3. A room is built in the shape of the region between two semicircles with the same center and parallel diameters. The farthest distance between two points with a clear line of sight is $12 \mathrm{~m}$. What is the area (in $\mathrm{m}^{2}$ ) of the room?
Solution:
$18 \pi$
The maximal distance is as shown in the figure. Call the radii $R$ and $r, R>r$. Then $R^{2}-r^{2}=6^{2}$ by the Pythagorean theorem, so the area is $(\pi / 2) \cdot\left(R^{2}-r^{2}\right)=18 \pi$.