# Maximum Volume

1. Oct 22, 2007

### Reshma

I am a little stuck on this problem

If the total surface area (including the area of the top and bottom ends) of a cylinder is to be kept fixed (=A), what is its maximum possible volume?

For such cylinders of fixed total area, plot Volume(V) v/s Radius(R) clearly indicating the values of R for which the volume is maximum and zero.

The total surface area will be $A = 2\pi R(L + R)$ where L is the length of the cylinder. Here, A = constant and I have to determine the maximum possible volume. I don't know how to proceed, should I express the volume in terms of the area and do something?

Kindly guide me on this...

2. Oct 22, 2007

### Gokul43201

Staff Emeritus
Fixing the area, A, sets up a relationship between the radius R and length L. This then allows you to write down an equation for the volume that involves only one variable (either L or R). From here, it's just the matter of taking a derivative, etc.

3. Oct 22, 2007

### Reshma

Thank you so much, I will try it and post my solution soon.

4. Oct 26, 2007

### Reshma

Surface area A = constant.
$$A = 2\pi R (L + R)$$

$$L = {{A - 2\pi R^2}\over 2\pi R}$$

$$V = \pi R^2 L$$

Putting the value of L:
$$V = {AR\over 2} - \pi R^3$$

Solving for dV/dR = 0 for maximum:
$$R = \sqrt{{A\over 6\pi}}$$

$$V_{max} = {A\over 2}\sqrt{{A\over 6\pi}} - \pi \left({A \over 6\pi}\right)^{3/2}$$

V = 0 for $$R = \sqrt{{A\over 2\pi}}$$

Should I assume some value for A when I plot V v/s R?

Last edited: Oct 26, 2007