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2021 AMC 10B Problems/Problem 7: Difference between revisions

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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==
[asy]
<asy>
/* diagram made by samrocksnature */
pair A=(10,0);
pair A=(10,0);
pair B=(-10,0);
pair B=(-10,0);
Line 14: Line 15:
dot((0,7));
dot((0,7));
draw((0,7)--(0,0));
draw((0,7)--(0,0));
label("<math>7</math>",(0,3.5),E);
label("$7$",(0,3.5),E);
label("<math>l</math>",(-9,0),S);
label("$l$",(-9,0),S);
[/asy]
</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>
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> ~ samrocksnature

Revision as of 18:53, 11 February 2021

Problem

In a plane, four circles with radii $1,3,5,$ and $7$ are tangent to line $l$ at the same point $A,$ but they may be on either side of $l$. Region $S$ consists of all the points that lie inside exactly one of the four circles. What is the maximum possible area of region $S$?

$\textbf{(A) }24\pi \qquad \textbf{(B) }32\pi \qquad \textbf{(C) }64\pi \qquad \textbf{(D) }65\pi \qquad \textbf{(E) }84\pi$

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. $49 \pi + (25-9) \pi=65 \pi \rightarrow \boxed{D}$ ~ samrocksnature

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