Question

一扇装饰窗由一个长方形和两端的半圆组成。 $AD$ 与 $AB$ 的比为 $3:2$ ，且 $AB=30$ 英寸。长方形面积与两个半圆面积之和的比是多少？

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?

$\ 2:3$

$\ 3:2$

$\ 6:\pi$

$\ 9:\pi$

$\ 30 :\pi$

答案：C

知识点：圆面积矩形面积比

难度评级：1240

解答：

两个半圆合起来是一个直径为 $d=30$ 的圆，其半径为 $\dfrac{d}{2}$ 。因此两个半圆的总面积为 $(\dfrac{d}{2})^2 \cdot \pi = \dfrac{\pi d^2}{4}.$

因为 $AD:AB=3:2$ 且 $AB=d$ ，所以 $AD = \dfrac 32 d$ 。长方形面积为 $\dfrac 32 d^2$ 。所求比值为 $\dfrac{\dfrac 32 d^2}{\dfrac{\pi d^2}{4}} = \dfrac{6}{\pi}$ 。

所以正确答案是 C。

Combining the semicircles would make a circle of diameter $d=30.$ This would make the radius equal to $\dfrac{d}{2}.$ Therefore, the combined area of the semicircles is $(\dfrac{d}{2})^2 \cdot \pi = \dfrac{\pi d^2}{4}.$

Side $AD = \dfrac 32 d$ because $AD:AB=3:2$ and $AB=d.$ The area of the rectangle is therefore $\dfrac 32 d^2.$ The ratio of the area of the rectangle to the area of the semicircles is $\dfrac{\dfrac 32 d^2}{\dfrac{\pi d^2}{4}} = \dfrac{6}{\pi}.$

Thus, the answer is C .

2010 AMC 8 第 18 题

答案：C

难度评级：1240

解答：

其他年份的第 18 题