Difference between revisions of "2008 AIME I Problems/Problem 5"
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The path is a circle with radius equal to the slant height of the cone, which is <math>\sqrt {r^{2} + h^{2}}</math>. Thus, the length of the path is <math>2\pi\sqrt {r^{2} + h^{2}}</math>. | The path is a circle with radius equal to the slant height of the cone, which is <math>\sqrt {r^{2} + h^{2}}</math>. Thus, the length of the path is <math>2\pi\sqrt {r^{2} + h^{2}}</math>. | ||
| − | Also, the length of the path is 17 times the circumference of the base, which is <math>34r\pi</math>. Setting these equal gives <math>\sqrt {r^{2} + h^{2}} = 17r</math>, or <math>h^{2} = 288r^{2}</math>. Thus, <math>\dfrac{ | + | Also, the length of the path is 17 times the circumference of the base, which is <math>34r\pi</math>. Setting these equal gives <math>\sqrt {r^{2} + h^{2}} = 17r</math>, or <math>h^{2} = 288r^{2}</math>. Thus, <math>\dfrac{h^{2}}{r^{2}} = 288</math>, and <math>\dfrac{h}{r} = 12\sqrt {2}</math>, giving an answer of <math>12 + 2 = \boxed{014}</math>. |
== See also == | == See also == | ||
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[[Category:Intermediate Geometry Problems]] | [[Category:Intermediate Geometry Problems]] | ||
| + | {{MAA Notice}} | ||
Latest revision as of 19:18, 4 July 2013
Problem
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 
.
Solution
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 
.
See also
| 2008 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 4 |
Followed by Problem 6 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
