A1. A cylinder is 4 cm high, with a radius of 1 cm. From point $P$ on the lower base to point $Q$, which lies on the upper base directly above point $P$, we stretch a string along the lateral surface of the cylinder, wrapping it around the cylinder exactly once. How many centimeters long is the shortest string that can be stretched in this way?
(A) $2 \pi$
(B) $4 \pi$
(C) $\pi \sqrt{2}$
(D) $2 \sqrt{\pi^{2}+4}$
(E) $\sqrt{2 \pi^{2}+4}$
A1. If we cut the cylinder along the vertical line $P Q$ and unfold it, we get a rectangle with side lengths of $4 \mathrm{~cm}$ and $2 \pi \mathrm{cm}$. The problem then asks for the length of the shortest path between the opposite vertices of the rectangle. This is equal to the length of the diagonal, which is $\sqrt{(2 \pi)^{2}+4^{2}}=$ $2 \sqrt{\pi^{2}+4} \mathrm{~cm}$.