A rain gutter is to be constructed of aluminum sheets 12 inc

A rain gutter is to be constructed of aluminum sheets 12 inches wide. After making off a length of 4 inches from each edge, this length is bent up at an angle theta. (See the illustration) The area A of the opening as a function of theta is given by A = 16 sin theta (cos theta + 1) 0 degree

Solution

cos 2theta+cos theta=0

2cos²theta+cos theta-1=0

2cos²theta+2cos theta-cos theta-1=0

(2cos theta-1)(cos theta+1)=0

cos theta=1/2 ,-1

theta=pi/3

Area=16sin pi/3(cos pi/3 +1) =8sqrt3(3/2) = 12sqrt3 square inches