AMC10 2015 B

AMC10 2015 B · Q9

AMC10 2015 B · Q9. It mainly tests Circle theorems, Area & perimeter.

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 of radius $\frac{3}{2}$ and center $(0, \frac{3}{2})$ 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?

下面的阴影区域称为鲨鱼鳍弯刀，是达·芬奇研究过的图形。它被半径为 3、圆心为 (0,0) 的圆的第一象限部分、半径为 $\frac{3}{2}$、圆心为 $(0, \frac{3}{2})$ 的圆的第一象限部分，以及从 (0,0) 到 (3,0) 的线段所包围。鲨鱼鳍弯刀的面积是多少？

(A) $\frac{4\pi}{5}$ $\frac{4\pi}{5}$

(B) $\frac{9\pi}{8}$ $\frac{9\pi}{8}$

(D) $\frac{7\pi}{5}$ $\frac{7\pi}{5}$

(E) $\frac{3\pi}{2}$ $\frac{3\pi}{2}$

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): The shaded area is obtained by subtracting the area of the semicircle from the area of the quarter circle: $\frac{1}{4}\pi\cdot 3^2-\frac{1}{2}\pi\left(\frac{3}{2}\right)^2=\frac{9\pi}{4}-\frac{9\pi}{8}=\frac{9\pi}{8}.$

答案（B）：阴影部分的面积等于四分之一圆的面积减去半圆的面积： $\frac{1}{4}\pi\cdot 3^2-\frac{1}{2}\pi\left(\frac{3}{2}\right)^2=\frac{9\pi}{4}-\frac{9\pi}{8}=\frac{9\pi}{8}.$

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