# Math Help - shortest path around cylinder

1. ## shortest path around cylinder

An ant starts at a point $P$ on the bottom edge of a right circular cylinder of radius $R$ and height $H$. If the ant makes $n$ complete circuits around the cylinder and finishes at a point at the top edge directly above its starting point, find, with justification, the length of its shortest possible path.

2. Originally Posted by scipa
An ant starts at a point $P$ on the bottom edge of a right circular cylinder of radius $R$ and height $H$. If the ant makes $n$ complete circuits around the cylinder and finishes at a point at the top edge directly above its starting point, find, with justification, the length of its shortest possible path.
Slit the cylinder vertically from the ants starting point and flatten out. Now take n copies of the flattened cylinder and place them side by side.

The shortest path on the cylinder is equivalent to a diagonal on the rectangle made out of n flattened copies of the cylinder.

RonL