Question

下图显示一个大的未涂色圆，内部有若干较小的未涂色圆和阴影圆。大未涂色圆内部的面积中，阴影部分占几分之几？

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

$\dfrac{1}{4}$

$\dfrac{11}{36}$

$\dfrac{1}{3}$

$\dfrac{19}{36}$

$\dfrac{5}{9}$

答案：B

知识点：圆面积面积比

难度评级：1370

小提示：

圆面积与半径的平方成正比

Circle areas are proportional to the square of the radius

大提示：

用最小圆的面积作为单位来数阴影面积

Count shaded area in units of the smallest circle’s area

视频讲解：

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文字解答：

不妨设每个最小圆的半径为 $1$ 。

这样有 $3$ 个阴影单位圆，总面积为 $3 \cdot 1^2 \pi = 3 \pi$ 。

另外还有一个半径为 $4$ 的阴影圆，其中挖去了两个半径为 $2$ 的未涂色圆。

这部分额外阴影面积为 $\begin{align*} 4^2 \pi - 2 \cdot 2^2 \pi &= 16 \pi - 8 \pi \\ &= 8 \pi\text{。} \end{align*}$

阴影总面积为 $8\pi + 3\pi = 11\pi\text{。}$ 大未涂色圆的面积为 $6^2\pi = 36\pi\text{。}$ 所以所求分数为 $\dfrac{11\pi}{36\pi} = \dfrac{11}{36}\text{。}$

所以正确答案是 B。

Let each smallest circle have radius $1.$

This means that there are $3$ shaded unit circles, which total to $3 \cdot 1^2 \pi = 3 \pi$ area.

There is also a shaded circle with radius $4$ with two unshaded circles of radius $2$ inside.

This gives us an extra shaded area of $\begin{align*} 4^2 \pi - 2 \cdot 2^2 \pi &= 16 \pi - 8 \pi \\ &= 8 \pi. \end{align*}$

The total shaded area is $8\pi + 3\pi = 11\pi.$ The area of the large unshaded circle is $6^2\pi = 36\pi.$ Therefore, the desired fraction is $\dfrac{11\pi}{36\pi} = \dfrac{11}{36}.$

Thus, B is the correct answer.

2023 AMC 8 第 12 题