Billy the Goat

Billy the goat is tied to the corner of Patty�s barn. The barn is 20 x 40 feet, and the rope is 50 feet long. No trees or other obstructions are in the way. What is the available area of grass that Billy can eat? (Draw a good picture of this area first.)

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Billy’s (unchewable) rope is anchored at B.
He can range through
¾ of a circle, radius 50,
¼ of a circle, radius 30, and
¼ of a circle, radius 10.
¾pi(50)2 + ¼pi(302) + ¼pi(102)
= pi(1875 + 225 + 25)
= 2125pi ≈ 6676 ft2.

This is about the same as a square 82 feet on a side or a bit more than 8 times the area of the barn.

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