Difference between revisions of "2021 AMC 10B Problems/Problem 7"
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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== | ||
− | D | + | [asy] |
+ | 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("<math>7</math>",(0,3.5),E); | ||
+ | label("<math>l</math>",(-9,0),S); | ||
+ | [/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> |
Revision as of 18:53, 11 February 2021
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] 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("",(0,3.5),E); label("",(-9,0),S); [/asy] After a bit of wishful thinking and inspection, we find that the above configuration maximizes our area.