AMC10 2005 B

AMC10 2005 B · Q8

AMC10 2005 B · Q8. It mainly tests Circle theorems, Area & perimeter.

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 1/2 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?

一个8英尺乘10英尺的地板用1英尺乘1英尺的正方形瓷砖铺成。每块瓷砖上有四个以瓷砖每个角为中心、半径1/2英尺的白色四分之一圆。瓷砖的其余部分是遮荫的。地板的遮荫面积有多少平方英尺？

(A) $80 - 20\pi$ $80 - 20\pi$

(B) $60 - 10\pi$ $60 - 10\pi$

(D) $60 + 10\pi$ $60 + 10\pi$

(E) $80 + 10\pi$ $80 + 10\pi$

Answer

Correct choice: (A)

正确答案：（A）

Solution

(A) The four white quarter circles in each tile have the same area as a whole circle of radius \$1/2\$, that is, \$\pi(1/2)^2=\pi/4\$ square feet. So the area of the shaded portion of each tile is \$1-\pi/4\$ square feet. Since there are \$8\cdot10=80\$ tiles in the entire floor, the area of the total shaded region in square feet is \[ 80\left(1-\frac{\pi}{4}\right)=80-20\pi. \]

（A）每块瓷砖上的四个白色四分之一圆的总面积等于半径为 \$1/2\$ 的整圆面积，即 \$\pi(1/2)^2=\pi/4\$ 平方英尺。因此，每块瓷砖阴影部分的面积为 \$1-\pi/4\$ 平方英尺。由于整个地面共有 \$8\cdot10=80\$ 块瓷砖，所以阴影总面积（单位：平方英尺）为 \[ 80\left(1-\frac{\pi}{4}\right)=80-20\pi. \]

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