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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

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/
```
Associated Topics:
High School Calculus

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