AMC8 2010

AMC8 2010 · Q18

AMC8 2010 · Q18. It mainly tests Ratios & proportions, Area & perimeter.

A decorative window is made up of a rectangle with semicircles on either end. The ratio of $AD$ to $AB$ is $3:2$ and $AB = 30$ inches. What is the ratio of the area of the rectangle to the combined areas of the semicircles?

一个装饰窗户由一个矩形两端各有一个半圆组成。$AD$ 与 $AB$ 的比值为 $3:2$，且 $AB = 30$ 英寸。矩形面积与两个半圆面积之和的比值为多少？

(A) $2:3$ $2:3$

(B) $3:2$ $3:2$

(D) $9:\pi$ $9:\pi$

(E) $30:\pi$ $30:\pi$

Answer

Correct choice: (C)

正确答案：（C）

Solution

We can set a proportion: \[\dfrac{AD}{AB}=\dfrac{3}{2}\] We substitute $AB$ with 30 and solve for $AD$. \[\dfrac{AD}{30}=\dfrac{3}{2}\] \[AD=45\] We calculate the combined area of semicircle by putting together semicircle $AB$ and $CD$ to get a circle with radius $15$. Thus, the area is $225\pi$. The area of the rectangle is $30\cdot 45=1350$. We calculate the ratio: \[\dfrac{1350}{225\pi}=\dfrac{6}{\pi}\Rightarrow\boxed{\textbf{(C)}\ 6:\pi}\]

设比例： \[ \dfrac{AD}{AB} = \dfrac{3}{2} \] 代入 $AB = 30$ 解得 $AD$： \[ \dfrac{AD}{30} = \dfrac{3}{2} \] \[ AD = 45 \] 两个半圆合成一个半径为 15 的圆，面积为 $225\pi$。矩形面积为 $30 \cdot 45 = 1350$。比值为： \[ \dfrac{1350}{225\pi} = \dfrac{6}{\pi} \Rightarrow \boxed{\textbf{(C)}\ 6:\pi} \]

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