A unicorn is tethered by a $20$-foot silver rope to the base of a magician's [cylindrical](https://artofproblemsolving.com/wiki/index.php/Cylinder) tower whose [radius](https://artofproblemsolving.com/wiki/index.php/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](https://artofproblemsolving.com/wiki/index.php/Positive) [integers](https://artofproblemsolving.com/wiki/index.php/Integer), and $c$ is prime. Find $a+b+c.$
Looking from an overhead view, call the [center](https://artofproblemsolving.com/wiki/index.php/Center) of the [circle](https://artofproblemsolving.com/wiki/index.php/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](https://artofproblemsolving.com/wiki/index.php/Right_triangle) because $OB$ is a radius and $BA$ is a [tangent line](https://artofproblemsolving.com/wiki/index.php/Tangent_line) at point $B$.  We use the [Pythagorean Theorem](https://artofproblemsolving.com/wiki/index.php/Pythagorean_Theorem) to find the horizontal component of $AB$ has length $4\sqrt{5}$.  


Now look at a side view and "unroll" the cylinder to be a flat surface. Let $C$ be the bottom tether of the rope, let $D$ be the point on the ground below $A$, and let $E$ be the point directly below $B$.  [Triangles](https://artofproblemsolving.com/wiki/index.php/Triangle) $\triangle CDA$ and $\triangle CEB$ are [similar](https://artofproblemsolving.com/wiki/index.php/Similar) [right triangles](https://artofproblemsolving.com/wiki/index.php/Right_triangle).  By the Pythagorean Theorem $CD=8\cdot\sqrt{6}$.  
Let $x$ be the length of $CB$. \[\frac{CA}{CD}=\frac{CB}{CE}\implies \frac{20}{8\sqrt{6}}=\frac{x}{8\sqrt{6}-4\sqrt{5}}\implies x=\frac{60-\sqrt{750}}{3}\]
Therefore $a=60, b=750, c=3, a+b+c=\boxed{813}$.