Question

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 $a + b + c.$

答案

Solution:

The rope runs from its anchor $A$ at the base of the tower, hugs the wall up to a point $P,$ then goes straight to its end $Q,$ which is at height $4$ and at distance $8 + 4 = 12$ from the tower's axis. Unroll the cylinder's wall into a plane: a taut rope becomes a single straight segment of length $20$ rising $4$ feet, so its horizontal projection has length $\sqrt{20^2 - 4^2} = 8\sqrt{6},$ and every piece of the rope has the same ratio $\frac{20}{8\sqrt{6}} = \frac{5}{2\sqrt{6}}$ of length to horizontal projection.

Viewed from above, the free portion $PQ$ is tangent to the circle of radius $8$ at $P$ from a point at distance $12,$ so its horizontal projection has length $\sqrt{12^2 - 8^2} = 4\sqrt{5}.$ Therefore $PQ = \frac{5}{2\sqrt{6}} \cdot 4\sqrt{5} = \frac{10\sqrt{5}}{\sqrt{6}} = \frac{5\sqrt{30}}{3}.$

The rope touching the tower has length $20 - \frac{5\sqrt{30}}{3} = \frac{60 - \sqrt{750}}{3},$ and $c = 3$ is prime, so $a + b + c = 60 + 750 + 3 = 813.$

2004 AIME I Problem 14

Solution:

Problem 14 in Other Years