# 2001 AMC 10 Problems/Problem 21

## Problem

A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has diameter $10$ and altitude $12$, and the axes of the cylinder and cone coincide. Find the radius of the cylinder. $\textbf{(A)}\ \frac{8}3\qquad\textbf{(B)}\ \frac{30}{11}\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ \frac{25}{8}\qquad\textbf{(E)}\ \frac{7}{2}$

## Solution 2 $[asy] draw((5,0)--(-5,0)--(0,12)--cycle); unitsize(.75cm); draw((-30/11,0)--(-30/11,60/11)); draw((-30/11,60/11)--(30/11,60/11)); draw((30/11,60/11)--(30/11,0)); draw((0,0)--(0,12)); label("2r",(0,30/11),E); label("12-2r",(0,80/11),E); label("2r",(0,60/11),S); label("10",(0,0),S); label("A",(0,12),N); label("B",(-5,0),SW); label("C",(5,0),SE); label("D",(-30/11,60/11),W); label("E",(30/11,60/11),E); [/asy]$

Let the diameter of the cylinder be $2r$. Examining the cross section of the cone and cylinder, we find two similar triangles. Hence, $\frac{12-2r}{12}=\frac{2r}{10}$ which we solve to find $r=\frac{30}{11}$. Our answer is $\boxed{\textbf{(B)}\ \frac{30}{11}}$.

## Solution 3 (Very similar to solution 2 but explained more)

We are asked to find the radius of the cylinder, or $r$ so we can look for similarity. We know that $\angle BEF = \angle BDA$ and $\angle FBE = \angle ABD$, thus we have similarity between $\triangle BFE$ and $\triangle BAD$ by $AA$ similarity.

Therefore, we can create an equation to find the length of the desired side. We know that: $\frac{BE}{BD}=\frac{FE}{AD}.$

Plugging in yields: $\frac{12-2r}{12}=\frac{r}{5}.$

Cross multiplying and simplifying gives: $5(12-2r)=12r$ $\Downarrow$ $r=\frac{30}{11}.$

Since the problem asks us to find the radius of the cylinder, we are done and the radius of the cylinder is $\boxed{\textbf{(B)}\ \frac{30}{11}}$.

~etvat

## Trivia

This problem appeared in AoPS's Introduction to Geometry as a challenge problem.

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 