Inscribed Cone

Date: 3/11/96 at 18:47:33
From: Edward Tang
Subject: Calc problem

Hi - 

My Calculus teacher and I are having problems with this question: 

"Find the volume of the largest right circular cone that can be 
inscribed in a sphere of radius 'r'"

It seems hard to relate 'r' with anything on the cone...... 
If that connection can be made, then the derivative can be found 
and the problem solved......

Thanks!

- Ed

Date: 6/21/96 at 11:48:14
From: Doctor Jerry
Subject: Re: Calc problem

Start by drawing a triangle inscribed in a circle of radius r.  
Draw it as a somewhat slender isoceles triangle, with the base 
between the equal sides placed on the bottom.  Locate the center 
of the circle.  Draw a line from the center to the lower left 
vertex and a line from the center to the base and perpendicular 
to it.  You see a small right triangle. Let its lower base be called R.  
This is the base radius of the inscribed cone.

The height of the inscribed cone is r plus the vertical side of the 
small right triangle.  This side is sqrt(r^2-R^2).  

So, the volume of the cone, in terms of r and R, is
V=(pi/3) R^2 (r+sqrt(r^2-R^2)).

This function can be maximized without much trouble. 
I get R=2*sqrt(2) r/3.

-Doctor Jerry,  The Math Forum