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

$[asy] /* Settings */ import three; defaultpen(fontsize(10)+linewidth(0.62)); currentprojection = perspective(-2,-50,15); size(200); /* Variables */ real x = 20 - ((750)^.5)/3, CE = 8*(6^.5) - 4*(5^.5), CD = 8*(6^.5), h = 4*CE/CD; pair Cxy = 8*expi((3*pi)/2-CE/8); triple Oxy = (0,0,0), A=(4*5^.5,-8,4), B=(0,-8,h), C=(Cxy.x,Cxy.y,0), D=(A.x,A.y,0), E=(B.x,B.y,0), O=(O.x,O.y,h); pair L = 8*expi(pi+0.05), R = 8*expi(-0.22); /* left and right cylinder lines, numbers from trial/error */ /* Drawing */ draw(B--A--D--E--B--C); draw(circle(Oxy,8)); draw(circle(O,8)); draw((L.x,L.y,0)--(L.x,L.y,h)); draw((R.x,R.y,0)--(R.x,R.y,h)); draw(O--B--(A.x,A.y,h)--cycle,dashed); /* Labeling */ label("$A$",A,NE); dot(A); label("$B$",B,NW); dot(B); label("$C$",C,W); dot(C); label("$D$",D,E); dot(D); label("$E$",E,S); dot(E); label("$O$",O,NW); dot(O); pair O1 = (25,4), A1=O1+(4*sqrt(5),-8), B1=O1+(0,-8); draw(circle(O1,8)); draw(O1--A1--B1--O1); label("$A$",A1,SE);label("$B$",B1,SW);label("$O$",O1,N); label("$8$",O1--B1,W);label("$4$",(5*A1+O1)/6,0.8*(1,1)); label("$8$",O1--A1,0.8*(-0.5,2));label("$4\sqrt{5}$",B1/2+A1/2,(0,-1)); [/asy]$

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 at point $B$ . We use the Pythagorean Theorem to find the horizontal component of $AB$ has length $4\sqrt{5}$ .

$[asy] defaultpen(fontsize(10)+linewidth(0.62)); pair A=(-4*sqrt(5),4), B=(0,4*(8*sqrt(6)-4*sqrt(5))/(8*sqrt(6))), C=(8*sqrt(6)-4*sqrt(5),0), D=(-4*sqrt(5),0), E=(0,0); draw(A--C--D--A);draw(B--E); label("$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("$4\sqrt{5}$",D/2+E/2,(0,-1));label("$8\sqrt{6}-4\sqrt{5}$",C/2+E/2,(0,-1)); label("$4$",D/2+A/2,(-1,0));label("$x$",C/2+B/2,(1,0.5));label("$20-x$",0.7*A+0.3*B,(1,0.5)); dot(A^^B^^C^^D^^E); [/asy]$

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 $\triangle CDA$ and $\triangle CEB$ are similar right triangles. 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 $a=60, b=750, c=3, a+b+c=\boxed{813}$ .

Solution 2

Note that by Power of a Point, the point the unicorn is at has power $4 \cdot 20 = 80$ which implies that the tangent from that point to the tower is of length $\sqrt{80}=4\sqrt{5},$ however this is length of the rope projected into 2-D. If we let $\theta$ be the angle between the horizontal and the rope, we have that $\cos\theta=\frac{1}{5}$ which implies that $\sin\theta=\frac{2\sqrt{6}}{5}.$ Note that the portion of rope not on the tower is $4\sqrt{5} \cdot \frac{5}{2\sqrt{6}}= \frac{5\sqrt{30}}{3},$ the requested length of rope is $20-\frac{5\sqrt{30}}{3}=\frac{60-\sqrt{750}}{3}$ thus the requested sum is $\boxed{813}.$

~ Dhillonr25

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

Contents

Problem

Solution

Solution 2

See also