💡First, I need to determine the ratio of the height to the breadth of the wall. I'll start by converting the given breadth from centimeters to meters for consistency with the volume measurement.Next, I'll express the height and length of the wall in terms of the breadth using the given relationships. The height is some multiple of the breadth, and the length is eight times the height.With these expressions, I'll substitute into the volume formula for a rectangular prism, which is length multiplied by breadth multiplied by height. This will give me an equation involving the unknown ratio.After substituting the known values, I'll solve for the ratio by isolating it on one side of the equation. This involves simplifying the equation and taking the square root to find the value of the ratio.Finally, I'll present the ratio in the required format.

💡To determine the volume of the open box created from the metallic sheet, follow these steps:1. **Calculate the new length of the box:** - Original length = 48 m - Subtract twice the side length of the square (3 m) from the original length. - New length = 48 m - 2 * 3 m = 42 m2. **Calculate the new width of the box:** - Original width = 36 m - Subtract twice the side length of the square (3 m) from the original width. - New width = 36 m - 2 * 3 m = 30 m3. **Determine the height of the box:** - The height is equal to the side length of the square cut out. - Height = 3 m4. **Calculate the volume of the box:** - Volume = Length * Width * Height - Volume = 42 m * 30 m * 3 m = 3780 m³

💡Okay, so I have this sequence problem to solve. Let me try to understand what it's asking. The sequence {a_n} is defined with a_1 = 1, and for n ≥ 2, a_n is given by the formula (2S_n²)/(2S_n - 1). Here, S_n is the sum of the first n terms of the sequence, right? So S_n = a_1 + a_2 + ... + a_n.The problem has two parts. The first part asks me to prove that the sequence {1/S_n} is an arithmetic sequence. The second part wants me to find the general formula for {a_n}.Alright, let's start with part (1). I need to show that {1/S_n} is arithmetic. An arithmetic sequence has a constant difference between consecutive terms. So, if I can show that 1/S_n - 1/S_{n-1} is a constant for all n ≥ 2, then I'm done.Given that a_n = (2S_n²)/(2S_n - 1) for n ≥ 2. Hmm, let's see. Since S_n = S_{n-1} + a_n, I can substitute a_n into this equation.So, S_n = S_{n-1} + (2S_n²)/(2S_n - 1). Let me write that down:S_n = S_{n-1} + (2S_n²)/(2S_n - 1)Hmm, this looks a bit complicated. Maybe I can rearrange this equation to solve for S_n in terms of S_{n-1}.Let me subtract S_{n-1} from both sides:S_n - S_{n-1} = (2S_n²)/(2S_n - 1)But S_n - S_{n-1} is just a_n, which is equal to (2S_n²)/(2S_n - 1). Wait, that's the original definition. Maybe I need a different approach.Alternatively, maybe I can express S_n in terms of S_{n-1}. Let me denote S_n as S and S_{n-1} as s for simplicity.So, S = s + (2S²)/(2S - 1)Let me multiply both sides by (2S - 1) to eliminate the denominator:(2S - 1)S = (2S - 1)s + 2S²Expanding the left side:2S² - S = (2S - 1)s + 2S²Subtract 2S² from both sides:-S = (2S - 1)sSo, -S = 2Ss - sLet me bring all terms to one side:- S - 2Ss + s = 0Factor out S:S(-1 - 2s) + s = 0Hmm, not sure if that's helpful. Maybe I can solve for 1/S in terms of 1/s.Let me rearrange the equation:- S = 2Ss - sBring all terms to the left:- S - 2Ss + s = 0Factor terms with S:S(-1 - 2s) + s = 0So, S(-1 - 2s) = -sDivide both sides by (-1 - 2s):S = (-s)/(-1 - 2s) = s/(1 + 2s)So, S = s/(1 + 2s)Hmm, that's interesting. So, S_n = S_{n-1}/(1 + 2S_{n-1})Let me take reciprocal of both sides:1/S_n = (1 + 2S_{n-1})/S_{n-1} = 1/S_{n-1} + 2Ah! So, 1/S_n = 1/S_{n-1} + 2That means the difference between consecutive terms of {1/S_n} is 2, which is constant. Therefore, {1/S_n} is an arithmetic sequence with common difference 2.Great, that proves part (1). Now, moving on to part (2), finding the general formula for {a_n}.Since we know that {1/S_n} is arithmetic with first term 1/S_1 = 1 (since S_1 = a_1 = 1) and common difference 2, we can write:1/S_n = 1 + 2(n - 1) = 2n - 1Therefore, S_n = 1/(2n - 1)Now, to find a_n, we can use the relation a_n = S_n - S_{n-1} for n ≥ 2.So, a_n = S_n - S_{n-1} = 1/(2n - 1) - 1/(2(n - 1) - 1) = 1/(2n - 1) - 1/(2n - 3)Let me compute this difference:1/(2n - 1) - 1/(2n - 3) = [(2n - 3) - (2n - 1)] / [(2n - 1)(2n - 3)] = (-2)/[(2n - 1)(2n - 3)]So, a_n = -2 / [(2n - 1)(2n - 3)] for n ≥ 2.But wait, for n = 1, a_1 = 1. So, the general formula is:a_n = 1 if n = 1,a_n = -2 / [(2n - 1)(2n - 3)] if n ≥ 2.Let me check this formula for n = 2.S_2 = a_1 + a_2 = 1 + a_2. But S_2 = 1/(2*2 - 1) = 1/3.So, 1 + a_2 = 1/3 => a_2 = -2/3.Using the formula, a_2 = -2 / [(4 - 1)(4 - 3)] = -2/(3*1) = -2/3. Correct.Similarly, for n = 3, S_3 = 1/5.So, a_3 = S_3 - S_2 = 1/5 - 1/3 = (3 - 5)/15 = -2/15.Using the formula, a_3 = -2 / [(6 - 1)(6 - 3)] = -2/(5*3) = -2/15. Correct.Seems like the formula works. So, I think I've got it right.

💡Alright, so I have this problem here about finding the smallest possible degree of a nonzero polynomial with rational coefficients that has all these numbers as roots: 1 + √3, 2 + √4, 3 + √5, all the way up to 100 + √102. Hmm, okay, let me try to unpack this step by step.First off, I know that if a polynomial has rational coefficients, then for any irrational root, its conjugate must also be a root. That's because if you have a root like a + √b, then a - √b must also be a root to ensure that the coefficients remain rational. So, this is a key point here.Looking at the given roots, they are of the form n + √(n+2), where n ranges from 1 to 100. So, for each n, we have a root n + √(n+2). Now, I need to figure out for how many of these roots, their conjugates will also be required to be roots of the polynomial.But wait, hold on. For some values of n, √(n+2) might actually be an integer, right? Because if n+2 is a perfect square, then √(n+2) is rational, making the entire root n + √(n+2) rational as well. In such cases, the conjugate would be the same as the original root, so we don't need to include an additional root. That means only when n+2 is not a perfect square do we need to include both the root and its conjugate.So, my next task is to figure out how many of these n+2 are perfect squares. Let's see, n ranges from 1 to 100, so n+2 ranges from 3 to 102. I need to count how many perfect squares are in this range.Starting from the smallest perfect square greater than or equal to 3, which is 2² = 4. Then, 3² = 9, 4² = 16, and so on, up to the largest perfect square less than or equal to 102. The square root of 102 is approximately 10.1, so the largest perfect square is 10² = 100.So, the perfect squares between 3 and 102 are 4, 9, 16, 25, 36, 49, 64, 81, and 100. Let me count them: 4, 9, 16, 25, 36, 49, 64, 81, 100. That's 9 perfect squares in total.This means that for 9 values of n, the root n + √(n+2) is actually rational because √(n+2) is an integer. For the remaining values of n, where n+2 is not a perfect square, the root is irrational, and thus we need both the root and its conjugate.So, how many irrational roots do we have? Since n ranges from 1 to 100, that's 100 roots in total. Out of these, 9 are rational, so 100 - 9 = 91 are irrational. But since each irrational root requires its conjugate to be included as well, each of these 91 irrational roots actually contributes 2 roots to the polynomial.Therefore, the total number of roots contributed by the irrational ones is 91 * 2 = 182. Adding the 9 rational roots, the total number of roots is 182 + 9 = 191.Since the polynomial must have all these roots, and each root corresponds to a linear factor in the polynomial, the smallest possible degree of such a polynomial is equal to the total number of roots, which is 191.Wait, let me double-check that. So, 100 total terms, 9 are rational, so 91 are irrational, each needing a conjugate, so 91 * 2 = 182, plus 9 gives 191. Yeah, that seems right.I think that's the answer. So, the smallest possible degree is 191.**Final Answer**The smallest possible degree of such a polynomial is boxed{191}.

💡First, determine the number of trees remaining after the typhoon by subtracting the number of trees that died from the initial number of trees.Next, calculate the number of trees Haley grew after the typhoon by subtracting the remaining trees after the typhoon from the current total number of trees.Finally, present the calculated number of trees grown after the typhoon as the final answer.

💡First, we need to find the smallest positive integer ( n ) greater than 1 that leaves a remainder of 1 when divided by 4, 5, and 6. This means:[n equiv 1 pmod{4}][n equiv 1 pmod{5}][n equiv 1 pmod{6}]To solve this, we can express ( n ) as:[n = k times text{LCM}(4, 5, 6) + 1]Calculating the least common multiple (LCM) of 4, 5, and 6:[text{LCM}(4, 5, 6) = 60]Thus,[n = 60k + 1]The smallest positive integer ( n ) greater than 1 is when ( k = 1 ):[n = 60 times 1 + 1 = 61]Finally, we determine the range in which 61 falls. It lies between 60 and 79.[boxed{D}]