2010 AMC 8 第 18 题

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

小提示：

用比值

AD:AB=3:2

，将长方形高度用

AB

表示

Use the ratio

AD:AB=3:2

to express the rectangle height in terms of

AB

.

大提示：

两个半圆合起来是一个整圆

The two semicircles combine to one full circle.

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}\text{。}

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

小提示：

大提示：

解答：

其他年份的第 18 题