answer：Alright, so I've got this problem here about Andrey, Boris, and Denis eating candies. It says they each have their own constant rate of eating candies. Let me try to break this down step by step.First, it mentions that while Andrey ate 4 candies, Boris managed to eat only 3. Okay, so that gives me a ratio of their eating rates. Andrey eats 4 candies in the same time Boris eats 3. So, Andrey's rate is faster than Boris's.Then it says Denis ate candies the fastest: he ate 7 candies while Andrey ate 6. Hmm, so Denis is faster than Andrey. So, in the time it takes Andrey to eat 6 candies, Denis eats 7. That gives me another ratio.In total, they ate 70 candies. I need to find out how many candies each of them ate.Let me think about how to approach this. Maybe I can set up some ratios based on the information given.From the first part, Andrey eats 4 candies while Boris eats 3. So, the ratio of Andrey's rate to Boris's rate is 4:3.From the second part, Denis eats 7 candies while Andrey eats 6. So, the ratio of Denis's rate to Andrey's rate is 7:6.Now, I need to combine these ratios so that I can express all three rates in terms of a common variable.Let me denote the rates as follows:- Let Andrey's rate be A candies per unit time.- Let Boris's rate be B candies per unit time.- Let Denis's rate be D candies per unit time.From the first ratio, A:B = 4:3. So, I can write A = (4/3)B.From the second ratio, D:A = 7:6. So, D = (7/6)A.But since A is already expressed in terms of B, I can substitute that in:D = (7/6)*(4/3)B = (28/18)B = (14/9)B.So now, all rates are expressed in terms of B:- A = (4/3)B- D = (14/9)BNow, I need to find the total number of candies eaten by all three. It says they ate 70 candies in total.Assuming they ate candies for the same amount of time, the total candies eaten would be the sum of each person's rate multiplied by time.Let me denote the time as t.So, total candies = A*t + B*t + D*t = (A + B + D)*t = 70.I can factor out t:t*(A + B + D) = 70.But I need to find t, or express t in terms of B.From above, A = (4/3)B and D = (14/9)B.So, A + B + D = (4/3)B + B + (14/9)B.Let me compute that:First, convert all to ninths to add them up:(4/3)B = (12/9)BB = (9/9)B(14/9)B remains as is.So, adding them up:(12/9)B + (9/9)B + (14/9)B = (12 + 9 + 14)/9 B = 35/9 B.So, A + B + D = (35/9)B.Therefore, t*(35/9)B = 70.I need to solve for t.But I have two variables here: t and B. I need another equation to solve for both.Wait, maybe I can express t in terms of B.From the first ratio, A = (4/3)B.But A is also equal to 4 candies per some time unit. Wait, no, A is the rate. So, if A = 4 candies per unit time, then B = 3 candies per unit time.Wait, maybe I misinterpreted the ratios.Let me go back.The problem says: "While Andrey ate 4 candies, Boris managed to eat only 3."So, in the same amount of time, Andrey ate 4, Boris ate 3.So, their rates are A = 4/t and B = 3/t, where t is the time taken.Similarly, "Denis ate candies the fastest: he ate 7 candies while Andrey ate 6."So, in the same amount of time, Denis ate 7, Andrey ate 6.So, their rates are D = 7/s and A = 6/s, where s is the time taken.But wait, the time s might be different from time t.This is getting a bit confusing.Maybe I need to find a common time frame to compare all three.Let me think.From the first scenario, Andrey eats 4 in time t, Boris eats 3 in time t.From the second scenario, Andrey eats 6 in time s, Denis eats 7 in time s.I need to relate these two scenarios.Maybe I can find the relationship between t and s.From the first scenario, A = 4/t.From the second scenario, A = 6/s.So, 4/t = 6/s.Therefore, s = (6/4)t = (3/2)t.So, the time s is 1.5 times the time t.Now, let's express D in terms of t.From the second scenario, D = 7/s = 7/(1.5t) = (7/1.5)/t = (14/3)/t.So, D = 14/(3t).Now, we have:A = 4/tB = 3/tD = 14/(3t)Now, the total candies eaten by all three is 70.Assuming they ate candies for the same amount of time, say T.So, total candies = A*T + B*T + D*T = (A + B + D)*T = 70.Substituting the expressions for A, B, D:(4/t + 3/t + 14/(3t)) * T = 70.Combine the terms:( (4 + 3) / t + 14/(3t) ) * T = 70(7/t + 14/(3t)) * T = 70Convert 7/t to 21/(3t) to have a common denominator:(21/(3t) + 14/(3t)) * T = 70(35/(3t)) * T = 70Now, T is the total time they spent eating.But I need to relate T to t.From the first scenario, t is the time it took Andrey to eat 4 candies.So, T is the total time, which might be a multiple of t.Let me assume that T = n*t, where n is the number of intervals of time t.Then, substituting T = n*t:(35/(3t)) * n*t = 70The t cancels out:35/3 * n = 70Solve for n:n = 70 * 3 / 35 = 2 * 3 = 6So, n = 6.Therefore, T = 6*t.Now, let's find the total candies each ate.Andrey's total candies = A*T = (4/t)*(6t) = 24Boris's total candies = B*T = (3/t)*(6t) = 18Denis's total candies = D*T = (14/(3t))*(6t) = 28So, Andrey ate 24 candies, Boris ate 18, and Denis ate 28.Let me check if these add up to 70:24 + 18 + 28 = 70Yes, that's correct.So, the final answer is:Andrey ate 24 candies, Boris ate 18 candies, and Denis ate 28 candies.

answer：Okay, so I have this problem about two candles. Both candles are the same height at the start, and they're lit at the same time. The first candle burns out in 5 hours, and the second one burns out in 4 hours. I need to figure out after how many hours the height of the first candle will be three times the height of the second candle. The options are given as A. 10/3, B. 12/5, C. 40/11, and D. 24/7.Alright, let's break this down. Both candles start at the same height, let's call that height 'H'. Since they burn at constant rates, I can model their heights as linear functions of time.First, I need to find the burning rates of each candle. The first candle burns out in 5 hours, so its burning rate is H divided by 5 hours. Similarly, the second candle burns out in 4 hours, so its burning rate is H divided by 4 hours.Let me write that down:- Burning rate of first candle: H / 5 per hour- Burning rate of second candle: H / 4 per hourNow, let's denote the time elapsed as 't' hours. After 't' hours, the height of the first candle will be its initial height minus the amount burned, which is (H / 5) * t. Similarly, the height of the second candle will be H minus (H / 4) * t.So, the height of the first candle after 't' hours is:Height1 = H - (H / 5) * tAnd the height of the second candle after 't' hours is:Height2 = H - (H / 4) * tThe problem states that at some time 't', the height of the first candle is three times the height of the second candle. So, we can set up the equation:Height1 = 3 * Height2Substituting the expressions we have:H - (H / 5) * t = 3 * [H - (H / 4) * t]Hmm, let me simplify this equation. First, I can factor out H on both sides since it's common to all terms. Let's divide both sides by H to make it simpler:1 - (1 / 5) * t = 3 * [1 - (1 / 4) * t]Now, let's expand the right side:1 - (1 / 5)t = 3 - (3 / 4)tOkay, now I need to solve for 't'. Let's get all the terms involving 't' on one side and constants on the other. Let me subtract 1 from both sides:- (1 / 5)t = 2 - (3 / 4)tNow, let's add (3 / 4)t to both sides to get all 't' terms on the left:- (1 / 5)t + (3 / 4)t = 2To combine these terms, I need a common denominator. The denominators are 5 and 4, so the least common denominator is 20. Let's convert the fractions:- (4 / 20)t + (15 / 20)t = 2Now, combine the terms:(11 / 20)t = 2To solve for 't', multiply both sides by 20:11t = 40Then, divide both sides by 11:t = 40 / 11Hmm, 40 divided by 11 is approximately 3.636 hours. Let me check the options given:A. 10/3 ≈ 3.333B. 12/5 = 2.4C. 40/11 ≈ 3.636D. 24/7 ≈ 3.428So, 40/11 is option C. That seems to be the correct answer.Wait, let me double-check my steps to make sure I didn't make a mistake.Starting from:1 - (1/5)t = 3 - (3/4)tSubtract 1:- (1/5)t = 2 - (3/4)tAdd (3/4)t:(3/4 - 1/5)t = 2Convert to common denominator:(15/20 - 4/20)t = 2(11/20)t = 2Multiply both sides by 20:11t = 40Divide by 11:t = 40/11Yes, that seems correct. So, the answer is 40/11 hours, which is option C.I think that's solid. I don't see any errors in my calculations. The key was setting up the correct equations based on the burning rates and then solving for 't' when one height is three times the other. It's important to remember that both candles start at the same height, so we can factor that out and simplify the equation.Another way to think about it is to consider the heights as fractions of the initial height. The first candle burns at 1/5 per hour, so after 't' hours, it's at (1 - t/5) height. The second burns at 1/4 per hour, so it's at (1 - t/4) height. Setting up the ratio as 3 times and solving gives the same result.I guess another check could be plugging the value back into the original height equations to see if the ratio holds.Let's try t = 40/11.Height1 = 1 - (1/5)*(40/11) = 1 - (8/11) = 3/11Height2 = 1 - (1/4)*(40/11) = 1 - (10/11) = 1/11Is 3/11 equal to 3*(1/11)? Yes, because 3*(1/11) is 3/11. So that checks out.Alright, I'm confident that 40/11 is the correct answer.**Final Answer**boxed{C}

answer：Okay, so I have this problem about a geometric sequence. Let me try to understand it step by step. The problem says that all the terms of the geometric sequence {a_n} are positive, and it gives me an equation: a₁₀a₁₁ + a₉a₁₂ = 2e⁵. Then, it's asking for the sum of the natural logarithms of the first 20 terms, which is ln a₁ + ln a₂ + … + ln a₂₀. Hmm, okay.First, I remember that in a geometric sequence, each term is a constant multiple of the previous one. That constant is called the common ratio, usually denoted by r. So, the nth term of a geometric sequence can be written as a_n = a₁ * r^(n-1). That might be useful here.Now, the problem gives me an equation involving terms a₁₀, a₁₁, a₉, and a₁₂. Let me write those out using the formula for the nth term.a₉ = a₁ * r^(8)a₁₀ = a₁ * r^(9)a₁₁ = a₁ * r^(10)a₁₂ = a₁ * r^(11)So, substituting these into the given equation:a₁₀a₁₁ + a₉a₁₂ = (a₁ * r^9)(a₁ * r^10) + (a₁ * r^8)(a₁ * r^11)Let me compute each product:First term: (a₁ * r^9)(a₁ * r^10) = a₁² * r^(9+10) = a₁² * r^19Second term: (a₁ * r^8)(a₁ * r^11) = a₁² * r^(8+11) = a₁² * r^19So, both terms are equal to a₁² * r^19. Therefore, the equation becomes:a₁² * r^19 + a₁² * r^19 = 2e⁵That simplifies to:2 * a₁² * r^19 = 2e⁵Dividing both sides by 2:a₁² * r^19 = e⁵Hmm, okay. So, a₁² * r^19 equals e⁵. That's one equation.Now, the problem is asking for the sum of the natural logarithms of the first 20 terms. Let me write that sum:ln a₁ + ln a₂ + … + ln a₂₀I remember that the sum of logarithms is the logarithm of the product. So, this sum is equal to:ln(a₁ * a₂ * … * a₂₀)So, I need to find the product of the first 20 terms of the geometric sequence and then take the natural logarithm of that product.In a geometric sequence, the product of the terms can be expressed in terms of the first term, the common ratio, and the number of terms. Let me recall the formula for the product of n terms of a geometric sequence.The product P of the first n terms of a geometric sequence is given by:P = (a₁^n) * r^(n(n-1)/2)So, in this case, n is 20, so:P = (a₁^20) * r^(20*19/2) = a₁^20 * r^(190)Therefore, the product of the first 20 terms is a₁^20 * r^190.So, the sum of the logarithms is:ln(a₁^20 * r^190) = ln(a₁^20) + ln(r^190) = 20 ln a₁ + 190 ln rHmm, okay. So, I need to find 20 ln a₁ + 190 ln r.But from earlier, I have that a₁² * r^19 = e⁵. Let me write that again:a₁² * r^19 = e⁵Taking natural logarithm on both sides:ln(a₁² * r^19) = ln(e⁵)Which simplifies to:ln(a₁²) + ln(r^19) = 5Which is:2 ln a₁ + 19 ln r = 5So, 2 ln a₁ + 19 ln r = 5.Now, I need to find 20 ln a₁ + 190 ln r. Hmm, that looks like 10 times (2 ln a₁ + 19 ln r). Let me check:10*(2 ln a₁ + 19 ln r) = 20 ln a₁ + 190 ln rYes, exactly. So, 20 ln a₁ + 190 ln r = 10*(2 ln a₁ + 19 ln r) = 10*5 = 50Therefore, the sum of the logarithms is 50.Wait, let me make sure I didn't skip any steps or make any mistakes.Starting from the given equation:a₁₀a₁₁ + a₉a₁₂ = 2e⁵Expressed in terms of a₁ and r, both products a₁₀a₁₁ and a₉a₁₂ equal a₁² r^19, so their sum is 2a₁² r^19 = 2e⁵, leading to a₁² r^19 = e⁵.Taking natural logs: 2 ln a₁ + 19 ln r = 5.Then, the sum of ln a₁ + ln a₂ + … + ln a₂₀ is ln(a₁a₂…a₂₀) = ln(a₁^20 r^190) = 20 ln a₁ + 190 ln r.Factor out 10: 10*(2 ln a₁ + 19 ln r) = 10*5 = 50.Yes, that seems correct. So, the answer is 50.**Final Answer**boxed{50}

answer：Okay, so I have this problem where I need to find constants A, B, and C such that the fraction (frac{5x^2}{(x - 4)(x - 2)^2}) can be expressed as (frac{A}{x - 4} + frac{B}{x - 2} + frac{C}{(x - 2)^2}). Hmm, I think this is a partial fractions problem. I remember that partial fractions are used to break down complicated rational expressions into simpler ones that are easier to work with, especially for integration or other operations.Alright, so the first step is to set up the equation. The given expression is already in the form of partial fractions, so I need to find the values of A, B, and C. To do this, I think I should multiply both sides of the equation by the denominator of the original fraction, which is ((x - 4)(x - 2)^2). That should eliminate the denominators and make it easier to solve for A, B, and C.Let me write that out:[frac{5x^2}{(x - 4)(x - 2)^2} = frac{A}{x - 4} + frac{B}{x - 2} + frac{C}{(x - 2)^2}]Multiplying both sides by ((x - 4)(x - 2)^2):[5x^2 = A(x - 2)^2 + B(x - 4)(x - 2) + C(x - 4)]Okay, so now I have an equation without denominators. The next step is to find the values of A, B, and C. I remember that one method is to choose specific values of x that simplify the equation. For example, if I set (x = 4), the term with A will remain, but the terms with B and C will become zero because of the factors ((x - 4)). Similarly, if I set (x = 2), the terms with C will remain, but the others will become zero because of ((x - 2)^2).Let me try that. First, let me set (x = 4):[5(4)^2 = A(4 - 2)^2 + B(4 - 4)(4 - 2) + C(4 - 4)]Simplifying:[5 times 16 = A times 2^2 + B times 0 times 2 + C times 0][80 = 4A][A = frac{80}{4} = 20]Great, so A is 20. Now, let me set (x = 2) to find C:[5(2)^2 = A(2 - 2)^2 + B(2 - 4)(2 - 2) + C(2 - 4)]Simplifying:[5 times 4 = A times 0^2 + B times (-2) times 0 + C times (-2)][20 = 0 + 0 - 2C][-2C = 20][C = frac{20}{-2} = -10]Alright, so C is -10. Now I have A and C, but I still need to find B. Since I can't set x to another value that will eliminate all variables except B, I think I need to substitute A and C back into the equation and then compare coefficients.Let me write the equation again with A = 20 and C = -10:[5x^2 = 20(x - 2)^2 + B(x - 4)(x - 2) - 10(x - 4)]Now, I need to expand each term on the right-hand side.First, expand (20(x - 2)^2):[20(x^2 - 4x + 4) = 20x^2 - 80x + 80]Next, expand (B(x - 4)(x - 2)):Let me first multiply out ((x - 4)(x - 2)):[(x - 4)(x - 2) = x^2 - 2x - 4x + 8 = x^2 - 6x + 8]So, multiplying by B:[B(x^2 - 6x + 8) = Bx^2 - 6Bx + 8B]Lastly, expand (-10(x - 4)):[-10x + 40]Now, let's combine all these expanded terms:[20x^2 - 80x + 80 + Bx^2 - 6Bx + 8B - 10x + 40]Combine like terms:- The (x^2) terms: (20x^2 + Bx^2 = (20 + B)x^2)- The x terms: (-80x - 6Bx - 10x = (-80 - 6B - 10)x = (-90 - 6B)x)- The constant terms: (80 + 8B + 40 = 120 + 8B)So, the right-hand side becomes:[(20 + B)x^2 + (-90 - 6B)x + (120 + 8B)]Now, the left-hand side is (5x^2), so we can set up the equation:[5x^2 = (20 + B)x^2 + (-90 - 6B)x + (120 + 8B)]To solve for B, we can equate the coefficients of corresponding powers of x on both sides.First, for (x^2):[5 = 20 + B][B = 5 - 20][B = -15]Wait, let me check that. If 5 = 20 + B, then subtracting 20 from both sides gives B = -15. That seems correct.Now, let's check the x term and the constant term to make sure everything is consistent.For the x term:Left-hand side has 0x, so:[0 = -90 - 6B][-90 - 6B = 0][-6B = 90][B = -15]Okay, that's consistent with what we found earlier.For the constant term:Left-hand side has 0, so:[0 = 120 + 8B][120 + 8B = 0][8B = -120][B = -15]Again, consistent. So, B is indeed -15.Therefore, the constants are A = 20, B = -15, and C = -10.Let me just recap what I did:1. I multiplied both sides by the denominator to eliminate fractions.2. I substituted specific x-values to solve for A and C directly.3. Then, I substituted A and C back into the equation and expanded everything.4. I combined like terms and set up equations by equating coefficients of corresponding powers of x.5. I solved for B and checked the consistency with the other terms.It all seems to check out. I don't think I made any calculation errors, but just to be thorough, let me plug the values back into the original partial fractions expression and see if it simplifies back to the original fraction.So, substituting A = 20, B = -15, and C = -10:[frac{20}{x - 4} + frac{-15}{x - 2} + frac{-10}{(x - 2)^2}]Let me combine these terms over the common denominator ((x - 4)(x - 2)^2):First term: (frac{20}{x - 4} = frac{20(x - 2)^2}{(x - 4)(x - 2)^2})Second term: (frac{-15}{x - 2} = frac{-15(x - 4)(x - 2)}{(x - 4)(x - 2)^2})Third term: (frac{-10}{(x - 2)^2} = frac{-10(x - 4)}{(x - 4)(x - 2)^2})Now, combine them:[frac{20(x - 2)^2 - 15(x - 4)(x - 2) - 10(x - 4)}{(x - 4)(x - 2)^2}]Let me expand each numerator term:First term: (20(x - 2)^2 = 20(x^2 - 4x + 4) = 20x^2 - 80x + 80)Second term: (-15(x - 4)(x - 2)). Let's expand ((x - 4)(x - 2)) first:[(x - 4)(x - 2) = x^2 - 2x - 4x + 8 = x^2 - 6x + 8]Multiply by -15:[-15x^2 + 90x - 120]Third term: (-10(x - 4) = -10x + 40)Now, add all these together:First term: (20x^2 - 80x + 80)Second term: (-15x^2 + 90x - 120)Third term: (-10x + 40)Combine like terms:- (20x^2 - 15x^2 = 5x^2)- (-80x + 90x - 10x = 0x)- (80 - 120 + 40 = 0)So, the numerator simplifies to (5x^2 + 0x + 0 = 5x^2), which matches the original numerator. Therefore, the partial fraction decomposition is correct.I think I did everything right. I found A, B, and C by plugging in suitable x-values and then solving for the remaining constant by equating coefficients. Then, I double-checked by plugging the constants back into the partial fractions and confirming that it simplifies to the original expression. So, I'm confident that A = 20, B = -15, and C = -10 are the correct constants.**Final Answer**The constants are (boxed{20}), (boxed{-15}), and (boxed{-10}).**Final Answer**The constants are (boxed{20}), (boxed{-15}), and (boxed{-10}).

answer：Alright, so I have this problem: What is the remainder when 3 to the power of 1995 is divided by 7? Hmm, that's a pretty big exponent. I remember something about modular arithmetic and maybe patterns in powers. Let me think about how to approach this.First, I recall that when dealing with exponents in modular arithmetic, especially with prime numbers, Fermat's Little Theorem might be useful. Fermat's Little Theorem says that if p is a prime number, then for any integer a not divisible by p, a^(p-1) is congruent to 1 modulo p. Since 7 is a prime number, and 3 is not divisible by 7, this theorem should apply here.So, applying Fermat's Little Theorem, 3^(7-1) = 3^6 should be congruent to 1 modulo 7. Let me check that: 3^1 is 3 mod 7, 3^2 is 9 which is 2 mod 7, 3^3 is 6 mod 7, 3^4 is 18 which is 4 mod 7, 3^5 is 12 which is 5 mod 7, and 3^6 is 15 which is 1 mod 7. Okay, that checks out. So, 3^6 ≡ 1 mod 7.Now, since 3^6 is congruent to 1, I can use this to simplify 3^1995. The idea is to express 1995 as a multiple of 6 plus some remainder. That way, I can break down the exponent into manageable parts.Let me divide 1995 by 6 to find the quotient and the remainder. So, 6 times 332 is 1992. Subtracting that from 1995 gives a remainder of 3. So, 1995 = 6*332 + 3. Therefore, 3^1995 can be written as 3^(6*332 + 3), which is the same as (3^6)^332 * 3^3.From Fermat's Little Theorem, I know that 3^6 ≡ 1 mod 7. So, (3^6)^332 ≡ 1^332 ≡ 1 mod 7. That simplifies the expression to 1 * 3^3 mod 7, which is just 3^3 mod 7.Now, calculating 3^3 is straightforward: 3*3 is 9, and 9*3 is 27. So, 3^3 is 27. Now, I need to find 27 mod 7. Dividing 27 by 7 gives 3 with a remainder of 6, because 7*3 is 21, and 27 - 21 is 6. Therefore, 27 mod 7 is 6.Putting it all together, 3^1995 mod 7 is equal to 6. So, the remainder when 3^1995 is divided by 7 is 6.Wait, let me double-check my steps to make sure I didn't make a mistake. I used Fermat's Little Theorem correctly, breaking down the exponent into multiples of 6. The division of 1995 by 6 gave me a quotient of 332 and a remainder of 3, which seems right. Then, I correctly applied the theorem to simplify the expression, and my calculation of 3^3 and its modulus with 7 also checks out. Yeah, I think that's solid.I guess another way to look at it is by observing the pattern of powers of 3 modulo 7. Let me list them out:- 3^1 mod 7 = 3- 3^2 mod 7 = 9 mod 7 = 2- 3^3 mod 7 = 6- 3^4 mod 7 = 18 mod 7 = 4- 3^5 mod 7 = 12 mod 7 = 5- 3^6 mod 7 = 1And then it repeats because 3^6 is 1, so 3^7 would be 3 again, and so on. So, the cycle length here is 6. That means every 6 exponents, the pattern repeats. Therefore, to find 3^1995 mod 7, I can find where 1995 falls in this cycle.Since the cycle is every 6 exponents, I can divide 1995 by 6 and find the remainder. As I did earlier, 1995 divided by 6 is 332 with a remainder of 3. So, 3^1995 mod 7 is the same as 3^3 mod 7, which is 6. Yep, that confirms my earlier result.I think I'm confident now that the remainder is indeed 6. It makes sense because both methods—the theorem and the pattern observation—lead to the same conclusion. So, I don't think I made any errors here. It's always good to cross-verify using different approaches to ensure the answer is correct.In summary, by applying Fermat's Little Theorem and recognizing the cyclical pattern of powers modulo 7, I was able to simplify the problem and find that the remainder when 3^1995 is divided by 7 is 6.

answer：Okay, so I have this problem where there's a block of silver that's 72 cubic centimeters, and it's being melted down and drawn into a wire. The wire has a diameter that starts at 1 millimeter at one end and increases linearly to 3 millimeters at the other end. I need to find out how long this wire is, in meters.Alright, first, I know that when you melt something and reshape it, the volume stays the same. So the volume of the silver block is equal to the volume of the wire. That makes sense because melting and drawing it into a wire doesn't add or remove any silver; it just changes the shape.Now, the tricky part is figuring out the volume of this wire. It's not a regular cylinder because the diameter changes from 1mm to 3mm. So, it's like a cone that's been stretched out or something? Maybe it's a shape called a "frustum" of a cone. A frustum is like a cone with the tip cut off, right? So, if I imagine the wire as part of a cone that's been cut, that might work.Let me recall the formula for the volume of a frustum of a cone. I think it's something like:[ V = frac{1}{3} pi h (R^2 + R r + r^2) ]Where:- ( V ) is the volume,- ( h ) is the height (which would be the length of the wire in this case),- ( R ) is the radius of the larger base,- ( r ) is the radius of the smaller base.Okay, so I need to find ( h ), the length of the wire. I know the volume ( V ) is 72 cm³. The diameters are given as 1mm and 3mm, so I need to convert those to radii in centimeters because the volume is in cubic centimeters.First, let's convert the diameters to radii:- The smaller diameter is 1mm, so the radius is ( frac{1}{2} ) mm, which is 0.5mm. To convert that to centimeters, since 1cm = 10mm, 0.5mm is 0.05cm.- The larger diameter is 3mm, so the radius is ( frac{3}{2} ) mm, which is 1.5mm. Converting to centimeters, that's 0.15cm.So, ( r = 0.05 ) cm and ( R = 0.15 ) cm.Now, plug these values into the frustum volume formula:[ 72 = frac{1}{3} pi h (0.15^2 + 0.05 times 0.15 + 0.05^2) ]Let me calculate the terms inside the parentheses first:- ( 0.15^2 = 0.0225 )- ( 0.05 times 0.15 = 0.0075 )- ( 0.05^2 = 0.0025 )Adding these together:[ 0.0225 + 0.0075 + 0.0025 = 0.0325 ]So, the equation simplifies to:[ 72 = frac{1}{3} pi h times 0.0325 ]Multiply ( frac{1}{3} ) by 0.0325:[ frac{1}{3} times 0.0325 = 0.0108333... ]So now, the equation is:[ 72 = 0.0108333... times pi times h ]Let me compute ( 0.0108333... times pi ):[ 0.0108333... times pi approx 0.03407 ]So, now we have:[ 72 = 0.03407 times h ]To solve for ( h ), divide both sides by 0.03407:[ h = frac{72}{0.03407} ]Calculating that:[ h approx 2113.2 text{ cm} ]Wait, that seems really long. 2113 centimeters is over 21 meters. That doesn't seem right. Did I make a mistake somewhere?Let me double-check my calculations.First, the radii:- 1mm diameter is 0.1cm diameter, so radius is 0.05cm. Correct.- 3mm diameter is 0.3cm diameter, so radius is 0.15cm. Correct.Volume formula:[ V = frac{1}{3} pi h (R^2 + R r + r^2) ]Plugging in the values:[ 72 = frac{1}{3} pi h (0.15^2 + 0.05 times 0.15 + 0.05^2) ]Calculating inside the parentheses:- ( 0.15^2 = 0.0225 )- ( 0.05 times 0.15 = 0.0075 )- ( 0.05^2 = 0.0025 )Adding them up:[ 0.0225 + 0.0075 + 0.0025 = 0.0325 ]So, that's correct.Then,[ 72 = frac{1}{3} pi h times 0.0325 ]Calculating ( frac{1}{3} times 0.0325 ):[ frac{0.0325}{3} approx 0.0108333 ]So,[ 72 = 0.0108333 times pi times h ]Calculating ( 0.0108333 times pi ):[ 0.0108333 times 3.1416 approx 0.03407 ]So,[ 72 = 0.03407 times h ]Then,[ h = frac{72}{0.03407} approx 2113.2 text{ cm} ]Hmm, 2113 cm is 21.13 meters. That seems too long for a wire made from 72 cm³ of silver. Maybe the formula I used isn't correct?Wait, maybe I confused the formula for the frustum with something else. Let me think again.Alternatively, I've heard that when the diameter changes linearly, the volume can be calculated by averaging the cross-sectional areas. So, maybe the volume is the average of the two circular areas multiplied by the length.Let me try that approach.The cross-sectional area at one end is ( pi r_1^2 ) and at the other end is ( pi r_2^2 ). The average area would be ( frac{pi r_1^2 + pi r_2^2}{2} ).So, the volume would be:[ V = text{Average Area} times text{Length} = frac{pi r_1^2 + pi r_2^2}{2} times h ]Let me plug in the values:[ 72 = frac{pi (0.05)^2 + pi (0.15)^2}{2} times h ]Calculating the areas:- ( pi (0.05)^2 = pi times 0.0025 approx 0.007854 ) cm²- ( pi (0.15)^2 = pi times 0.0225 approx 0.070686 ) cm²Adding them together:[ 0.007854 + 0.070686 = 0.07854 ) cm²Divide by 2:[ frac{0.07854}{2} = 0.03927 ) cm²So, the equation becomes:[ 72 = 0.03927 times h ]Solving for ( h ):[ h = frac{72}{0.03927} approx 1833.6 text{ cm} ]Which is approximately 18.34 meters. Still, that's over 18 meters, which seems extremely long for a wire made from 72 cm³ of silver.Wait, maybe I'm misunderstanding the shape. If the diameter varies linearly, is it actually a cone or a frustum? Or is it something else?Alternatively, perhaps the wire is a cylinder with a linearly varying diameter, which would make it a conical shape. But in that case, the volume formula for a cone is:[ V = frac{1}{3} pi r^2 h ]But here, the radius changes from 0.05cm to 0.15cm over the length ( h ). So, maybe I need to integrate the area over the length.Let me think about calculus. If the diameter varies linearly, the radius as a function of position ( x ) along the wire can be expressed as:[ r(x) = r_1 + left( frac{r_2 - r_1}{h} right) x ]Where ( x ) ranges from 0 to ( h ).Then, the volume would be the integral of the cross-sectional area from 0 to ( h ):[ V = int_{0}^{h} pi [r(x)]^2 dx ]Substituting ( r(x) ):[ V = pi int_{0}^{h} left[ r_1 + left( frac{r_2 - r_1}{h} right) x right]^2 dx ]Expanding the square:[ V = pi int_{0}^{h} left[ r_1^2 + 2 r_1 left( frac{r_2 - r_1}{h} right) x + left( frac{r_2 - r_1}{h} right)^2 x^2 right] dx ]Integrating term by term:1. ( int_{0}^{h} r_1^2 dx = r_1^2 h )2. ( int_{0}^{h} 2 r_1 left( frac{r_2 - r_1}{h} right) x dx = 2 r_1 left( frac{r_2 - r_1}{h} right) times frac{h^2}{2} = r_1 (r_2 - r_1) h )3. ( int_{0}^{h} left( frac{r_2 - r_1}{h} right)^2 x^2 dx = left( frac{r_2 - r_1}{h} right)^2 times frac{h^3}{3} = frac{(r_2 - r_1)^2 h}{3} )Adding them together:[ V = pi left[ r_1^2 h + r_1 (r_2 - r_1) h + frac{(r_2 - r_1)^2 h}{3} right] ]Factor out ( h ):[ V = pi h left[ r_1^2 + r_1 (r_2 - r_1) + frac{(r_2 - r_1)^2}{3} right] ]Simplify inside the brackets:First, expand ( r_1 (r_2 - r_1) ):[ r_1 r_2 - r_1^2 ]So, the expression becomes:[ r_1^2 + r_1 r_2 - r_1^2 + frac{(r_2 - r_1)^2}{3} ]Simplify ( r_1^2 - r_1^2 ):[ r_1 r_2 + frac{(r_2 - r_1)^2}{3} ]So,[ V = pi h left[ r_1 r_2 + frac{(r_2 - r_1)^2}{3} right] ]Let me plug in the values:( r_1 = 0.05 ) cm, ( r_2 = 0.15 ) cm.First, calculate ( r_1 r_2 ):[ 0.05 times 0.15 = 0.0075 ) cm²Next, calculate ( (r_2 - r_1)^2 ):[ (0.15 - 0.05)^2 = (0.10)^2 = 0.01 ) cm²Then, divide by 3:[ frac{0.01}{3} approx 0.003333 ) cm²Add them together:[ 0.0075 + 0.003333 approx 0.010833 ) cm²So, the volume equation becomes:[ 72 = pi h times 0.010833 ]Solving for ( h ):[ h = frac{72}{pi times 0.010833} ]Calculate the denominator:[ pi times 0.010833 approx 0.03407 )So,[ h = frac{72}{0.03407} approx 2113.2 text{ cm} ]Again, I get approximately 2113 cm, which is 21.13 meters. That seems really long. Is this correct?Wait, maybe I should consider the units more carefully. The volume is 72 cm³, and the cross-sectional area is in cm², so when I multiply by length in cm, I get cm³, which matches.But intuitively, 21 meters seems too long for a wire made from 72 cm³ of silver. Maybe I made a mistake in the formula.Let me check the formula for the volume of a conical frustum again. It is:[ V = frac{1}{3} pi h (R^2 + R r + r^2) ]Which is what I used initially. So, plugging in the numbers:[ V = frac{1}{3} pi h (0.15^2 + 0.05 times 0.15 + 0.05^2) ]Which simplifies to:[ V = frac{1}{3} pi h times 0.0325 ]So,[ 72 = frac{1}{3} pi h times 0.0325 ]Which leads to:[ h = frac{72 times 3}{pi times 0.0325} ]Calculating that:[ h = frac{216}{0.1021} approx 2113.2 text{ cm} ]Same result. So, it seems consistent.But to double-check, let's consider the average cross-sectional area.The average radius is ( frac{0.05 + 0.15}{2} = 0.10 ) cm.So, the average cross-sectional area is:[ pi (0.10)^2 = 0.0314 ) cm²Then, the volume would be:[ V = 0.0314 times h ]Set equal to 72 cm³:[ 72 = 0.0314 times h ][ h = frac{72}{0.0314} approx 2292.3 text{ cm} approx 22.92 text{ meters} ]Hmm, this is different from the previous result. So, using the average radius gives a different length.This inconsistency suggests that maybe the average cross-sectional area method isn't accurate for a linearly changing diameter. Perhaps the frustum formula is the correct one.Alternatively, maybe I should think of it as a series of infinitesimal cylinders along the length, each with a slightly increasing radius, and integrate their volumes.Wait, that's essentially what I did earlier with the calculus approach, and it gave me 21.13 meters.But why does the average radius method give a different answer? Maybe because the cross-sectional area doesn't increase linearly with radius; it increases with the square of the radius. So, the average area isn't just the area of the average radius.That makes sense. So, the average cross-sectional area isn't simply ( pi (text{average radius})^2 ), because area depends on the square of the radius. Therefore, the average area is actually greater than ( pi (text{average radius})^2 ).So, going back, the frustum formula is more accurate because it accounts for the squared terms in the volume calculation.Therefore, despite the large length, 21.13 meters seems to be the correct answer based on the frustum volume formula.But just to be thorough, let me check online for the volume of a conical frustum to confirm the formula.[Imagines checking a reference]Yes, the formula is indeed:[ V = frac{1}{3} pi h (R^2 + R r + r^2) ]So, my initial approach was correct.Therefore, the length of the wire is approximately 21.13 meters.Wait, but in the initial problem, the user mentioned that the diameter varies linearly. So, is the wire a frustum of a cone? Or is it something else?Thinking about it, if the diameter varies linearly along the length, then yes, it's a frustum of a cone. If it were a cylinder, the diameter would be constant. If it were a cone, one end would have zero diameter. Here, both ends have non-zero diameters, and the diameter increases linearly, so it's a frustum.Therefore, using the frustum formula is appropriate.So, after carefully going through the calculations multiple times and verifying the formulas, I conclude that the length of the wire is approximately 21.13 meters.