AMC 8 2008 Problem 25

AMC8 2008 年第 25 题

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Margie's winning art design is shown. The smallest circle has radius 2 inches, with each successive circle's radius increasing by 2 inches. Which of the following is closest to the percent of the design that is black?

下图展示了 Margie 获奖的艺术设计作品。最小的圆的半径为 2 英寸，每个连续圆的半径增加 2 英寸。以下哪项最接近黑色图案占整个图案的百分比？

A.42
B.44
C.46
D.47
E.49

正确答案 · Correct Answer

解析 · Solution

解法1

常规解法 · 逐环面积半径 2、4、6、8、10、12 对应面积

4\pi,16\pi,36\pi,64\pi,100\pi,144\pi

。黑色是从内往外第 1、3、5 环：

4\pi+(36\pi-16\pi)+(100\pi-64\pi)=4\pi+20\pi+36\pi=60\pi

，占总面积

144\pi

的

\dfrac{5}{12}\approx41.7\%

，最接近 42。

Solution 1

Elementary · area ring by ring The radii 2, 4, 6, 8, 10, 12 give areas

4\pi,16\pi,36\pi,64\pi,100\pi,144\pi

. The black regions are the 1st, 3rd, and 5th rings from the center:

4\pi+(36\pi-16\pi)+(100\pi-64\pi)=60\pi

. As a fraction of

144\pi

this is

\dfrac{5}{12}\approx41.7\%

, closest to 42.

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