2015 AMC 10B Problem 9

Problem 9 of 25EasierGeometry

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)

(3,0).

What is the area of the shark’s fin falcata?

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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.

Tagged: circle area · sector · area decomposition