Cylinder Volume and Surface Area

Date: 04/14/99 at 21:18:13
From: Long Nguyen
Subject: Maxima and Minima - Cylinder Volume

Hi,

My name is Long and I stumbled across a question that I just can't 
figure out. Can you please help?

You have to make a cylinder that holds the greatest volume when the 
entire surface area of the cylinder (including the area of both the 
circles on the top and bottom) is 600 cm squared. You can make the 
cylinder any way you want but it has to be a cylinder. 

I am having a lot of difficulty with this and I hope you can help. 
Please explan to me how to solve this and please show me in 
mathematical terms, and can you tell me how we would know that the 
answer you give me is the maximum volume it can hold?

Thank you very much,

Long Nguyen

Date: 04/16/99 at 16:00:17
From: Doctor Jeremiah
Subject: Re: Maxima and Minima - Cylinder Volume

Dear Long:

It's a hard question.  Figuring out the maximum volume requires 
calculus.

The surface area of a cylinder is
A = 2*Top + Side
A = 2*CircleArea + Rectangle
A = 2*CircleArea + CircleCircumference*Height
600 = 2*(pi*r*r) + (2*pi*r*h)

The Volume of a cylinder is
V = Top*Height
V = CircleArea*Height
V = (pi*r*r)*h

To find the maximum volume we must have one variable (r), so we must 
solve the surface area for h and substitute.

A = 2*Top + Side
A = 2*CircleArea + Rectangle
A = 2*CircleArea + CircleCircumference*Height
600 = 2*(pi*r*r) + (2*pi*r*h)
600 - 2*(pi*r*r) = 2*pi*r*h
(600 - 2*pi*r*r)/(2*pi*r) = h

V = (pi*r*r)*h <== h = (600 - 2*pi*r*r)/(2*pi*r)
V = (pi*r*r)*(600 - 2*pi*r*r)/(2*pi*r)
V = (pi*r*r)*600/(2*pi*r) - (pi*r*r)*(2*pi*r*r)/(2*pi*r)
V = (2*pi*r)*r*300/(2*pi*r) - (2*pi*r)*r*(pi*r*r)/(2*pi*r)
V = r*300 - r*(pi*r*r)
V = 300r - pi*r^3

Now to find a minimum or maximum of V we must set V to 0. A zero slope 
means a maximum or a minumum. Then differentiate with respect to r.

dV/dr = d(300r - pi*r^3)/dr

After differentiating you know what r equals. Plug that into the 
Volume equation V = 300r - pi*r^3 and find the maximum volume.

If you need more help, please write back.

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