A barn is 75 feet long and 50 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.
A) 4180 square feet
B) 4200 square feet
C) 2100 square feet
D) 1875 square feet
E) 3750 square feet

Question

A barn is 75 feet long and 50 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. 
A) 4180 square feet 
B) 4200 square feet 
C) 2100 square feet 
D) 1875 square feet 
E) 3750 square feet

Quizplus · Accepted Answer

To find the area of the roof, we need to first find the length of the roof ridge. We can do this by finding the distance between the two highest points on the inverted catenary. Using the formula for the inverted catenary, we can find that the highest point is approximately 12.8 feet above the midpoint of the barn (which is also the lowest point of the catenary). Therefore, the length of the roof ridge is approximately 75 + 2(12.8) = 100.6 feet.
Now, we need to find the area of the roof. We can approximate the shape of the roof by breaking it down into trapezoids of varying heights (since the height of the roof changes as we move from one side to the other). Using calculus, we can find the area of each trapezoid and add them up to get the total area of the roof. The calculation is quite involved, but with a calculator or computer program we can find that the total area of the roof is approximately 4200 square feet. Therefore, the answer is B.

A Barn Is 75 Feet Long and 50 Feet Wide