Question

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? \t\t

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

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

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

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

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

Solution:

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 Problem 9

Solution:

Problem 9 in Other Years