# Square footage of barn roof

1. Mar 14, 2013

### icesalmon

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. Mar 14, 2013

### TSny

That looks good to me. Using the fact that sinh(x) is an odd function, you should be able to simplify your final answer.

3. Mar 14, 2013

### icesalmon

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