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

(Adding a solution)
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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>
 
<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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==Solution==
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==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>.
  
 
Now, we consider the second disk. The part it sweeps is comprised of <math>4</math> quarter circles with radius <math>2</math> and <math>4</math> rectangles with a side lengths of <math>2</math> and <math>s</math>. When we add it all together, <math>2A=8s+4\pi\implies A=4s+2\pi</math>. <math>8s-20+\pi=4s+2\pi</math> so <math>s=5+\frac{\pi}{4}</math>. Finally, <math>5+1+4=\boxed{\textbf{(A) } 10}</math>.
 
Now, we consider the second disk. The part it sweeps is comprised of <math>4</math> quarter circles with radius <math>2</math> and <math>4</math> rectangles with a side lengths of <math>2</math> and <math>s</math>. When we add it all together, <math>2A=8s+4\pi\implies A=4s+2\pi</math>. <math>8s-20+\pi=4s+2\pi</math> so <math>s=5+\frac{\pi}{4}</math>. Finally, <math>5+1+4=\boxed{\textbf{(A) } 10}</math>.
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 +
 +
== Solution 2 ==
 +
The area of the region covered by the first disk is
 +
<cmath>
 +
\begin{align*}
 +
A & = s^2 - \left( s - 4 \right)^2 - \left( 2^2 - \pi 1^2 \right) \\
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& = 8 s - 20 + \pi .
 +
\end{align*}
 +
</cmath>
 +
 +
The area of the region covered by the second disk is
 +
<cmath>
 +
\begin{align*}
 +
2 A & = 4 \cdot 2 s + \pi 2^2 \\
 +
& = 8 s +  4\pi .
 +
\end{align*}
 +
</cmath>
 +
 +
These two equations jointly imply <math>s = 5 + \frac{1 \cdot \pi}{4}</math>.
 +
 +
Therefore, the answer is <math>\boxed{\textbf{(A) }10}</math>.
 +
 +
~Steven Chen (www.professorchenedu.com)
  
 
~MathFun1000 (Inspired by Way Tan)
 
~MathFun1000 (Inspired by Way Tan)

Revision as of 20:20, 25 November 2021

$19$. A disk of radius $1$ rolls all the way around in 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 $a + \dfrac{b\pi}{c}$, 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, $5+1+4=\boxed{\textbf{(A) } 10}$.


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)

~MathFun1000 (Inspired by Way Tan)

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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