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Difference between revisions of "2021 Fall AMC 10A Problems/Problem 19"

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<math>19</math>. A disk of radius <math>1</math> rolls all the way around in the inside of a square of side length <math>s>4</math> and sweeps out a region of area <math>A</math>. A second disk of radius <math>1</math> rolls all the way around the outside of the same square and sweeps out a region of area <math>2A</math>. The value of <math>s</math> can be written as <math>a + \dfrac{b\pi}{c}</math>, where <math>a,b,</math> and <math>c</math> are positive integers and <math>b</math> and <math>c</math> are relatively prime. What is <math>a+b+c?</math>
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==Problem==
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A disk of radius <math>1</math> rolls all the way around the inside of a square of side length <math>s>4</math> and sweeps out a region of area <math>A</math>. A second disk of radius <math>1</math> rolls all the way around the outside of the same square and sweeps out a region of area <math>2A</math>. The value of <math>s</math> can be written as <math>a+\frac{b\pi}{c}</math>, where <math>a,b</math>, and <math>c</math> are positive integers and <math>b</math> and <math>c</math> are relatively prime. What is <math>a+b+c</math>?
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<math>\textbf{(A)} ~10\qquad\textbf{(B)} ~11\qquad\textbf{(C)} ~12\qquad\textbf{(D)} ~13\qquad\textbf{(E)} ~14</math>
  
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<math>\textbf{(A) }10\qquad\textbf{(B) }11\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }14</math>
 
 
==Solution 1==
 
==Solution 1==
 
The side length of the inner square traced out by the disk with radius <math>1</math> is <math>s-4</math>. However, there is a little triangle piece at each corner where the disk never sweeps out. The combined area of these <math>4</math> pieces is <math>(1+1)^2-\pi\cdot1^2=4-\pi</math>. As a result, <math>A=s^2-(s-4)^2-(4-\pi)=8s-20+\pi</math>.
 
The side length of the inner square traced out by the disk with radius <math>1</math> is <math>s-4</math>. However, there is a little triangle piece at each corner where the disk never sweeps out. The combined area of these <math>4</math> pieces is <math>(1+1)^2-\pi\cdot1^2=4-\pi</math>. As a result, <math>A=s^2-(s-4)^2-(4-\pi)=8s-20+\pi</math>.

Revision as of 15:10, 28 November 2021

Problem

A disk of radius $1$ rolls all the way around the inside of a square of side length $s>4$ and sweeps out a region of area $A$. A second disk of radius $1$ rolls all the way around the outside of the same square and sweeps out a region of area $2A$. The value of $s$ can be written as $a+\frac{b\pi}{c}$, where $a,b$, and $c$ are positive integers and $b$ and $c$ are relatively prime. What is $a+b+c$?

$\textbf{(A)} ~10\qquad\textbf{(B)} ~11\qquad\textbf{(C)} ~12\qquad\textbf{(D)} ~13\qquad\textbf{(E)} ~14$

Solution 1

The side length of the inner square traced out by the disk with radius $1$ is $s-4$. However, there is a little triangle piece at each corner where the disk never sweeps out. The combined area of these $4$ pieces is $(1+1)^2-\pi\cdot1^2=4-\pi$. As a result, $A=s^2-(s-4)^2-(4-\pi)=8s-20+\pi$.

Now, we consider the second disk. The part it sweeps is comprised of $4$ quarter circles with radius $2$ and $4$ rectangles with a side lengths of $2$ and $s$. When we add it all together, $2A=8s+4\pi\implies A=4s+2\pi$. $8s-20+\pi=4s+2\pi$ so $s=5+\frac{\pi}{4}$. Finally, $5+1+4=\boxed{\textbf{(A) } 10}$.

~MathFun1000 (Inspired by Way Tan)

Solution 2

The area of the region covered by the first disk is \begin{align*} A & = s^2 - \left( s - 4 \right)^2 - \left( 2^2 - \pi 1^2 \right) \\ & = 8 s - 20 + \pi . \end{align*}

The area of the region covered by the second disk is \begin{align*} 2 A & = 4 \cdot 2 s + \pi 2^2 \\ & = 8 s + 4\pi . \end{align*}

These two equations jointly imply $s = 5 + \frac{1 \cdot \pi}{4}$.

Therefore, the answer is $\boxed{\textbf{(A) }10}$.

~Steven Chen (www.professorchenedu.com)

See Also

2021 Fall AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 18 		Followed by
Problem 20
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

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