Mock AIME 2 2006-2007/Problem 7

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Problem

A right circular cone of base radius $\displaystyle 17$ cm and slant height $\displaystyle 34$ cm is given. $\displaystyle P$ is a point on the circumference of the base and the shortest path from $\displaystyle P$ around the cone and back is drawn (see diagram). If the length of this path is $\displaystyle m\sqrt{n},$ where $\displaystyle m$ and $\displaystyle n$ are relatively prime positive integers, find $\displaystyle m+n.$