Rectangle $PQRS$ lies in a plane with $PQ=RS=3$ and $QR=SP=4$. The rectangle is rotated $90^\circ$ clockwise about $R$, then rotated $90^\circ$ clockwise about the point $S$ moved to after the first rotation. What is the length of the path traveled by point $P$?

1. In the first rotation, $P$ travels a quarter-circle centered at $R$ with radius $PR=\sqrt{3^2+4^2}=5$, so this arc has length $\frac{5\pi}{2}$.
2. In the second rotation, $P'$ travels a quarter-circle centered at the moved point $S'$ with radius $P'S'=PS=4$, so this arc has length $2\pi$.
3. The total path length is $\frac{5\pi}{2}+2\pi=\frac{9\pi}{2}$.