Chapter 7.6, Problem 102AYU

Constructing a Rain Gutter A rain gutter is to be constructed of aluminum sheets 12 inches wide. After marking off a length of 4 inches from each edge, the builder bends this length up at an angle $\theta$. See the illustration. The area $A$ of the opening as a function of $\theta$ is given by

$A\left(\theta \right)\text{ = 16sin}\theta \left(\cos \theta \text{ + 1}\right)$, ${\text{0}}^{\circ }\text{ < }\theta {\text{ < 90}}^{\circ }$

(a) In calculus, you will be asked to find the angle that maximizes A by solving the equation

$\cos \left(2\theta \right)\text{ + cos}\theta {\text{ = 0, 0}}^{\circ }\text{ < }\theta {\text{ < 90}}^{\circ }$

Solve this equation for $\theta$.

(b) What is the maximum area $A$ of the opening?

(c) Graph $A\text{ = }A\left(\theta \right),{\text{ 0}}^{\circ }\le \theta \le {\text{ 90}}^{\circ }$ and find the angle $\theta$ that maximizes the area $A$. Also find the maximum area.