Difference between revisions of "2021 AMC 10B Problems/Problem 7"

(Solution)
(Video Solution by TheBeautyofMath)
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<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>
 
==Solution==
 
==Solution==
[asy]
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<asy>
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/* diagram made by samrocksnature */
 
pair A=(10,0);
 
pair A=(10,0);
 
pair B=(-10,0);
 
pair B=(-10,0);
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dot((0,7));
 
dot((0,7));
 
draw((0,7)--(0,0));
 
draw((0,7)--(0,0));
label("<math>7</math>",(0,3.5),E);
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label("$7$",(0,3.5),E);
label("<math>l</math>",(-9,0),S);
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label("$l$",(-9,0),S);
[/asy]
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</asy>
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>
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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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~ samrocksnature
 +
 
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== Video Solution by OmegaLearn (Area of Circles and Logic) ==
 +
https://youtu.be/yPIFmrJvUxM
 +
 
 +
~ pi_is_3.14
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 +
==Video Solution by TheBeautyofMath==
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https://youtu.be/GYpAm8v1h-U?t=206
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 +
~IceMatrix
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==Video Solution by Interstigation==
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https://youtu.be/DvpN56Ob6Zw?t=555
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 +
~Interstigation
 +
 
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{{AMC10 box|year=2021|ab=B|num-b=6|num-a=8}}

Revision as of 01:07, 23 February 2021

Problem

In a plane, four circles with radii $1,3,5,$ and $7$ are tangent to line $l$ at the same point $A,$ but they may be on either side of $l$. 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

[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("$l$",(-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

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

2021 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 6
Followed by
Problem 8
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All AMC 10 Problems and Solutions