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

defaultpen(fontsize(8));
pair A=(4*sqrt(5),8), B=(0,8), O=(0,0);
draw(circle((0,0),8));
draw(O--A--B--O);
label("A",A,(1,1));label("B",B,(-1,1));label("O",O,(-1,-1));
label("<math>8</math>",A/3,(0.5,-1));label("<math>4</math>",5*A/6,(0.5,-1));
label("<math>8</math>",B/2,(-1,0));label("<math>4\sqrt{5}</math>",B/2+A/2,(0,1)); (Error making remote request. Unknown error_msg)

Looking from an overhead view, call the center of the circle , the tether point to the unicorn and the last point where the rope touches the tower . is a right triangle because is a radius and is a tangent line at point . We use the Pythagorean Theorem to find the horizontal component of has length .

defaultpen(fontsize(8));
pair A=(-4*sqrt(5),4), B=(0,4*(8*sqrt(6)-4*sqrt(5))/(8*sqrt(6))), C=(8*sqrt(6)-4*sqrt(5),0), D=(-4*sqrt(5),0), E=(0,0);
draw(A--C--D--A);draw(B--E);
label("A",A,(-1,1));label("B",B,(1,1));label("C",C,(1,0));label("D",D,(-1,-1));label("E",E,(0,-1));
label("<math>4\sqrt{5}</math>",D/2+E/2,(0,-1));label("<math>8\sqrt{6}-4\sqrt{5}</math>",C/2+E/2,(0,-1));
label("<math>4</math>",D/2+A/2,(-1,0));label("<math>x</math>",C/2+B/2,(1,0.5));label("<math>20-x</math>",0.7*A+0.3*B,(1,0.5));
dot(A^^B^^C^^D^^E);
 (Error making remote request. Unknown error_msg)

Now look at a side view and "unroll" the cylinder to be a flat surface. Let be the bottom tether of the rope, let be the point on the ground below , and let be the point directly below . Triangles and are similar right triangles. By the Pythagorean Theorem .

Let be the length of .

Therefore .