AMC10 2002 A

AMC10 2002 A · Q19

AMC10 2002 A · Q19. It mainly tests Circle theorems, Area & perimeter.

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?

Spot 的狗屋有一个边长一码的正六边形底座。他被一根两码长的绳子拴在一个顶点上。Spot 能到达的狗屋外部区域面积有多少平方码？

(A) (2/3)$\pi$ (2/3)$\pi$

(B) 2$\pi$ 2$\pi$

(D) (8/3)$\pi$ (8/3)$\pi$

(E) 3$\pi$ 3$\pi$

Answer

Correct choice: (E)

正确答案：（E）

Solution

(E) Spot can go anywhere in a $240^\circ$ sector of radius two yards and can cover a $60^\circ$ sector of radius one yard around each of the adjoining corners. The total area is $\pi(2)^2\cdot\frac{240}{360}+2\left(\pi(1)^2\cdot\frac{60}{360}\right)=3\pi.$

（E）Spot 可以在半径为 2 码、圆心角为 $240^\circ$ 的扇形区域内移动，并且可以在相邻的每个角附近覆盖一个半径为 1 码、圆心角为 $60^\circ$ 的扇形区域。总面积为 $\pi(2)^2\cdot\frac{240}{360}+2\left(\pi(1)^2\cdot\frac{60}{360}\right)=3\pi.$

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