Difference between revisions of "2021 AMC 10B Problems/Problem 7"
(Created page with "no one knows yet") |
Cellsecret (talk | contribs) (→Video Solution by TheBeautyofMath) |
||
| (10 intermediate revisions by 6 users not shown) | |||
| Line 1: | Line 1: | ||
| − | + | ==Problem== | |
| − | one | + | 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>? |
| − | + | ||
| − | + | <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== | ||
| + | <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 <math>49 \pi + (25-9) \pi=65 \pi \rightarrow \boxed{\textbf{(D)}}</math> | ||
| + | |||
| + | ~ 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 | ||
| + | |||
| + | {{AMC10 box|year=2021|ab=B|num-b=6|num-a=8}} | ||
Revision as of 01:07, 23 February 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
![[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]](http://latex.artofproblemsolving.com/5/4/8/5481e45f8fbf33deba5c7afb818793e2e33f4269.png)
After a bit of wishful thinking and inspection, we find that the above configuration maximizes our area, which is 
~ samrocksnature
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
| 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 | ||
