Answer
(a) See explanations.
(b) See graph.
(c) $\theta=\frac{\pi}{3}$
Work Step by Step
(a) Use the figure given in the problem, the cross-sectional area of the gutter the area of a trapezoid with one base of $10cm$, height of $10sin\theta$ and another base of $10+2\times10cos\theta$, thus we have the gutter area
$A(\theta)=\frac{1}{2}\times10sin\theta(10+10+2\times10cos\theta)=100sin\theta+100sin\theta cos\theta$
(b) See graph.
(c) Based on the graph, we can find the maximum of the cross section area of $129.9cm^2$ at $\theta=\frac{\pi}{3}$
