Question

Spot's doghouse has a regular hexagonal base that measures one yard on each side. He is tethered to a vertex with a two-yard rope. What is the area, in square yards, of the region outside the doghouse that Spot can reach?

$\dfrac{2}{3}\pi$

$2\pi$

$\dfrac{5}{2}\pi$

$\dfrac{8}{3}\pi$

$3\pi$

Solution:

At the tether vertex the hexagon blocks its $120^\circ$ interior angle, leaving a $240^\circ$ sector of radius $2:$ area $\dfrac{240}{360}\pi(2)^2=\dfrac{8\pi}{3}.$

Wrapping around each of the two adjacent vertices, $1$ yard of rope remains and sweeps a $60^\circ$ sector: $2\cdot\dfrac{60}{360}\pi(1)^2=\dfrac{\pi}{3}.$ The total is $\dfrac{8\pi}{3}+\dfrac{\pi}{3}=3\pi.$

Thus, the correct answer is E.

2002 AMC 10A Problem 19

Solution:

Problem 19 in Other Years