2008 AIME I Problems/Problem 5
A right circular cone has base radius and height . 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 complete rotations. The value of can be written in the form , where and are positive integers and is not divisible by the square of any prime. Find .
The path is a circle with radius equal to the slant height of the cone, which is . Thus, the length of the path is .
Also, the length of the path is 17 times the circumference of the base, which is . Setting these equal gives , or . Thus, , and , giving an answer of .
|2008 AIME I (Problems • Answer Key • Resources)|
|1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15|
|All AIME Problems and Solutions|