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 19:53, 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. 
