Question

下图阴影区域称为鲨鱼鳍形弓月，是 Leonardo da Vinci 研究过的图形。它由第一象限内圆心为 $(0,0)$ 、半径为 $3$ 的圆弧，第一象限内圆心为 $(0,\tfrac{3}{2})$ 、半径为 $\tfrac{3}{2}$ 的圆弧，以及从 $(0,0)$ 到 $(3,0)$ 的线段围成。该鲨鱼鳍形弓月的面积是多少？

The shaded region below is called a shark's fin falcata, a figure studied by Leonardo da Vinci. It is bounded by the portion of the circle of radius $3$ and center $(0,0)$ that lies in the first quadrant, the portion of the circle with radius $\tfrac{3}{2}$ and center $(0,\tfrac{3}{2})$ that lies in the first quadrant, and the line segment from $(0,0)$ to $(3,0).$ What is the area of the shark's fin falcata?

$\dfrac{4\pi}{5}$

$\dfrac{9\pi}{8}$

$\dfrac{4\pi}{3}$

$\dfrac{7\pi}{5}$

$\dfrac{3\pi}{2}$

答案：B

知识点：圆面积扇形面积分割

难度评级：1020

解答：

外边界是半径为 $3$ 的四分之一圆，所以面积为 $\frac14\pi\cdot3^2=\frac{9\pi}{4}$ 。

内边界是半径为 $\frac32$ 的圆的右半部分，所以面积为 $\frac12\pi\left(\frac32\right)^2=\frac{9\pi}{8}$ 。

阴影面积就是二者之差， $\frac{9\pi}{4}-\frac{9\pi}{8}=\frac{9\pi}{8}$ 。

所以正确答案是 B。

The larger boundary is a quarter circle of radius $3$ , so its area is $\frac14\pi\cdot3^2=\frac{9\pi}{4}$ .

The inner boundary is the right half of a circle of radius $\frac32$ , so its area is $\frac12\pi\left(\frac32\right)^2=\frac{9\pi}{8}$ .

The shaded area is the difference, $\frac{9\pi}{4}-\frac{9\pi}{8}=\frac{9\pi}{8}$ .

Thus, the correct answer is B.

2015 AMC 10B 第 9 题

答案：B

难度评级：1020

解答：

其他年份的第 9 题