Question

Spot 的狗屋有一个正六边形底面，每边长一码。他被一根两码长的绳子拴在一个顶点上。Spot 在狗屋外能到达的区域面积是多少平方码？

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?

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

$2\pi$

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

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

$3\pi$

答案：E

知识点：扇形正多边形圆面积

难度评级：1600

解答：

在拴点处，狗屋挡住了 $120^\circ$ 的内角，剩下半径为 $2$ 的 $240^\circ$ 扇形，面积为 $\dfrac{240}{360}\pi(2)^2=\dfrac{8\pi}{3}$ 。

绕过两个相邻顶点时，各剩 $1$ 码绳长，每处扫过 $60^\circ$ 扇形，总面积为 $2\cdot\dfrac{60}{360}\pi(1)^2=\dfrac{\pi}{3}$ 。总面积为 $\dfrac{8\pi}{3}+\dfrac{\pi}{3}=3\pi$ 。

所以正确答案是 E。

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 第 19 题

答案：E

难度评级：1600

解答：

其他年份的第 19 题