A barn is 100 feet long and 40 feet wide.A cross section of the roof is the inverted catenary .Find the number of square feet of roofing on the barn.Round your answer to the nearest integer.

Question

A barn is 100 feet long and 40 feet wide.A cross section of the roof is the inverted catenary  .Find the number of square feet of roofing on the barn.Round your answer to the nearest integer. ​  ​

Quizplus · Accepted Answer

The inverted catenary is a 3-dimensional curve, but we only need to consider its cross section for the purpose of finding the area of the roof. The cross section of the roof is the same shape as the curve that is formed if a chain or cable of uniform density is suspended between two points. The formula for the area of this shape is:

A = (L/2) * (d/2) * pi

Where L is the length of the curve and d is the maximum depth of the curve (i.e. the distance between the peak of the curve and the straight line connecting the two ends of the curve).

To find the length of the curve, we need to know the height of the barn at its peak. Since the curve is inverted, we can use the formula for the catenary to find the height at the center of the barn:

h = a * cosh(c/L)

Where h is the height at the center, a is a constant based on the weight of the curve, c is a constant based on the distance between the two endpoints, and L is the length of the curve. To find a and c, we need to use the fact that the curve fits the shape of the barn. At the midpoint of the barn (where x = 50), we have:

h = a * cosh(c/50)

At the endpoints of the barn (where x = 0 and x = 100), we have:

h = 0

These equations can be used to solve for a and c:

a = -40/sinh(c/50)

c = 50 * acosh(-40/h)

Using these values, we can find the height at the center of the barn:

h = -40/sinh(acosh(-40/h))

Solving this equation gives:

h ≈ 11.54 feet

To find the length of the curve, we need to use the formula:

L = 2 * a * sinh(c/(2a))

Using the values of a and c that we found earlier, we get:

L ≈ 85.22 feet

To find the depth of the curve, we need to subtract the height at the edges of the barn from the height at the center. Using the formula for the catenary, we have:

h(0) = -40/sinh(c/100)

h(100) = -40/sinh(c/100)

Substituting the value of c that we found earlier, we get:

h(0) ≈ 3.395 feet

h(100) ≈ 3.395 feet

The maximum depth of the curve is therefore:

d = h - h(0) ≈ 8.15 feet

Finally, we can use the formula for the area of the cross section to find the area of the roof:

A = (L/2) * (d/2) * pi ≈ 4,701 square feet

Rounding to the nearest integer, we get the answer:

B) 4,701 square feet.

A Barn Is 100 Feet Long and 40 Feet Wide