answer：Okay, so I have this sequence problem here. It starts with a₁ = 1, and then each subsequent term is defined by aₙ₊₁ = floor(aₙ + sqrt(aₙ) + 1/2). I need to find all n ≤ 2013 such that aₙ is a perfect square. Hmm, interesting.First, let me try to understand the recurrence relation. The next term is the floor of the current term plus the square root of the current term plus a half. So, it's like adding the square root of aₙ and then taking the floor after adding 1/2. That probably means that aₙ₊₁ is roughly aₙ plus sqrt(aₙ), but adjusted to the nearest integer.Since a₁ is 1, which is a perfect square, I wonder if the sequence stays as perfect squares or if it ever becomes a perfect square again. Let me compute the first few terms to see.Starting with a₁ = 1.a₂ = floor(1 + sqrt(1) + 1/2) = floor(1 + 1 + 0.5) = floor(2.5) = 2.Okay, a₂ is 2, which is not a perfect square.a₃ = floor(2 + sqrt(2) + 1/2). Let's compute sqrt(2) ≈ 1.4142. So, 2 + 1.4142 + 0.5 ≈ 3.9142. Taking the floor gives 3. So, a₃ = 3, not a perfect square.a₄ = floor(3 + sqrt(3) + 1/2). sqrt(3) ≈ 1.732. So, 3 + 1.732 + 0.5 ≈ 5.232. Floor is 5. a₄ = 5, not a perfect square.a₅ = floor(5 + sqrt(5) + 1/2). sqrt(5) ≈ 2.236. So, 5 + 2.236 + 0.5 ≈ 7.736. Floor is 7. a₅ = 7, not a perfect square.a₆ = floor(7 + sqrt(7) + 1/2). sqrt(7) ≈ 2.6458. So, 7 + 2.6458 + 0.5 ≈ 10.1458. Floor is 10. a₆ = 10, not a perfect square.a₇ = floor(10 + sqrt(10) + 1/2). sqrt(10) ≈ 3.1623. So, 10 + 3.1623 + 0.5 ≈ 13.6623. Floor is 13. a₇ = 13, not a perfect square.a₈ = floor(13 + sqrt(13) + 1/2). sqrt(13) ≈ 3.6055. So, 13 + 3.6055 + 0.5 ≈ 17.1055. Floor is 17. a₈ = 17, not a perfect square.a₉ = floor(17 + sqrt(17) + 1/2). sqrt(17) ≈ 4.1231. So, 17 + 4.1231 + 0.5 ≈ 21.6231. Floor is 21. a₉ = 21, not a perfect square.a₁₀ = floor(21 + sqrt(21) + 1/2). sqrt(21) ≈ 4.5837. So, 21 + 4.5837 + 0.5 ≈ 26.0837. Floor is 26. a₁₀ = 26, not a perfect square.Hmm, so far, only a₁ is a perfect square. Let me try a few more terms.a₁₁ = floor(26 + sqrt(26) + 1/2). sqrt(26) ≈ 5.099. So, 26 + 5.099 + 0.5 ≈ 31.599. Floor is 31. a₁₁ = 31, not a perfect square.a₁₂ = floor(31 + sqrt(31) + 1/2). sqrt(31) ≈ 5.56776. So, 31 + 5.56776 + 0.5 ≈ 37.06776. Floor is 37. a₁₂ = 37, not a perfect square.a₁₃ = floor(37 + sqrt(37) + 1/2). sqrt(37) ≈ 6.08276. So, 37 + 6.08276 + 0.5 ≈ 43.58276. Floor is 43. a₁₃ = 43, not a perfect square.a₁₄ = floor(43 + sqrt(43) + 1/2). sqrt(43) ≈ 6.5574. So, 43 + 6.5574 + 0.5 ≈ 50.0574. Floor is 50. a₁₄ = 50, not a perfect square.a₁₅ = floor(50 + sqrt(50) + 1/2). sqrt(50) ≈ 7.07107. So, 50 + 7.07107 + 0.5 ≈ 57.57107. Floor is 57. a₁₅ = 57, not a perfect square.a₁₆ = floor(57 + sqrt(57) + 1/2). sqrt(57) ≈ 7.5498. So, 57 + 7.5498 + 0.5 ≈ 65.0498. Floor is 65. a₁₆ = 65, not a perfect square.a₁₇ = floor(65 + sqrt(65) + 1/2). sqrt(65) ≈ 8.0623. So, 65 + 8.0623 + 0.5 ≈ 73.5623. Floor is 73. a₁₇ = 73, not a perfect square.a₁₈ = floor(73 + sqrt(73) + 1/2). sqrt(73) ≈ 8.544. So, 73 + 8.544 + 0.5 ≈ 82.044. Floor is 82. a₁₈ = 82, not a perfect square.a₁₉ = floor(82 + sqrt(82) + 1/2). sqrt(82) ≈ 9.0554. So, 82 + 9.0554 + 0.5 ≈ 91.5554. Floor is 91. a₁₉ = 91, not a perfect square.a₂₀ = floor(91 + sqrt(91) + 1/2). sqrt(91) ≈ 9.5394. So, 91 + 9.5394 + 0.5 ≈ 101.0394. Floor is 101. a₂₀ = 101, not a perfect square.Hmm, so up to a₂₀, only a₁ is a perfect square. Maybe I should check a few more terms, but it's getting tedious. Maybe there's a pattern or a formula here.Let me think about the recurrence relation again: aₙ₊₁ = floor(aₙ + sqrt(aₙ) + 1/2). So, it's approximately aₙ + sqrt(aₙ). If I ignore the floor and the 1/2 for a moment, it's like aₙ₊₁ ≈ aₙ + sqrt(aₙ). This is a kind of recursive sequence where each term increases by roughly the square root of the previous term.I wonder if this sequence grows quadratically or something else. Let me see: If aₙ is roughly k², then sqrt(aₙ) is k, so aₙ₊₁ ≈ k² + k. So, if aₙ is k², then aₙ₊₁ is roughly k² + k. Then, aₙ₊₂ would be roughly (k² + k) + sqrt(k² + k). Hmm, sqrt(k² + k) is approximately k + 1/2 for large k, right? Because (k + 1/2)² = k² + k + 1/4, which is close to k² + k.So, aₙ₊₂ ≈ (k² + k) + (k + 1/2) = k² + 2k + 1/2. Then, aₙ₊₃ ≈ (k² + 2k + 1/2) + sqrt(k² + 2k + 1/2). The sqrt term is approximately k + 1, since (k + 1)² = k² + 2k + 1. So, aₙ₊₃ ≈ k² + 2k + 1/2 + k + 1 = k² + 3k + 3/2.Continuing this way, each term adds roughly k, then k + 1, then k + 2, etc. So, the sequence seems to be increasing by approximately k each time, where k is the square root of the current term.But wait, if aₙ is roughly k², then aₙ₊₁ is roughly k² + k, aₙ₊₂ is roughly k² + 2k + 1, which is (k + 1)². Oh! So, does that mean that after a few steps, the sequence reaches the next perfect square?Wait, let's test this. Suppose aₙ = k². Then aₙ₊₁ = floor(k² + k + 1/2). Since k² + k + 1/2 is not an integer, the floor would be k² + k. Then, aₙ₊₂ = floor(k² + k + sqrt(k² + k) + 1/2). Now, sqrt(k² + k) is approximately k + 1/2, as I thought earlier. So, k² + k + (k + 1/2) + 1/2 = k² + 2k + 1, which is (k + 1)². So, aₙ₊₂ would be floor(k² + 2k + 1) = (k + 1)².Wait, so if aₙ is a perfect square, then two steps later, aₙ₊₂ is the next perfect square. But in our earlier calculations, starting from a₁ = 1, we had a₂ = 2, a₃ = 3, a₄ = 5, a₅ = 7, etc., which didn't reach 4, which is the next perfect square. Hmm, that contradicts my earlier thought.Wait, maybe I made a mistake in the approximation. Let's see: If aₙ = k², then aₙ₊₁ = floor(k² + k + 1/2). Since k² + k + 1/2 is not an integer, it's equal to k² + k + 0.5, so the floor is k² + k. Then, aₙ₊₂ = floor(k² + k + sqrt(k² + k) + 1/2). Now, sqrt(k² + k) is approximately k + 1/2 - 1/(8k) + ... using the binomial expansion. So, sqrt(k² + k) ≈ k + 1/2 - 1/(8k). Therefore, aₙ₊₂ ≈ k² + k + (k + 1/2 - 1/(8k)) + 1/2 = k² + 2k + 1 - 1/(8k). So, floor of that is k² + 2k, which is k(k + 2). Hmm, not exactly (k + 1)².Wait, but k² + 2k is still less than (k + 1)² = k² + 2k + 1. So, aₙ₊₂ is k² + 2k, not a perfect square. Then, aₙ₊₃ = floor(k² + 2k + sqrt(k² + 2k) + 1/2). Now, sqrt(k² + 2k) is approximately k + 1 - 1/(2k). So, sqrt(k² + 2k) ≈ k + 1 - 1/(2k). Therefore, aₙ₊₃ ≈ k² + 2k + (k + 1 - 1/(2k)) + 1/2 = k² + 3k + 1.5 - 1/(2k). The floor of that is k² + 3k + 1, which is still not a perfect square.Wait, so maybe the sequence doesn't reach the next perfect square in two steps? Or maybe it takes more steps? Let's see with k=1: a₁=1, a₂=2, a₃=3, a₄=5, a₅=7, a₆=10, a₇=13, a₈=17, a₉=21, a₁₀=26, a₁₁=31, a₁₂=37, a₁₃=43, a₁₄=50, a₁₅=57, a₁₆=65, a₁₇=73, a₁₈=82, a₁₉=91, a₂₀=101.Looking at these numbers, I don't see any perfect squares except a₁=1. So, maybe the only n where aₙ is a perfect square is n=1? But let me check for larger k.Suppose k=2: aₙ=4. Then aₙ₊₁ = floor(4 + 2 + 0.5) = floor(6.5) = 6. Then aₙ₊₂ = floor(6 + sqrt(6) + 0.5) ≈ floor(6 + 2.449 + 0.5) = floor(8.949) = 8. Then aₙ₊₃ = floor(8 + 2.828 + 0.5) ≈ floor(11.328) = 11. Then aₙ₊₄ = floor(11 + 3.316 + 0.5) ≈ floor(14.816) = 14. Then aₙ₊₅ = floor(14 + 3.741 + 0.5) ≈ floor(18.241) = 18. Then aₙ₊₆ = floor(18 + 4.242 + 0.5) ≈ floor(22.742) = 22. Then aₙ₊₇ = floor(22 + 4.690 + 0.5) ≈ floor(27.190) = 27. Then aₙ₊₈ = floor(27 + 5.196 + 0.5) ≈ floor(32.696) = 32. Then aₙ₊₉ = floor(32 + 5.657 + 0.5) ≈ floor(38.157) = 38. Then aₙ₊₁₀ = floor(38 + 6.164 + 0.5) ≈ floor(44.664) = 44.So, starting from aₙ=4, the sequence goes 4,6,8,11,14,18,22,27,32,38,44,... None of these are perfect squares except the starting point. So, it seems that once you start from a perfect square, the next terms are not perfect squares.Wait, but in the initial sequence starting from 1, we never reach 4, which is the next perfect square. So, maybe the only perfect square in the sequence is a₁=1.But let me think again. Suppose aₙ is a perfect square, say m². Then aₙ₊₁ = floor(m² + m + 0.5). Since m² + m + 0.5 is not an integer, it's equal to m² + m + 0.5, so the floor is m² + m. Then, aₙ₊₂ = floor(m² + m + sqrt(m² + m) + 0.5). Now, sqrt(m² + m) is approximately m + 0.5 - 1/(8m). So, aₙ₊₂ ≈ m² + m + m + 0.5 - 1/(8m) + 0.5 = m² + 2m + 1 - 1/(8m). So, floor of that is m² + 2m, which is m(m + 2). Not a perfect square.Then, aₙ₊₃ = floor(m² + 2m + sqrt(m² + 2m) + 0.5). sqrt(m² + 2m) ≈ m + 1 - 1/(2m). So, aₙ₊₃ ≈ m² + 2m + m + 1 - 1/(2m) + 0.5 = m² + 3m + 1.5 - 1/(2m). Floor is m² + 3m + 1. Not a perfect square.Continuing this, it seems that the sequence doesn't reach the next perfect square, which would be (m + 1)² = m² + 2m + 1. But our aₙ₊₂ is m² + 2m, which is one less than (m + 1)². So, it's just below the next perfect square.Therefore, unless the sequence somehow jumps over the perfect square, it won't land on it. But from the calculations, it seems that it doesn't. So, maybe the only perfect square in the sequence is a₁=1.But wait, let me check for m=0. If m=0, then aₙ=0, but our sequence starts at 1, so m=0 is not relevant.Alternatively, maybe there's a point where aₙ is a perfect square again. Let me see if there's a k such that aₙ = k², and then aₙ₊₁ = floor(k² + k + 0.5) = k² + k, and then aₙ₊₂ = floor(k² + k + sqrt(k² + k) + 0.5). If sqrt(k² + k) is an integer, then k² + k must be a perfect square. Let me see if k² + k is a perfect square for any k>1.Suppose k² + k = m². Then, m² - k² - k = 0. Rearranging, m² = k² + k. Let's see if this equation has integer solutions for k>1.Rewriting: m² = k(k + 1). Since k and k + 1 are consecutive integers, they are coprime. Therefore, for their product to be a perfect square, both k and k + 1 must be perfect squares. But the only two consecutive perfect squares are 0 and 1. So, k=0 and k +1=1. But k=0 is not in our sequence. Therefore, there are no k>1 such that k² + k is a perfect square. Hence, aₙ₊₁ cannot be a perfect square if aₙ is a perfect square, except when aₙ=1.Therefore, the only n where aₙ is a perfect square is n=1.Wait, but let me double-check. Suppose aₙ is a perfect square, say m². Then aₙ₊₁ = m² + m. Then, aₙ₊₂ = m² + m + sqrt(m² + m) + 0.5, floored. If sqrt(m² + m) is not an integer, then aₙ₊₂ is not a perfect square. But we saw that m² + m is not a perfect square for m>1, so sqrt(m² + m) is irrational, hence aₙ₊₂ is not a perfect square.Therefore, the only perfect square in the sequence is a₁=1.So, the answer is n=1.

answer：Okay, so I have this problem about perfect squares and the number of divisors. Let me try to understand it step by step. First, the problem states that if a number ( N ) is expressed in its prime factorization form, which is ( N = p_1^{alpha} p_2^{beta} p_3^{gamma} ldots p_k^{lambda} ), where ( p_1, p_2, p_3, ldots, p_k ) are prime numbers and ( alpha, beta, gamma, ldots, lambda ) are their respective exponents, then the number of divisors ( n ) of ( N ) is given by the formula ( n = (alpha + 1)(beta + 1)(gamma + 1) ldots (lambda + 1) ).Now, I need to prove that a necessary and sufficient condition for ( N ) to be a perfect square is that the number of divisors ( n ) is odd. Hmm, okay. So, I need to show two things: 1. If ( N ) is a perfect square, then ( n ) is odd.2. If ( n ) is odd, then ( N ) is a perfect square.Let me start by recalling what a perfect square is. A perfect square is a number that can be expressed as some integer squared. In terms of prime factorization, this means that all the exponents in the prime factorization must be even numbers. For example, ( 36 = 2^2 times 3^2 ) is a perfect square because both exponents are even.So, if ( N ) is a perfect square, then each exponent ( alpha, beta, gamma, ldots, lambda ) must be even. Let's see what this implies for the number of divisors ( n ). Each term in the formula for ( n ) is ( (text{exponent} + 1) ). If each exponent is even, then each ( (text{exponent} + 1) ) will be odd because an even number plus one is odd. For example, if ( alpha = 2 ), then ( alpha + 1 = 3 ), which is odd. Now, if we multiply several odd numbers together, the result is still odd. Because multiplying two odd numbers gives an odd number, and multiplying another odd number to that result still gives an odd number. So, if all the ( (text{exponent} + 1) ) terms are odd, their product ( n ) will be odd. That takes care of the first part: if ( N ) is a perfect square, then ( n ) is odd. Now, for the converse: if ( n ) is odd, then ( N ) must be a perfect square. Let's think about this. If ( n ) is odd, then each ( (text{exponent} + 1) ) must be odd because the product of numbers is odd only if each factor is odd. If each ( (text{exponent} + 1) ) is odd, then each exponent must be even because an odd number minus one is even. For example, if ( alpha + 1 = 3 ), then ( alpha = 2 ), which is even. Therefore, if all exponents ( alpha, beta, gamma, ldots, lambda ) are even, ( N ) must be a perfect square. So, putting it all together, ( N ) is a perfect square if and only if the number of its divisors ( n ) is odd. Wait, let me double-check if there are any exceptions or special cases. What if ( N = 1 )? The number 1 is considered a perfect square because ( 1 = 1^2 ). Its prime factorization is empty, so the number of divisors is 1, which is odd. That fits the condition. What about a prime number like 2? Its number of divisors is 2, which is even, and 2 is not a perfect square. That also fits. Another example: ( N = 4 ), which is ( 2^2 ). The number of divisors is ( (2 + 1) = 3 ), which is odd. Correct. And ( N = 6 ), which is ( 2^1 times 3^1 ). The number of divisors is ( (1 + 1)(1 + 1) = 4 ), which is even, and 6 is not a perfect square. So, all these examples seem to confirm the condition. I think the reasoning holds. I guess the key idea is that the parity (odd or even nature) of the number of divisors directly relates to the parity of the exponents in the prime factorization. If all exponents are even, the number of divisors is odd, and vice versa. Therefore, the necessary and sufficient condition for ( N ) to be a perfect square is that the number of its divisors ( n ) is odd.

answer：Okay, so I have this problem here with two parts, both involving functions and their derivatives. Let me try to work through them step by step.Starting with part (I): We're given f(x) = ln x - a x, and we need to find the intervals where f(x) is strictly increasing. Hmm, I remember that to find where a function is increasing or decreasing, we look at its first derivative. If the derivative is positive, the function is increasing; if it's negative, the function is decreasing.So, let's find f'(x). The derivative of ln x is 1/x, and the derivative of -a x is -a. So, f'(x) = 1/x - a. That simplifies to (1 - a x)/x.Now, we need to figure out where this derivative is positive. So, (1 - a x)/x > 0. Since x is in the domain of ln x, x must be greater than 0. So, x > 0.Therefore, the sign of the derivative depends on the numerator: 1 - a x. So, 1 - a x > 0 implies that a x < 1, which means x < 1/a, provided that a is positive. If a is negative, then 1 - a x will always be positive because -a x becomes positive, making 1 - a x larger than 1, which is positive.Wait, let me think again. If a is positive, then 1 - a x > 0 implies x < 1/a. If a is negative, then 1 - a x is 1 plus a positive number, so it's always positive, meaning f'(x) is positive for all x > 0.So, summarizing:- If a ≤ 0, f'(x) is always positive, so f(x) is strictly increasing on (0, ∞).- If a > 0, f'(x) is positive only when x < 1/a, so f(x) is strictly increasing on (0, 1/a).Okay, that makes sense. So, that's part (I) done.Moving on to part (II): We're given F(x) = f(x) + a x² + a x. Let's substitute f(x) into this: F(x) = ln x - a x + a x² + a x. Simplifying, the -a x and +a x cancel out, so F(x) = ln x + a x².Now, we need to determine if F(x) has any extreme values, which are maxima or minima. Again, we'll look at the first derivative to find critical points and then determine if they're maxima or minima.So, let's compute F'(x). The derivative of ln x is 1/x, and the derivative of a x² is 2a x. So, F'(x) = 1/x + 2a x.To find critical points, set F'(x) = 0: 1/x + 2a x = 0.Let's solve for x. Multiply both sides by x to eliminate the denominator: 1 + 2a x² = 0.So, 2a x² = -1, which implies x² = -1/(2a). Now, x² must be positive, so -1/(2a) must be positive. That means that a must be negative because 2a is negative, making -1/(2a) positive.So, if a is negative, we have x² = -1/(2a), so x = sqrt(-1/(2a)). Since x must be positive (because of the ln x term), we take the positive square root: x = sqrt(-1/(2a)).If a is non-negative, then -1/(2a) is negative or undefined, so there are no real solutions, meaning F'(x) never equals zero. Therefore, F(x) has no critical points when a ≥ 0.Now, let's analyze the behavior of F'(x) to determine if the critical point is a maximum or minimum.Case 1: a > 0.In this case, F'(x) = 1/x + 2a x. Both terms are positive for x > 0, so F'(x) is always positive. Therefore, F(x) is strictly increasing on (0, ∞), so no extreme values.Case 2: a = 0.Then, F(x) = ln x + 0 = ln x. The derivative is 1/x, which is positive for all x > 0, so again, F(x) is strictly increasing, no extreme values.Case 3: a < 0.We have a critical point at x = sqrt(-1/(2a)). Let's test the intervals around this point.For x < sqrt(-1/(2a)):Choose a test point, say x = sqrt(-1/(2a))/2.Compute F'(x) = 1/x + 2a x.Since a is negative, 2a x is negative. But x is small, so 1/x is large positive, and 2a x is negative but small in magnitude. So, overall, F'(x) is positive.For x > sqrt(-1/(2a)):Choose a test point, say x = sqrt(-1/(2a)) * 2.Compute F'(x) = 1/x + 2a x.Here, x is large, so 1/x is small positive, and 2a x is negative and large in magnitude. So, overall, F'(x) is negative.Therefore, the function F(x) increases before x = sqrt(-1/(2a)) and decreases after that point. So, the critical point is a maximum.Now, let's find the value of F at this critical point.F(x) = ln x + a x².Substitute x = sqrt(-1/(2a)).First, ln(sqrt(-1/(2a))) = (1/2) ln(-1/(2a)).Second, a x² = a * (-1/(2a)) = -1/2.So, F(x) = (1/2) ln(-1/(2a)) - 1/2.Therefore, the maximum value is (1/2) ln(-1/(2a)) - 1/2.So, summarizing:- If a ≥ 0, F(x) has no extreme values.- If a < 0, F(x) has a maximum at x = sqrt(-1/(2a)) with value (1/2) ln(-1/(2a)) - 1/2.Wait, let me double-check the calculation for F(x) at the critical point.F(x) = ln x + a x².x = sqrt(-1/(2a)).So, x² = (-1)/(2a).Therefore, a x² = a * (-1)/(2a) = -1/2.And ln x = ln(sqrt(-1/(2a))) = (1/2) ln(-1/(2a)).So, yes, F(x) = (1/2) ln(-1/(2a)) - 1/2.That seems correct.So, putting it all together, that's the solution.

answer：First, calculate the number of balloons that blew up in the first half-hour by finding one-fifth of the total number of balloons:1/5 * 200 = 40 balloons.Next, determine the total number of balloons that have blown up by subtracting the remaining intact balloons from the total:200 - 80 = 120 balloons.Subtract the number of balloons that blew up in the first half-hour from the total number of blown-up balloons to find how many blew up in the second hour:120 - 40 = 80 balloons.Finally, establish the ratio of balloons blown up in the second hour to those blown up in the first half-hour:80 : 40 = 2 : 1.

answer：Okay, so I have this problem here that says: "3z varies inversely as the cube of x. When z=8, x=2. Find z when x=4." And there are multiple-choice answers: A)1, B)2, C)3, D)4, E)6.Alright, let's break this down. First, I need to understand what it means for 3z to vary inversely as the cube of x. From what I remember, inverse variation means that as one quantity increases, the other decreases proportionally. Specifically, if A varies inversely as B, then A = k/B, where k is a constant.So, applying that to this problem: 3z varies inversely as the cube of x. That should translate to an equation like 3z = k / x³, right? Yeah, that sounds right. So, 3z equals some constant k divided by x cubed.Now, they give me a specific case where z is 8 and x is 2. I can use this information to find the constant k. Let me plug those values into the equation.So, if z is 8, then 3z is 3*8, which is 24. And x is 2, so x cubed is 2³, which is 8. So, plugging into the equation: 24 = k / 8.To solve for k, I can multiply both sides by 8. That gives me k = 24 * 8. Let me calculate that: 24 times 8 is 192. So, k is 192.Now that I have the constant k, I can write the general equation for this relationship: 3z = 192 / x³.The question asks me to find z when x is 4. So, I need to plug x = 4 into this equation and solve for z.First, let's compute x cubed when x is 4. That's 4³, which is 64. So, the equation becomes 3z = 192 / 64.Let me calculate 192 divided by 64. Hmm, 64 goes into 192 exactly 3 times because 64*3 is 192. So, 192 / 64 is 3.Now, the equation is 3z = 3. To solve for z, I divide both sides by 3. So, z = 3 / 3, which is 1.Wait, that seems straightforward. Let me double-check my steps to make sure I didn't make a mistake.1. Start with the inverse variation: 3z = k / x³.2. Plug in z=8 and x=2: 24 = k / 8.3. Solve for k: k = 24*8 = 192.4. Now, the equation is 3z = 192 / x³.5. Plug in x=4: 3z = 192 / 64 = 3.6. Solve for z: z = 1.Everything checks out. So, when x is 4, z is 1.Looking back at the multiple-choice options: A)1, B)2, C)3, D)4, E)6. So, the correct answer should be A)1.I think I did everything correctly. I translated the inverse variation into an equation, found the constant using the given values, and then used that constant to find z when x changed. The steps make sense, and the calculations seem right. I don't see any errors in my reasoning.Just to be extra sure, let me consider if there's another way to approach this problem. Sometimes, with inverse variation, you can set up a proportion without finding k explicitly.So, if 3z varies inversely as x³, then (3z₁) * (x₁³) = (3z₂) * (x₂³). That's another way to express the inverse variation because the product remains constant.Using this proportion, let's plug in the known values. When z₁=8 and x₁=2, and we need to find z₂ when x₂=4.So, (3*8) * (2³) = (3*z₂) * (4³).Calculating each part:3*8 = 24.2³ = 8.So, left side is 24*8 = 192.On the right side, 4³ is 64, so we have (3*z₂)*64.So, the equation is 192 = (3*z₂)*64.To solve for z₂, divide both sides by 64: 192 / 64 = 3*z₂.192 divided by 64 is 3, so 3 = 3*z₂.Divide both sides by 3: z₂ = 1.Same result! So, using this proportion method also gives me z=1 when x=4. That reinforces that my initial answer was correct.I think I'm confident now that the answer is A)1.**Final Answer**boxed{A}

answer：Okay, so I have this problem where there are two congruent squares, ABCD and EFGH, each with a side length of 20. They overlap to form a 20 by 35 rectangle called AEHD. I need to find the percentage of the area of rectangle AEHD that is shaded and also find the unshaded area. The answer choices are given, so that might help if I get stuck.First, let me visualize what's happening here. There are two squares, each 20 units on each side. When they overlap, they form a rectangle that is 20 units in one side and 35 units in the other. So, the rectangle is longer than the squares. That means the squares must be overlapping along their lengths to create this longer rectangle.Since each square is 20 units long, and the rectangle is 35 units long, I can figure out how much they overlap. If I have two squares placed side by side without overlapping, the total length would be 20 + 20 = 40 units. But the rectangle is only 35 units long, so the overlap must be 40 - 35 = 5 units. So, the squares overlap by 5 units along their length.Now, the area of the rectangle AEHD is 20 multiplied by 35, which is 700 square units. That's straightforward.Next, I need to find the shaded area. I think the shaded area refers to the overlapping region of the two squares within the rectangle. Since the overlap is 5 units along the length, and the width is still 20 units, the area of the overlap (shaded region) should be 5 multiplied by 20, which is 100 square units.So, the shaded area is 100 square units, and the total area of the rectangle is 700 square units. To find the percentage of the area that is shaded, I can divide the shaded area by the total area and then multiply by 100. That would be (100 / 700) * 100, which simplifies to approximately 14.29%.Now, for the unshaded area. Since the total area is 700 and the shaded area is 100, subtracting the shaded area from the total gives me the unshaded area. So, 700 - 100 = 600 square units.Let me double-check my calculations to make sure I didn't make a mistake. The area of the rectangle is definitely 20 * 35 = 700. The overlap is 5 units, so 5 * 20 = 100 for the shaded area. Then, 100 / 700 is roughly 14.29%, and 700 - 100 is 600. That seems correct.Looking at the answer choices, option B is 14.29% and 600. That matches my calculations. So, I think that's the right answer.**Final Answer**boxed{B}