AMC8 2023

AMC8 2023 · Q12

AMC8 2023 · Q12. It mainly tests Area & perimeter.

The figure below shows a large white circle with a number of smaller white and shaded circles in its interior. What fraction of the interior of the large white circle is shaded?

下图显示一个大的白色圆圈，内部有多个较小的白色和阴影圆圈。大的白色圆圈内部的阴影部分占几分之几？

(A) \frac{1}{4} \frac{1}{4}

(B) \frac{11}{36} \frac{11}{36}

(D) \frac{19}{36} \frac{19}{36}

(E) \frac{5}{9} \frac{5}{9}

Answer

Correct choice: (B)

正确答案：（B）

Solution

The large circle has radius 3 (in the standard diagram units), so its area is \[ \pi \cdot 3^2 = 9\pi. \] The shaded area consists of: - three small shaded circles, each with radius $\frac{1}{2}$, so each has area \[ \pi \left(\frac{1}{2}\right)^2 = \frac{\pi}{4}. \] Their total area is \[ 3 \times \frac{\pi}{4} = \frac{3\pi}{4}. \] - the equivalent of two full shaded circles of radius 1 (or the shaded regions that together contribute the same area as two such circles), giving area \[ 2 \times \pi \cdot 1^2 = 2\pi. \] The total shaded area is therefore \[ \frac{3\pi}{4} + 2\pi = \frac{3\pi}{4} + \frac{8\pi}{4} = \frac{11\pi}{4}. \] The fraction of the large circle that is shaded is \[ \frac{\frac{11\pi}{4}}{9\pi} = \frac{11}{4} \times \frac{1}{9} = \frac{11}{36}. \]

大圆的半径为3（标准图单位），其面积为 \[\pi \cdot 3^2 = 9\pi.\] 阴影面积包括： - 三个小阴影圆，每个半径$\frac{1}{2}$，每个面积 \[\pi \left(\frac{1}{2}\right)^2 = \frac{\pi}{4}.\] 总面积为 \[3 \times \frac{\pi}{4} = \frac{3\pi}{4}.\] - 等效于两个半径为1的完整阴影圆（或阴影区域的总面积相当于两个这样的圆），面积为 \[2 \times \pi \cdot 1^2 = 2\pi.\] 因此，总阴影面积为 \[\frac{3\pi}{4} + 2\pi = \frac{3\pi}{4} + \frac{8\pi}{4} = \frac{11\pi}{4}.\] 大圆的阴影分数为 \[\frac{\frac{11\pi}{4}}{9\pi} = \frac{11}{4} \times \frac{1}{9} = \frac{11}{36}.\]

Topics

GE.007 Area & perimeter