2004 AIME I Problems/Problem 14
Problem
A unicorn is tethered by a 20-foot silver rope to the base of a magician's cylindrical tower whose radius is 8 feet. The rope is attached to the tower at ground level and to the unicorn at a height of 4 feet. The unicorn has pulled the rope taut, the end of the rope is 4 feet from the nearest point on the tower, and the length of the rope that is touching the tower is feet, where
and
are positive integers, and
is prime. Find
Solution
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Looking from an overhead view, call the center of the circle O, the tether point to the unicorn A, and the last point where the rope touches the tower B. is a right triangle because OB is a radius and BA is a tangent line of point B. We use the Pythagorean theorem to find the horizontal component of
. Now looking at a side view and "unrolling" the cylinder to be a flat surface, call the bottom tether of the rope C, the point on the ground below A D, and the point directly above B and 4 feet off the ground E. Triangles CDA and AEB are similar right triangles. By the Pythagorean theorem
. Let the length of
. CA/CD=AB/AE:
.
, Therefore
.
See also
2004 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |