2008 AIME I Problems/Problem 5

Problem

A right circular cone has base radius $r$ and height $h$ . The cone lies on its side on a flat table. As the cone rolls on the surface of the table without slipping, the point where the cone's base meets the table traces a circular arc centered at the point where the vertex touches the table. The cone first returns to its original position on the table after making $17$ complete rotations. The value of $h/r$ can be written in the form $m\sqrt {n}$ , where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m + n$ .

Solution

The path is a circle with radius equal to the slant height of the cone, which is $\sqrt {r^{2} + h^{2}}$ . Thus, the length of the path is $2\pi\sqrt {r^{2} + h^{2}}$ .

Also, the length of the path is 17 times the circumference of the base, which is $34r\pi$ . Setting these equal gives $\sqrt {r^{2} + h^{2}} = 17r$ , or $h^{2} = 288r^{2}$ . Thus, $\dfrac{h^{2}}{r^{2}} = 288$ , and $\dfrac{h}{r} = 12\sqrt {2}$ , giving an answer of $12 + 2 = \boxed{014}$ .

2008 AIME I (Problems • Answer Key • Resources)
Preceded by Problem 4	Followed by Problem 6
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2008 AIME I Problems/Problem 5

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