Difference between revisions of "2021 AMC 10B Problems/Problem 7"
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==Solution 1 (Intuition by Graph)== | ==Solution 1 (Intuition by Graph)== | ||
<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((0,7)); | dot((0,7)); | ||
draw((0,7)--(0,0)); | draw((0,7)--(0,0)); | ||
| Line 19: | Line 19: | ||
label("$\ell$",(-9,0),S); | label("$\ell$",(-9,0),S); | ||
</asy> | </asy> | ||
| − | 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= | + | 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=\boxed{\textbf{(D) }65\pi}.</math> |
~ samrocksnature | ~ samrocksnature | ||
Revision as of 14:48, 20 July 2021
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 1 (Intuition by Graph)
![[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((0,7)); draw((0,7)--(0,0)); label("$7$",(0,3.5),E); label("$\ell$",(-9,0),S); [/asy]](http://latex.artofproblemsolving.com/a/2/4/a24c437e2200d6cab4cb6de489afb4b03c58efcf.png)
After a bit of wishful thinking and inspection, we find that the above configuration maximizes our area, which is 
~ samrocksnature
Solution 2 (Intuition in Words)
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 the desired region, we draw the circle with radius
south to 
by intuition. - 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. The diagram in Solution 1 shows one possible configuration of the four circles.
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 in Solution 1 shows one possible configuration of the four circles.
Together, the answer is 
~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
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. 
