MathDB
Disk Rolling

Source: 2022 AMC 10A #19

November 11, 2021
AMCAMC 10AMC 10 A

Problem Statement

A disk of radius 11 rolls all the way around the inside of a square of side length s>4s>4 and sweeps out a region of area AA. A second disk of radius 11 rolls all the way around the outside of the same square and sweeps out a region of area 2A2A. The value of ss can be written as a+bπca+\frac{b\pi}{c}, where a,ba,b, and cc are positive integers and bb and cc are relatively prime. What is a+b+ca+b+c?
<spanclass=latexbold>(A)</span> 10<spanclass=latexbold>(B)</span> 11<spanclass=latexbold>(C)</span> 12<spanclass=latexbold>(D)</span> 13<spanclass=latexbold>(E)</span> 14<span class='latex-bold'>(A)</span> ~10\qquad<span class='latex-bold'>(B)</span> ~11\qquad<span class='latex-bold'>(C)</span> ~12\qquad<span class='latex-bold'>(D)</span> ~13\qquad<span class='latex-bold'>(E)</span> ~14