MathDB
Shark's Fin Falcata

Source: 2015 AMC 10b #9

February 26, 2015
geometryAsymptoteLaTeXAMC

Problem Statement

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 33 and center (0,0)(0,0) that lies in the first quadrant, the portion of the circle with radius 32\tfrac{3}{2} and center (0,32)(0,\tfrac{3}{2}) that lies in the first quadrant, and the line segment from (0,0)(0,0) to (3,0)(3,0). What is the area of the shark's fin falcata? [asy] import cse5;pathpen=black;pointpen=black; size(1.5inch); D(MP("x",(3.5,0),S)--(0,0)--MP("\frac{3}{2}",(0,3/2),W)--MP("y",(0,3.5),W)); path P=(0,0)--MP("3",(3,0),S)..(3*dir(45))..MP("3",(0,3),W)--(0,3)..(3/2,3/2)..cycle; draw(P,linewidth(2)); fill(P,gray); [/asy] <spanclass=latexbold>(A)</span>4π5<spanclass=latexbold>(B)</span>9π8<spanclass=latexbold>(C)</span>4π3<spanclass=latexbold>(D)</span>7π5<spanclass=latexbold>(E)</span>3π2<span class='latex-bold'>(A) </span> \dfrac{4\pi}{5} \qquad<span class='latex-bold'>(B) </span> \dfrac{9\pi}{8} \qquad<span class='latex-bold'>(C) </span> \dfrac{4\pi}{3} \qquad<span class='latex-bold'>(D) </span> \dfrac{7\pi}{5} \qquad<span class='latex-bold'>(E) </span> \dfrac{3\pi}{2}