Minimizing the Surface Area of a Can

Date: 05/22/2000 at 03:42:43
From: Chris Donges
Subject: Surface area

What coke can dimensions would use the least amount of aluminum 
possible while still holding 375 ml? I'm not sure where to start -  
please help.

Date: 05/22/2000 at 08:06:25
From: Doctor Jerry
Subject: Re: Surface area

Hi Chris,

I'll have to make some assumptions. If the coke can can be thought of 
as a cylinder with two circular ends, then you can write two formulas:

     V = pi*r^2*h

and

     S = 2*pi*r^2 + 2*pi*r*h.

The second is the areas of the two ends plus the area of the cylinder.

If r and h are in cm, you can convert 375 ml into cubic centimeters 
and then you'll have an equation like:

     375*k = 2*pi*r^2 + 2*pi*r*h.

You can solve this for h in terms of r. Substitute this into the 
formula V = pi*r^2*h, which now will be entirely in terms of r. Now, 
just maximize V. You can do this with calculus or graphing.

- Doctor Jerry, The Math Forum
  http://mathforum.org/dr.math/