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Difference between revisions of "2021 AMC 10B Problems/Problem 7"
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==Problem==
 
==Problem==
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In a plane, four circles with radii <math>1,3,5,</math> and <math>7</math> are tangent to line <math>l</math> at the same point <math>A,</math> but they may be on either side of <math>l</math>. Region <math>S</math> consists of all the points that lie inside exactly one of the four circles. What is the maximum possible area of region <math>S</math>?
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In a plane, four circles with radii <math>1,3,5,</math> and <math>7</math> are tangent to line <math>\ell</math> at the same point <math>A,</math> but they may be on either side of <math>\ell</math>. Region <math>S</math> consists of all the points that lie inside exactly one of the four circles. What is the maximum possible area of region <math>S</math>?
  
 
<math>\textbf{(A) }24\pi \qquad \textbf{(B) }32\pi \qquad \textbf{(C) }64\pi \qquad \textbf{(D) }65\pi \qquad \textbf{(E) }84\pi</math>
 
<math>\textbf{(A) }24\pi \qquad \textbf{(B) }32\pi \qquad \textbf{(C) }64\pi \qquad \textbf{(D) }65\pi \qquad \textbf{(E) }84\pi</math>
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==Solution==
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==Solution 1==
 
<asy>
 
<asy>
 
/* diagram made by samrocksnature */
 
/* diagram made by samrocksnature */
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draw((0,7)--(0,0));
 
draw((0,7)--(0,0));
 
label("$7$",(0,3.5),E);
 
label("$7$",(0,3.5),E);
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label("$l$",(-9,0),S);
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label("$\ell$",(-9,0),S);
 
</asy>
 
</asy>
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After a bit of wishful thinking and inspection, we find that the above configuration maximizes our area. <math>49 \pi + (25-9) \pi=65 \pi \rightarrow \boxed{D}</math> ~ samrocksnature
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After a bit of wishful thinking and inspection, we find that the above configuration maximizes our area, which is <math>49 \pi + (25-9) \pi=65 \pi \rightarrow \boxed{\textbf{(D)}}</math>  
  
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{{AMC10 box|year=2021|ab=B|before=[[2021 AMC 10A]]|after=[[2022 AMC 10A]]}}
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~ samrocksnature
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 +
==Solution 2 (Explains Solution 1 Using Intuition)==
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Suppose that line <math>\ell</math> is horizontal, and each circle lies either north or south to <math>\ell.</math> We construct the circles one by one:
 +
<ol style="margin-left: 1.5em;">
 +
  <li>Without the loss of generality, we draw the circle with radius <math>7</math> north to <math>\ell.</math></li><p>
 +
  <li>To maximize the area of the desired region, we draw the circle with radius <math>5</math> south to <math>\ell</math> by intuition.</li><p>
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  <li>Now, we need to subtract the circle with radius <math>3</math> <i><b>at least</b></i>. The optimal situation is that the circle with radius <math>3</math> encompasses the circle with radius <math>1,</math> in which we do not need to subtract more. That is, the two smallest circles are on the same side of <math>\ell,</math> but can be on either side. The diagram in Solution 1 shows one possible configuration of the four circles.</li><p>
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</ol>
 +
 
 +
Together, the answer is <math>7^2\pi+5^2\pi-3^2\pi=\boxed{\textbf{(D) }65\pi}.</math>
 +
 
 +
~MRENTHUSIASM
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 +
== Video Solution by OmegaLearn (Area of Circles and Logic) ==
 +
https://youtu.be/yPIFmrJvUxM
 +
 
 +
~ pi_is_3.14
 +
 
 +
==Video Solution by TheBeautyofMath==
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https://youtu.be/GYpAm8v1h-U?t=206
 +
 
 +
~IceMatrix
 +
==Video Solution by Interstigation==
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https://youtu.be/DvpN56Ob6Zw?t=555
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 +
~Interstigation
 +
 
 +
==See Also==
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{{AMC10 box|year=2021|ab=B|num-b=6|num-a=8}}
 +
{{MAA Notice}}

Revision as of 04:39, 10 April 2021

Problem

In a plane, four circles with radii $1,3,5,$ and $7$ are tangent to line $\ell$ at the same point $A,$ but they may be on either side of $\ell$. Region $S$ consists of all the points that lie inside exactly one of the four circles. What is the maximum possible area of region $S$?

$\textbf{(A) }24\pi \qquad \textbf{(B) }32\pi \qquad \textbf{(C) }64\pi \qquad \textbf{(D) }65\pi \qquad \textbf{(E) }84\pi$

Solution 1

[asy] /* diagram made by samrocksnature */ pair A=(10,0); pair B=(-10,0); draw(A--B); draw(circle((0,-1),1)); draw(circle((0,-3),3)); draw(circle((0,-5),5)); draw(circle((0,7),7)); dot((0,7)); draw((0,7)--(0,0)); label("$7$",(0,3.5),E); label("$\ell$",(-9,0),S); [/asy] After a bit of wishful thinking and inspection, we find that the above configuration maximizes our area, which is $49 \pi + (25-9) \pi=65 \pi \rightarrow \boxed{\textbf{(D)}}$

~ samrocksnature

Solution 2 (Explains Solution 1 Using Intuition)

Suppose that line $\ell$ is horizontal, and each circle lies either north or south to $\ell.$ We construct the circles one by one:

  1. Without the loss of generality, we draw the circle with radius $7$ north to $\ell.$

  2. To maximize the area of the desired region, we draw the circle with radius $5$ south to $\ell$ by intuition.

  3. Now, we need to subtract the circle with radius $3$ at least. The optimal situation is that the circle with radius $3$ encompasses the circle with radius $1,$ in which we do not need to subtract more. That is, the two smallest circles are on the same side of $\ell,$ but can be on either side. The diagram in Solution 1 shows one possible configuration of the four circles.

Together, the answer is $7^2\pi+5^2\pi-3^2\pi=\boxed{\textbf{(D) }65\pi}.$

~MRENTHUSIASM

Video Solution by OmegaLearn (Area of Circles and Logic)

https://youtu.be/yPIFmrJvUxM

~ pi_is_3.14

Video Solution by TheBeautyofMath

https://youtu.be/GYpAm8v1h-U?t=206

~IceMatrix

Video Solution by Interstigation

https://youtu.be/DvpN56Ob6Zw?t=555

~Interstigation

See Also

2021 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 6 		Followed by
Problem 8
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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

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