2008 AMC 8 第 25 题

Question

玛吉获奖的艺术设计如图所示。最小的圆半径为

2

英寸，每个后续圆的半径都增加

2

英寸。这个设计中大约百分之多少被涂上阴影？

Margie’s winning art design is shown. The smallest circle has radius

2

inches, with each successive circle’s radius increasing by

2

inches. Approximately what percent of the design is shaded?

\ 42

\ 44

\ 45

\ 46

\ 48

设计的总面积是最大圆的面积

The total design area is the area of the largest circle.

阴影区域是半径每次增加

2

的交替圆盘或圆环

The shaded regions are alternating disks or annuli with radii increasing by

2.

最大圆的半径为

12

，所以整个设计的面积为

12^2\pi=144\pi

。

阴影部分的面积为

2^2\pi=4\pi

、

(6^2-4^2)\pi=20\pi

和

(10^2-8^2)\pi=36\pi

。

阴影总面积为

4\pi+20\pi+36\pi=60\pi

，所以阴影比例为

\dfrac{60\pi}{144\pi}=\dfrac{5}{12}

，约为

42\%

。

所以正确答案是 A。

The largest circle has radius

12

, so the entire design has area

12^2\pi=144\pi

.

The shaded parts have areas

2^2\pi=4\pi

,

(6^2-4^2)\pi=20\pi

, and

(10^2-8^2)\pi=36\pi

.

The total shaded area is

4\pi+20\pi+36\pi=60\pi

, so the shaded fraction is

\dfrac{60\pi}{144\pi}=\dfrac{5}{12}

, which is about

42\%

.

Thus, A is the correct answer.

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