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"I attach my pet snake, Earl, to one corner of my barn with a leash. The barn is square, with sides of length 10, and the leash has a length of twenty, which wraps around the barn. I would like to make sure that I am being humane to Earl, and would therefore like to know that area of my lawn he can traverse while on the leash. What is this area?"
bron

Solution

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We see that since this is at the corner of the barn, the snake is free to travel in a three quarter arc around this point such that it does not intersect the barn. On the corners, we notice that the leash will bend, and basically act as a shorter leash xed at the other two corners. Thus, there is a leash of length 10 for each of two quarter circles, which sum to half of an arc:

  • $A$=$\frac{3}{4}\pi 20^2+\frac{1}{2}\pi 10^2$=$350\pi$

bron