AMC8 2008

AMC8 2008 · Q25

AMC8 2008 · Q25. It mainly tests Percent, Area & perimeter.

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 black?

玛吉获奖的艺术设计如图所示。最小的圆半径为2英寸，每一个后续圆的半径增加2英寸。设计中大约百分之多少是黑色的？

(A) 42 42

(B) 44 44

(D) 46 46

(E) 48 48

Answer

Correct choice: (A)

正确答案：（A）

Solution

Answer (A): $\begin{array}{|c|c|c|} \hline \text{Circle #} & \text{Radius} & \text{Area} \\ \hline 1 & 2 & 4\pi \\ \hline 2 & 4 & 16\pi \\ \hline 3 & 6 & 36\pi \\ \hline 4 & 8 & 64\pi \\ \hline 5 & 10 & 100\pi \\ \hline 6 & 12 & 144\pi \\ \hline \end{array}$ The total black area is $4\pi + (36-16)\pi + (100-64)\pi = 60\pi\ \text{in}^2.$ So the percent of the design that is black is $100 \times \frac{60\pi}{144\pi} = 100 \times \frac{5}{12}$ or about $42\%.$

答案（A）： $\begin{array}{|c|c|c|} \hline \text{圆编号} & \text{半径} & \text{面积} \\ \hline 1 & 2 & 4\pi \\ \hline 2 & 4 & 16\pi \\ \hline 3 & 6 & 36\pi \\ \hline 4 & 8 & 64\pi \\ \hline 5 & 10 & 100\pi \\ \hline 6 & 12 & 144\pi \\ \hline \end{array}$ 黑色部分的总面积为 $4\pi + (36-16)\pi + (100-64)\pi = 60\pi\ \text{in}^2.$ 因此，该图案中黑色所占的百分比为 $100 \times \frac{60\pi}{144\pi} = 100 \times \frac{5}{12}$，约为 $42\%.$

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