Difference between revisions of "2021 AMC 10B Problems/Problem 7"
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==Problem== | ==Problem== | ||
| − | In a plane, four circles with radii <math>1,3,5,</math> and <math>7</math> are tangent to line <math> | + | 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> | ||
| + | |||
==Solution== | ==Solution== | ||
| + | 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 region <math>S,</math> we draw the circle with radius <math>5</math> south to <math>\ell.</math></li><p> | ||
| + | <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. </li> | ||
| + | </ol> | ||
| + | The diagram below shows one possible configuration of the four circles: | ||
<asy> | <asy> | ||
| − | /* diagram made by samrocksnature */ | + | /* diagram made by samrocksnature, edited by MRENTHUSIASM */ |
pair A=(10,0); | pair A=(10,0); | ||
pair B=(-10,0); | pair B=(-10,0); | ||
draw(A--B); | draw(A--B); | ||
| − | + | filldraw(circle((0,7),7),yellow); | |
| − | + | filldraw(circle((0,-5),5),yellow); | |
| − | + | filldraw(circle((0,-3),3),white); | |
| − | + | filldraw(circle((0,-1),1),white); | |
| − | dot((0, | + | dot((0,0)); |
| − | + | label("$A$",(0,0),(0,1.5)); | |
| − | + | label("$\ell$",(10,0),(1.5,0)); | |
| − | label("$ | ||
</asy> | </asy> | ||
| − | + | Together, the answer is <math>\pi\cdot7^2+\pi\cdot5^2-\pi\cdot3^2=\boxed{\textbf{(D) }65\pi}.</math> | |
| − | ~ samrocksnature | + | ~samrocksnature ~MRENTHUSIASM |
== Video Solution by OmegaLearn (Area of Circles and Logic) == | == Video Solution by OmegaLearn (Area of Circles and Logic) == | ||
| Line 27: | Line 34: | ||
~ pi_is_3.14 | ~ 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 | ||
| + | |||
| + | ==Video Solution== | ||
| + | https://youtu.be/3jC_yOKA7xE | ||
| + | ~Education, the Study of Everything | ||
| + | ==See Also== | ||
{{AMC10 box|year=2021|ab=B|num-b=6|num-a=8}} | {{AMC10 box|year=2021|ab=B|num-b=6|num-a=8}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 17:22, 16 August 2022
Contents
Problem
In a plane, four circles with radii 
and 
are tangent to line 
at the same point 
but they may be on either side of . Region 
consists of all the points that lie inside exactly one of the four circles. What is the maximum possible area of region ?
Solution
Suppose that line is horizontal, and each circle lies either north or south to 
We construct the circles one by one:
- Without the loss of generality, we draw the circle with radius
north to 
- To maximize the area of region
we draw the circle with radius 
south to - Now, we need to subtract the circle with radius
at least. The optimal situation is that the circle with radius encompasses the circle with radius 
in which we do not need to subtract more. That is, the two smallest circles are on the same side of 
but can be on either side.
we draw the circle with radius 
south to
at least. The optimal situation is that the circle with radius 
in which we do not need to subtract more. That is, the two smallest circles are on the same side of 
but can be on either side. The diagram below shows one possible configuration of the four circles:
![[asy] /* diagram made by samrocksnature, edited by MRENTHUSIASM */ pair A=(10,0); pair B=(-10,0); draw(A--B); filldraw(circle((0,7),7),yellow); filldraw(circle((0,-5),5),yellow); filldraw(circle((0,-3),3),white); filldraw(circle((0,-1),1),white); dot((0,0)); label("$A$",(0,0),(0,1.5)); label("$\ell$",(10,0),(1.5,0)); [/asy]](http://latex.artofproblemsolving.com/0/9/6/09622ff57f1014e51375086c90e7e2cf7dd463c6.png)
Together, the answer is 
~samrocksnature ~MRENTHUSIASM
Video Solution by OmegaLearn (Area of Circles and Logic)
~ 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
Video Solution
~Education, the Study of Everything
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. 
