sitting on a table, and I need to find the radius of the largest spherical ball that can be placed on top of the center of the torus while still touching the horizontal plane. Hmm, let me try to visualize this. First, the torus is formed by revolving a circle in the xz-plane around the z-axis. The circle has a center at (4.5, 0, 2) and a radius of 2. So, when this circle is revolved around the z-axis, it creates a torus. The inner radius of the torus is 3, and the outer radius is 6. That makes sense because the center of the generating circle is at 4.5 units from the z-axis, and the radius of the generating circle is 2, so the inner radius would be 4.5 - 2 = 2.5, but wait, the problem says the inner radius is 3. Hmm, maybe I need to double-check that.Wait, actually, the inner radius of the torus is the distance from the z-axis to the inner edge of the torus, which is the distance from the z-axis to the center of the generating circle minus the radius of the generating circle. So, the center is at 4.5, and the radius is 2, so 4.5 - 2 = 2.5. But the problem states the inner radius is 3. That seems contradictory. Maybe I'm misunderstanding something.Let me read the problem again: "A torus sits flat on a table with an inner radius of 3 and an outer radius of 6. The torus is formed by revolving a circle in the xz-plane centered at (4.5, 0, 2) with radius 2 about the z-axis." Hmm, so according to this, the inner radius is 3, but according to the generating circle, it should be 2.5. Maybe the problem is using a different definition of inner radius? Or perhaps it's considering the inner radius as the distance from the center of the torus to the inner edge, which would be different.Wait, no, the inner radius of a torus is typically the distance from the z-axis to the inner edge, which is the distance from the z-axis to the center of the generating circle minus the radius of the generating circle. So, if the center is at 4.5, and the radius is 2, then the inner radius should be 4.5 - 2 = 2.5, and the outer radius should be 4.5 + 2 = 6.5. But the problem says inner radius is 3 and outer radius is 6. That doesn't match. Maybe I'm missing something here.Wait, perhaps the generating circle is not in the xz-plane but in a different plane? No, the problem says it's in the xz-plane. Hmm, maybe the center is at (4.5, 0, 2), so the distance from the z-axis is 4.5, and the radius is 2, so the inner radius is 4.5 - 2 = 2.5, and the outer radius is 4.5 + 2 = 6.5. But the problem says inner radius is 3 and outer radius is 6. That suggests that the generating circle has a different radius or center.Wait, maybe the generating circle is not centered at (4.5, 0, 2). Let me check the problem again: "The torus is formed by revolving a circle in the xz-plane centered at (4.5, 0, 2) with radius 2 about the z-axis." So, that's correct. So, the inner radius should be 4.5 - 2 = 2.5, but the problem says 3. Maybe the problem is using a different definition or there's a typo. Hmm, perhaps I should proceed with the given information and see where it leads.So, regardless of that, I need to find the radius of the largest spherical ball that can be placed atop the center of the torus such that it still touches the horizontal plane. So, the sphere is sitting on top of the torus and also touching the table. Therefore, the center of the sphere must be at a height equal to its radius above the table. So, if the radius is r, the center is at (0, 0, r).Now, the sphere must also touch the torus. So, the distance between the center of the sphere and the center of the generating circle of the torus must be equal to the sum of their radii, right? Because when two spheres touch each other externally, the distance between their centers is equal to the sum of their radii.Wait, but in this case, the generating circle is part of the torus, which is a surface of revolution. So, the sphere must touch the torus at some point. The generating circle is at (4.5, 0, 2) with radius 2. So, the center of the generating circle is at (4.5, 0, 2). The sphere's center is at (0, 0, r). So, the distance between these two points is sqrt[(4.5)^2 + (r - 2)^2]. This distance should be equal to the sum of the radii of the sphere and the generating circle, which is r + 2.Wait, is that correct? Because the generating circle is part of the torus, but the sphere is touching the torus, not necessarily the generating circle. Hmm, maybe I need to think differently.Alternatively, perhaps the closest point on the torus to the sphere's center is along the line connecting the sphere's center and the generating circle's center. So, the distance between the sphere's center and the generating circle's center is sqrt[(4.5)^2 + (r - 2)^2], and this distance should be equal to the sum of the sphere's radius and the generating circle's radius, which is r + 2. So, setting up the equation:sqrt[(4.5)^2 + (r - 2)^2] = r + 2Squaring both sides:(4.5)^2 + (r - 2)^2 = (r + 2)^2Calculating (4.5)^2: 4.5 * 4.5 = 20.25Expanding (r - 2)^2: r^2 - 4r + 4Expanding (r + 2)^2: r^2 + 4r + 4So, substituting back:20.25 + r^2 - 4r + 4 = r^2 + 4r + 4Simplify:20.25 + r^2 - 4r + 4 = r^2 + 4r + 4Subtract r^2 and 4 from both sides:20.25 - 4r = 4rAdd 4r to both sides:20.25 = 8rSo, r = 20.25 / 8Calculating that: 20.25 divided by 8. Let's see, 20 divided by 8 is 2.5, and 0.25 divided by 8 is 0.03125, so total is 2.53125. As a fraction, 20.25 is 81/4, so 81/4 divided by 8 is 81/32. So, r = 81/32.Wait, 81 divided by 32 is 2.53125, which matches the decimal. So, the radius of the sphere is 81/32.But let me double-check if this makes sense. The sphere's center is at (0,0,81/32). The generating circle's center is at (4.5,0,2). The distance between them is sqrt[(4.5)^2 + (81/32 - 2)^2]. Let's compute 81/32 - 2: 81/32 - 64/32 = 17/32. So, the distance is sqrt[(20.25) + (17/32)^2]. Let's compute (17/32)^2: 289/1024 ≈ 0.282. So, total inside sqrt is 20.25 + 0.282 ≈ 20.532. The square root of that is approximately 4.531. Now, the sum of the radii is r + 2 = 81/32 + 64/32 = 145/32 ≈ 4.53125. So, that matches. Therefore, the calculation seems correct.So, the radius of the largest spherical ball that can be placed atop the center of the torus while still touching the horizontal plane is 81/32.

💡Okay, so I have this problem about figures made up of diamonds. The figures are labeled F₁, F₂, F₃, and so on. The problem says that each figure Fₙ for n ≥ 3 is constructed from Fₙ₋₁ by surrounding it with a square and placing one more diamond on each side of the new square than Fₙ₋₁ had on each side of its outside square. They also mention that F₃ has 13 diamonds, and they're asking how many diamonds are in F₂₀.First, I need to understand the pattern here. Let me try to visualize it. F₁ is just a single diamond. Then F₂ would be a square surrounding F₁, but how many diamonds does that add? Since F₁ is just one diamond, maybe F₂ adds four diamonds, one on each side of the square? So F₂ would have 1 + 4 = 5 diamonds.Wait, but the problem says that for n ≥ 3, each figure is constructed by surrounding the previous one with a square and adding one more diamond on each side than the previous figure had. So maybe the number of diamonds added each time increases by a certain amount.Let me think. If F₁ has 1 diamond, then F₂ would be a square around F₁. If each side of the square has one diamond, then the total number of diamonds added would be 4, but since the corners are shared, maybe it's 4 diamonds added, making F₂ have 5 diamonds.Then F₃ is constructed by surrounding F₂ with a square and placing one more diamond on each side than F₂ had. F₂ had one diamond on each side, so F₃ would have two diamonds on each side. So the number of diamonds added would be 4 sides times 2 diamonds, but again, the corners are shared, so maybe it's 4*(2) = 8 diamonds added. So F₃ would have 5 + 8 = 13 diamonds, which matches the problem statement.Okay, so the pattern seems to be that each time, when moving from Fₙ₋₁ to Fₙ, we add 4*(n-1) diamonds. Because for F₂, n=2, we added 4*(2-1)=4 diamonds. For F₃, n=3, we added 4*(3-1)=8 diamonds. So in general, the number of diamonds added when moving from Fₙ₋₁ to Fₙ is 4*(n-1).Therefore, the total number of diamonds in Fₙ would be the sum of all the diamonds added from F₁ up to Fₙ. So that would be 1 (for F₁) plus 4*(1) for F₂, plus 4*(2) for F₃, and so on, up to 4*(n-1) for Fₙ.So the total number of diamonds in Fₙ is 1 + 4*(1 + 2 + 3 + ... + (n-1)).I know that the sum of the first k integers is given by k*(k+1)/2. So in this case, the sum from 1 to (n-1) is (n-1)*n/2.Therefore, the total number of diamonds in Fₙ is 1 + 4*((n-1)*n/2) = 1 + 2*(n-1)*n.Simplifying that, it's 1 + 2n² - 2n.So the formula for the number of diamonds in Fₙ is 2n² - 2n + 1.Let me test this formula with the given information. For n=3, it should give 13 diamonds.Plugging in n=3: 2*(3)² - 2*(3) + 1 = 2*9 - 6 + 1 = 18 - 6 + 1 = 13. That's correct.For n=2: 2*(2)² - 2*(2) + 1 = 8 - 4 + 1 = 5. That's correct too.For n=1: 2*(1)² - 2*(1) + 1 = 2 - 2 + 1 = 1. Correct again.Okay, so the formula seems to work.Now, the problem asks for the number of diamonds in F₂₀. So I need to plug n=20 into the formula.Calculating that: 2*(20)² - 2*(20) + 1.First, 20 squared is 400. Multiply by 2: 800.Then, 2*20 is 40.So, 800 - 40 + 1 = 761.Therefore, F₂₀ has 761 diamonds.Wait, let me double-check my calculations to make sure I didn't make a mistake.2*(20)² = 2*400 = 800.2*(20) = 40.So, 800 - 40 = 760, plus 1 is 761. Yep, that seems right.So the answer should be 761, which is option E.**Final Answer**boxed{E}

💡Okay, so I have this problem here: I need to find the imaginary part of the complex number z, which is given as (3 - 4i) divided by (1 - i). The options are A: -1/2, B: 1/2, C: (1/2)i, and D: -(1/2)i. Hmm, let me think about how to approach this.First, I remember that when dealing with complex numbers in the denominator, it's often helpful to multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of (1 - i) is (1 + i). So, if I multiply both the numerator and the denominator by (1 + i), that should help me simplify the expression.Let me write that down:z = (3 - 4i) / (1 - i)Multiply numerator and denominator by (1 + i):z = [(3 - 4i)(1 + i)] / [(1 - i)(1 + i)]Now, I need to expand both the numerator and the denominator.Starting with the numerator: (3 - 4i)(1 + i)Using the distributive property (FOIL method):First: 3 * 1 = 3Outer: 3 * i = 3iInner: -4i * 1 = -4iLast: -4i * i = -4i²So, combining these terms:3 + 3i - 4i - 4i²Now, I know that i² = -1, so -4i² becomes -4*(-1) = 4.So, substituting that back in:3 + 3i - 4i + 4Combine like terms:3 + 4 = 73i - 4i = -iSo, the numerator simplifies to 7 - i.Now, let's work on the denominator: (1 - i)(1 + i)Again, using the distributive property:First: 1 * 1 = 1Outer: 1 * i = iInner: -i * 1 = -iLast: -i * i = -i²Combine these terms:1 + i - i - i²Simplify:The i and -i cancel out, so we have 1 - i²Again, since i² = -1, this becomes 1 - (-1) = 1 + 1 = 2.So, the denominator simplifies to 2.Putting it all together, z = (7 - i) / 2Which can be written as 7/2 - (1/2)i.So, the complex number z is 7/2 minus (1/2)i.Now, the question asks for the imaginary part of z. In a complex number of the form a + bi, the imaginary part is b. So, in this case, the imaginary part is -1/2.Looking back at the options, A is -1/2, which matches our result. So, the answer should be A.Wait, just to make sure I didn't make a mistake somewhere. Let me double-check the multiplication.Numerator: (3 - 4i)(1 + i)3*1 = 33*i = 3i-4i*1 = -4i-4i*i = -4i² = 4So, 3 + 3i -4i +4 = 7 - i. That seems correct.Denominator: (1 - i)(1 + i) = 1 - i² = 1 - (-1) = 2. That also seems correct.So, z = (7 - i)/2 = 7/2 - (1/2)i. So, the imaginary part is -1/2. Yeah, that seems right.I think I'm confident that the answer is A: -1/2.**Final Answer**boxed{A}

💡Okay, so I have this problem here: Given the function f(x) = 5x³ + 7, I need to find the value of x such that the inverse function f⁻¹(x) equals 2. Hmm, let me try to figure this out step by step.First, I remember that if f⁻¹(x) = 2, that means when I plug x into the inverse function, I get 2. But what does that tell me about x in terms of the original function f? I think it means that f(2) should equal x. Let me write that down: if f⁻¹(x) = 2, then f(2) = x. Yeah, that makes sense because the inverse function essentially reverses the original function.So, now I need to find f(2). The function f(x) is given as 5x³ + 7. Let me substitute x with 2 in this function. That would be 5*(2)³ + 7. Okay, calculating that step by step. First, 2 cubed is 8. Then, 5 times 8 is 40. Adding 7 to that gives me 47. So, f(2) equals 47.Wait, does that mean x is 47? Because f(2) = 47, and since f⁻¹(x) = 2, then x must be 47. Let me double-check that. If I plug 47 into the inverse function, I should get 2. To confirm, I can think about what f⁻¹(47) would be. Since f(2) = 47, then f⁻¹(47) should indeed be 2. That checks out.I think I got it right. So, the value of x that satisfies f⁻¹(x) = 2 is 47. But just to make sure I didn't make any calculation mistakes, let me go through the steps again. Starting with f(x) = 5x³ + 7, plugging in x = 2: 5*(2)³ + 7. 2³ is 8, 5*8 is 40, 40 + 7 is 47. Yep, that's correct.I guess another way to think about it is by understanding the relationship between a function and its inverse. The inverse function essentially undoes what the original function does. So, if f takes 2 and gives me 47, then f⁻¹ takes 47 and gives me back 2. That's consistent with what I found earlier.Is there another method to approach this problem? Maybe by finding the inverse function explicitly and then setting it equal to 2 to solve for x. Let me try that as a verification.To find the inverse function f⁻¹(x), I need to solve the equation y = 5x³ + 7 for x in terms of y. So, starting with y = 5x³ + 7, I can subtract 7 from both sides to get y - 7 = 5x³. Then, divide both sides by 5 to get (y - 7)/5 = x³. To solve for x, I take the cube root of both sides, so x = ∛[(y - 7)/5]. Therefore, the inverse function f⁻¹(y) = ∛[(y - 7)/5].Now, if I set f⁻¹(x) = 2, that means ∛[(x - 7)/5] = 2. To solve for x, I can cube both sides to eliminate the cube root. Cubing both sides gives me (x - 7)/5 = 8. Then, multiplying both sides by 5, I get x - 7 = 40. Adding 7 to both sides, I find x = 47. So, this method also leads me to the same answer, x = 47.This confirms that my initial approach was correct. It's always good to verify using different methods to ensure that the solution is accurate. Both methods—direct substitution and finding the inverse function—lead me to the same result, which is reassuring.I think I understand this concept better now. The key takeaway is that if the inverse function f⁻¹(x) equals a certain value, say 2, then x must be the result of applying the original function f to that value, which in this case is f(2). This relationship between a function and its inverse is crucial in solving such problems.Another thing I noticed is the importance of checking my work. By approaching the problem from two different angles—direct substitution and finding the inverse function—I was able to confirm that my answer was consistent. This helps prevent mistakes and ensures that I haven't overlooked anything.In summary, the problem required me to find x such that f⁻¹(x) = 2. By understanding the relationship between a function and its inverse, I determined that x must be f(2). Calculating f(2) using the given function f(x) = 5x³ + 7 gave me x = 47. Verifying this by finding the inverse function and solving for x also confirmed that the answer is correct. This exercise helped reinforce my understanding of inverse functions and their properties.