1998 AMC 12 第 16 题

Question

图形由一个圆和两个直径分别为

a

与

b

的半圆组成，三者的圆心共线。阴影区域面积与非阴影区域面积之比为

The figure shown is the union of a circle and two semicircles of diameters

a

and

b,

all of whose centers are collinear. The ratio of the area of the shaded region to that of the unshaded region is

\sqrt{\frac ab}

\frac ab

\frac{a^2}{b^2}

\frac{a+b}{2b}

\frac{a^2+2ab}{b^2+2ab}

用大半圆和一个小半圆分别表示两个区域的面积

Express each region using the large semicircle and one small semicircle

所得两个面积都含有

a+b

的一个公因式

Both resulting areas factor a common multiple of

a+b

阴影面积为

\begin{aligned} &\frac{\pi}{2}\left(\frac{a+b}{2}\right)^2 +\frac{\pi}{2}\left(\frac a2\right)^2\\ &\qquad-\frac{\pi}{2}\left(\frac b2\right)^2 =\frac{\pi a(a+b)}4 \end{aligned}\text{。}

同理，非阴影面积为

\frac{\pi b(a+b)}{4}

。二者之比为

\frac{a}{b}

，正确答案是 B。

The shaded area is

\begin{aligned} &\frac{\pi}{2}\left(\frac{a+b}{2}\right)^2 +\frac{\pi}{2}\left(\frac a2\right)^2\\ &\qquad-\frac{\pi}{2}\left(\frac b2\right)^2 =\frac{\pi a(a+b)}4. \end{aligned}

Similarly, the unshaded area is

\frac{\pi b(a+b)}{4}.

Their ratio is

\frac{a}{b},

so B is correct.

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