Question

两条不同直线通过三个同心圆的圆心，三个圆的半径分别为 $3$ 、 $2$ 、 $1$ 。图中阴影区域面积是非阴影区域面积的 $\dfrac{8}{13}$ 。两条直线所成锐角的弧度数是多少？（注： $\pi$ 弧度等于 $180$ 度。）

Two distinct lines pass through the center of three concentric circles of radii $3,$ $2,$ and $1.$ The area of the shaded region in the diagram is $\dfrac{8}{13}$ of the area of the unshaded region. What is the radian measure of the acute angle formed by the two lines? (Note: $\pi$ radians is $180$ degrees.)

$\dfrac{\pi}{8}$

$\dfrac{\pi}{7}$

$\dfrac{\pi}{6}$

$\dfrac{\pi}{5}$

$\dfrac{\pi}{4}$

答案：B

知识点：扇形圆面积圆环一次方程

难度评级：1880

解答：

设锐角为 $\theta$ 。阴影区域分三部分：单位圆中的两个锐角扇形总面积为 $\theta$ ；半径 $1$ 到 $2$ 的圆环中两个钝角扇形总面积为 $3(\pi - \theta)$ ；半径 $2$ 到 $3$ 的圆环中两个锐角扇形总面积为 $5\theta$ 。

相加得到阴影面积 $\theta + 3(\pi - \theta) + 5\theta = 3\pi + 3\theta.$

阴影面积是非阴影面积的 $\dfrac{8}{13}$ ，因此是总面积 $9\pi$ 的 $\dfrac{8}{21}$ 。于是解得 $\theta = \dfrac{\pi}{7}$ 。 $3\pi + 3\theta = \dfrac{8}{21}(9\pi) = \dfrac{24\pi}{7},$

所以正确答案是 B。

Let $\theta$ be the acute angle. The shaded region has three parts: two acute sectors of the unit disk with total area $\theta,$ two obtuse sectors of the ring between radii $1$ and $2$ with total area $3(\pi - \theta),$ and two acute sectors of the ring between radii $2$ and $3$ with total area $5\theta.$

Adding these gives a shaded area of $\theta + 3(\pi - \theta) + 5\theta = 3\pi + 3\theta.$

The shaded region is $\dfrac{8}{13}$ of the unshaded region, so it is $\dfrac{8}{21}$ of the total area $9\pi.$ Then $3\pi + 3\theta = \dfrac{8}{21}(9\pi) = \dfrac{24\pi}{7},$ which gives $\theta = \dfrac{\pi}{7}.$

Thus, the correct answer is B.

2004 AMC 10A 第 21 题

答案：B

难度评级：1880

解答：

其他年份的第 21 题