2005 PMWC Problems/Problem T7

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Problem

Skipper’s doghouse has a regular hexagonal base that measures one metre on each side. Skipper is tethered to a 2-metre rope which is fixed to a vertex. What is the area of the region outside the doghouse that Skipper can reach? Calculate an approximate answer by using $\pi=3.14$ or $\pi=22/7$ .

Solution

The big sector has a radius of 2 and is 2/3 of the total circle that it used to be a part of, and thus has an area of $4\pi*\frac{2}{3}=\frac{8}{3}\pi$ . The two little sectors each have radius 1 and are 1/6 the total area of their circle, so the area of each of those is $\frac{\pi}{6}$ . Adding, we get $\frac{8}{3}\pi+\frac{\pi}{6}+\frac{\pi}{6}=3\pi$ . Then approximate.

2005 PMWC (Problems)
Preceded by Problem T6	Followed by Problem T8
I: 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 T: 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10

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2005 PMWC Problems/Problem T7

From AoPSWiki

Problem

Solution

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