Applied Max/Min Problems

Date: 2/11/96 at 14:48:46
From: J.Brian Casey
Subject: Applied max/min problems

Here's the question. Find the largest possible volume of a right 
circular cylinder that is inscribed in a sphere of radius r. 

All I know is that I have to differentiate the volume equation of the 
cylinder, but I don't know how to find my constant that I have to plug 
into the volume equation. 

Date: 7/29/96 at 13:47:21
From: Doctor Alan
Subject: Re: Applied max/min problems

Hi Brian,

The solution to your question can be answered by taking the following 
steps:

1. Set up your volume equation (for your cylinder) in terms of R 
(sphere) and the angle formed by R and the line perpendicular to h 
going through the center of the sphere.

2. Differentiate this with respect to theta (the angle).

3. Find the roots to the equation. There will be two roots. If you are 
using Mathematica, as I am, you may have to plot the derivative and 
choose the starting point for FindRoot command carefully. Note: By 
inspection of a sketch you will notice that theta must be between 0 
and Pi/2.

Good luck!

-Doctor Alan,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/