Mock AIME 2 2006-2007 Problems/Problem 7

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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 minimum distance from the vertex $\displaystyle V$ to this path is $\displaystyle m\sqrt{n},$ where $\displaystyle m$ and $\displaystyle n$ are relatively prime positive integers, find $\displaystyle m+n.$

Mock AIME 2 2007 Problem7.jpg

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