2004 AIME I Problem 14

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 its nearest point on the tower, and the length of 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.