Expert: Steve Holleran - 8/20/2007

Question
Here's a good one for you, which I hope it's in your area of expertise. What is the maximum surface area for a cube of fixed volume? It might be too complicated if you allow all the sides to vary in the cube (partial derivatives may be required)? However as a basic starting point, assuming all the sides of the cube to be equal, we know Volume=axaxa=a^3 and surface area=6a^2. It's possible to equate the volume in terms of the surface area. And as the volume is fixed (or constant), if you differentiate it, it will be zero. Where do you go on from here? A few hints would be appreciated. Many thanks in advance for your assistance.

Answer
Hello Safa,

The problem here is that once you fix the volume, you automatically fix the surface area, because the cube has all edges the same dimension.

Once you say the volume is, let's say, a^3, then the only possible dimension for each edge is "a" , which means the only possible surface area is 6a^2.

The situation would be different for a rectangular solid, because the dimensions could vary, but not for a cube.

Steve Holleran