AMC12 2005 B

AMC12 2005 B · Q5

AMC12 2005 B · Q5. It mainly tests Area & perimeter, Geometry misc.

An $8$-foot by $10$-foot bathroom 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$10$英尺的浴室地面用边长为$1$英尺的正方形瓷砖铺设。每块瓷砖上有一个图案：在瓷砖的每个角以该角为圆心、半径为$1/2$英尺画一个白色的四分之一圆，共四个。瓷砖剩余部分为阴影。地面上共有多少平方英尺的面积是阴影部分？

(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

There are $80$ tiles. Each tile has $[\mbox{square} - 4 \cdot (\mbox{quarter circle})]$ shaded. Thus: \begin{align*} \mbox{shaded area} &= 80 \left( 1 - 4 \cdot \dfrac{1}{4} \cdot \pi \cdot \left(\dfrac{1}{2}\right)^2\right) \\ &= 80\left(1-\dfrac{1}{4}\pi\right) \\ &= \boxed{\textbf{(A) }80-20\pi}. \end{align*} 4 quarters of a circle is a circle so that may save you 0.5 seconds (very much lots) :)

共有 $80$ 块瓷砖。每块瓷砖的阴影面积为 $[\mbox{square} - 4 \cdot (\mbox{quarter circle})]$。因此： \begin{align*} \mbox{shaded area} &= 80 \left( 1 - 4 \cdot \dfrac{1}{4} \cdot \pi \cdot \left(\dfrac{1}{2}\right)^2\right) \\ &= 80\left(1-\dfrac{1}{4}\pi\right) \\ &= \boxed{\textbf{(A) }80-20\pi}. \end{align*} 4 quarters of a circle is a circle so that may save you 0.5 seconds (very much lots) :)

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