3440.12 – Billy the Goat

Billy is tied to the corner of Keisha’s barn. The barn is 40 x 20 feet. The rope is 50 feet long. There are no trees or other obstructions in the barnyard. What is the area of grass available for Billy to eat?

Solution

See the figure. The area consists of parts of three circles.

A=14π302+34π502+14π102=225π+1875π+25π=2125π\begin{aligned}A &= \dfrac{1}{4} \pi 30^2 + \dfrac{3}{4} \pi 50^2 + \dfrac{1}{4} \pi 10^2 \\&= 225 \pi + 1875 \pi + 25 \pi \\&= 2125 \pi\end{aligned}