Cylinder Problem

Date: 11/22/97 at 17:23:54
From: John Van Straalen
Subject: Geometry cylinder problem

The following question was brought up in my math class concerning the 
volume and surface are of a can.

Given an aluminum soft drink can with radius 3.25 cm, height 12 cm, 
volume 398.2 cubic cm, and surface area 311.4 square cm, is it 
possible to construct a can with a larger volume but with the 
same surface area? Can you construct a can with a smaller surface 
area but the same volume?

Is there a way to find the dimensions of the can with the largest 
volume but with the same surface area? Can you find the dimensions of 
the can with the smallest surface area but the same volume?

Any help, hints, or formulas that would help me answer these questions 
would be appreciated.

Date: 11/23/97 at 09:04:01
From: Doctor Jerry
Subject: Re: Geometry cylinder problem

Hi John,

These questions often come up in calculus.  They can be solved by 
graphing, although you may have to be content with an approximate 
answer.

Suppose the volume of the can is fixed and we want to choose the 
dimensions so that the surface area is a minimum.

So, V = pi*r^2*h, where V is fixed.  Note that this forces r and h to 
vary so that the product r^2*h is always equal to V/pi.  

Surface area S = 2pi*r*h+2pi*r^2 = lateral surface plus the two ends. 
Now, from the fact that r^2*h = V/pi, we can solve for h = V/(pi*r^2).  
So,

   S = 2pi*r*V/(pi*r^2)+2pi*r^2

Now you can graph S as a function of r and choose the low point.  
Looking at the graph, is there a high point, that is, is there a 
maximum surface area for a given volume?

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