2004 AMC 12A Problems/Problem 24

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Problem 24

A plane contains points $A$ and $B$ with $AB = 1$ . Let $S$ be the union of all disks of radius $1$ in the plane that cover $\overline{AB}$ . What is the area of $S$ ?

$\text {(A)} 2\pi + \sqrt3 \qquad \text {(B)} \frac {8\pi}{3} \qquad \text {(C)} 3\pi - \frac {\sqrt3}{2} \qquad \text {(D)} \...$

Solution

pair A=(-.5,0), B=(.5,0), C=(0,3**(.5)/2), D=(0,-3**(.5)/2);draw(arc(A,2,-60,60),blue);draw(arc(B,2,120,240),blue);draw(circl...

As the red circles move about segment $AB$ , they cover the area we are looking for. On the left side, the circle must move around pivoted on $B$ . On the right side, the circle must move pivoted on $A$ However, at the top and bottom, the circle must lie on both A and B, giving us our upper and lower bounds.

This egg-like shape is $S$ .

pair A=(-.5,0), B=(.5,0), C=(0,3**(.5)/2), D=(0,-3**(.5)/2);draw(arc(A,2,-60,60),blue);draw(arc(B,2,120,240),blue);draw(arc(C...

The area of the region can be found by dividing it into several sectors, namely

$\begin{align*}A &= 2(\mathrm{Blue\ Sector}) + 2(\mathrm{Red\ Sector}) - 2(\mathrm{Equilateral\ Triangle}) \\A &= 2\le...$

2004 AMC 12A (Problems • Resources)
Preceded by Problem 23	Followed by Problem 25
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2004 AMC 12A Problems/Problem 24

From AoPSWiki

Problem 24

Solution

See also