Shortest Path On a Cylinder

Consider cutting the cylinder along the two lines $x=10,y=0$ and $x=-10,y=0$ . You get two rectangles, each with side lengths of $10$ and $\frac 12 2 \pi \cdot 10 = 10\pi$ .

The points we want to connect lie on two opposite corners of these rectangles, so the shortest path between them is one rectangle's diagonal, which has a length of

$\sqrt{(10)^2+(10\pi)^2} = 10\sqrt{1+\pi^2} \approx \boxed{32.969}$ .