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?

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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}