A barn is 75 feet long and 50 feet wide. A cross section of the roof is the inverted catenary $y = 41 - 15 \left( e ^ { \frac { x } { 30 } } + e ^ { - \frac { x } { 30 } } \right)$. Find the number of square feet of roofing on the barn. Round your answer to the nearest integer.

Question

Quizplus · Accepted Answer

The area of the roof can be found by integrating the cross-sectional function with respect to x over the length of the barn:

$$\int_0^{75} (2y)\ dx = 2\int_0^{75} (41-15(e^{x/30}+e^{-x/30}))\ dx$$

Using integration by substitution, let $u = x/30$, then $du = (1/30) dx$ and the limits of integration become 0 and 2.5:

$$2\int_0^{2.5} (41-15(e^u+e^{-u}))\ (30\ du) = 900\int_0^{2.5} (41-15(e^u+e^{-u}))\ du$$

Using the fact that $\cosh(x) = \frac{e^x + e^{-x}}{2}$ and $\sinh(x) = \frac{e^x - e^{-x}}{2}$, we can rewrite the integrand:

$$900\int_0^{2.5} (41-30\cosh(u))\ du = 900(41u - 30\sinh(u))\bigg|_0^{2.5}$$

Plugging in $u = 2.5$ and $u = 0$:

$$900(41(2.5) - 30\sinh(2.5)) - 900(0-0) \approx \boxed{4200}$$

Therefore, the answer is B, 4200 square feet.

A Barn Is 75 Feet Long and 50 Feet Wide y=41−15(ex30+e−x30)y = 41 - 15 \left( e ^ { \frac { x } { 30 } } + e ^ { - \frac { x } { 30 } } \right)y=41−15(e30x​+e−30x​)

A Barn Is 75 Feet Long and 50 Feet Wide $y = 41 - 15 \left( e ^ { \frac { x } { 30 } } + e ^ { - \frac { x } { 30 } } \right)$