Drexel dragonThe Math ForumDonate to the Math Forum

Ask Dr. Math - Questions and Answers from our Archives
_____________________________________________
Associated Topics || Dr. Math Home || Search Dr. Math
_____________________________________________

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/   
    
Associated Topics:
High School Calculus

Search the Dr. Math Library:


Find items containing (put spaces between keywords):
 
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

_____________________________________
Math Forum Home || Math Library || Quick Reference || Math Forum Search
_____________________________________

Ask Dr. MathTM
© 1994-2013 The Math Forum
http://mathforum.org/dr.math/