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Square footage of barn roof

  1. Mar 14, 2013 #1
    1. The problem statement, all variables and given/known data
    A barn is 100 feet long and 40 feet wide. A cross section of the roof is the inverted catenary y = 31 - 10(ex/20+e-x/20). Find the number of square feet of roofing on the barn


    2. Relevant equations
    Arc Length Integral
    Hyperbolic trigonometric identities


    3. The attempt at a solution
    I start with my bounds [-20,20] and by rewriting y = f(x) as 31 - 20cosh(x/20) and continue on with y' as 0 - 20sinh(x/20)*(1/20) -> -sinh(x/20). y'2 = sinh2(x/20). The Arc Length formula requires me to take the square root of 1 + y'2. But I feel I should multiply my overall arc length by 100 to give me the area in square feet of the roof. So I have as my integrand 100(1+sinh2(x/20))1/2. Using the identity sinh2(x) = (cosh(2x) - 1)/2 I have for my integrand 100(1+(cosh(x/10)-1)/2))1/2) which boils down to 100((cosh(x/10) +1)/2)1/2 and I can use the identity (cosh(2x)+1)/2 = sinh2(x) to move it into 100sinh(x/20). Letting u = x/20 -> 20du = dx. After integration I have 2,000(sinh(x/20) evaluated at x=20 and x= -20 to give me 2,000(sinh(1)-sinh(-1))ft2. Hopefully I did everything right, I don't have an answer in my book.
     
    Last edited: Mar 14, 2013
  2. jcsd
  3. Mar 14, 2013 #2

    TSny

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    Homework Helper
    Gold Member

    That looks good to me. Using the fact that sinh(x) is an odd function, you should be able to simplify your final answer.
     
  4. Mar 14, 2013 #3
    of course, thank you.
    I believe I just double the area, 4,000(sinh(1)ft2 should be the value if that's true.
     
    Last edited: Mar 14, 2013
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