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 $\frac{a-\sqrt{b}}c$ feet, where $a, b,$ and $c$ are positive integers, and $c$ is prime. Find $a+b+c.$

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. $\triangle OAB$ 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 $AB=\sqrt{80}$ . 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 $CD=8*\sqrt{6}$ . Let the length of $x=AB$ . $\frac{CA}{CD}=\frac{AB}{AE}$ : $\frac{5}{2\sqrt{6}}=\frac{x}{\sqrt{80}}$ . $20-x=(60-\sqrt{750})/3$ , Therefore $a=60, b=750, c=3, a+b+c=813$ .

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
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2004 AIME I Problems/Problem 14

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