Question

一个 $8$ 英尺乘 $10$ 英尺的地面铺满 $1$ 英尺乘 $1$ 英尺的正方形瓷砖。每块瓷砖的图案由四个白色四分之一圆组成，每个圆半径为 $\dfrac12$ 英尺，圆心在瓷砖的四个角。瓷砖剩余部分为阴影。地面上阴影部分共有多少平方英尺？

An $8$ -foot by $10$ -foot floor is tiled with square tiles of size $1$ foot by $1$ foot. Each tile has a pattern consisting of four white quarter circles of radius $\dfrac12$ foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?

$80 - 20\pi$

$60 - 10\pi$

$80 - 10\pi$

$60 + 10\pi$

$80 + 10\pi$

答案：A

知识点：圆面积面积

难度评级：1130

解答：

每块瓷砖的四个四分之一圆合成一个半径为 $\dfrac12$ 的圆，面积为 $\pi\left(\dfrac12\right)^2 = \dfrac{\pi}{4}$ 。

每块瓷砖阴影面积为 $1 - \dfrac{\pi}{4}$ 。

共有 $8 \cdot 10 = 80$ 块瓷砖，所以总阴影面积为 $80\left(1 - \dfrac{\pi}{4}\right) = 80 - 20\pi.$

所以正确答案是 A。

The four quarter circles in a tile together form one full circle of radius $\dfrac12,$ with area $\pi\left(\dfrac12\right)^2 = \dfrac{\pi}{4}.$

So each tile has shaded area $1 - \dfrac{\pi}{4}$ square feet.

There are $8 \cdot 10 = 80$ tiles, so the total shaded area is $80\left(1 - \dfrac{\pi}{4}\right) = 80 - 20\pi.$

Thus, the correct answer is A.

2005 AMC 12B 第 5 题

答案：A

难度评级：1130

解答：

其他年份的第 5 题