Maximizing the Volume of a Cylinder

Date: 12/3/95 at 20:48:25
From: Anonymous
Subject: Optimization - Calculus

Find the dimensions of the cylinder of maximum volume that can be 
inscribed in a cone having a diameter of 40cm and a height of 30cm.
Show that the maximum area of the cylinder is 4/9 the volume of the 
cone. (ANSWER: h=10 and r=40/3)

My problem is that I cannot find a relationship between the necessary
equations and the boundaries.

Date: 1/15/96 at 9:54:50
From: Doctor Ethan
Subject: Re: Optimization - Calculus

Hello,

Here is your answer.

Here is a drawing of a cutaway of the problem.

                           _
        /\                  |
       /  \                 |
      /    \                |
     /      \               |
    /________\              |30 cm
   /|        |\             |
  / |        | \            |
 /  |       x|  \           |
/   |        |   \          |
*******************        _|
         |-y-|  

|__________________|
          40cm

I hope that you can understand my notation. 

So according to this drawing, we are calling the radius of the 
cylinder y, and the height we are calling x.

So the Volume of the cylinder is

V = Pi y^2 * x

Now we just need to find a way to relate x and y.

Do you think that you can do that??

Why don't you try, and then once you have put that value 
for y into the equation, take the maximum.

Hope that helps.

-Doctor Ethan,  The Geometry Forum