How Can I Determine the Minimum Volume of a Cube Given the Surface Area?

[member 392791]

Suppose you are given a problem to find the dimensions for the maximum volume of a cube given the surface area. These problems involve 2 equations, taking the derivative and setting it equal to zero (local minimum or maximum) and substituting the 2nd equation to find the parameters. However, suppose I wanted to know the dimensions with the minimum volume, how can I go about doing that? Clearly using the same method will result in a maximum? When doing these, since setting a derivative to zero gives a local max or min, how is one to know which one will be found?

 

[mfb]

There is no minimum - or 0, if you like. In principle, setting the derivative to 0 gives all extremal values. If the system has a proper minimum, it will show up there. Otherwise, you can check the boundary conditions of your problem (here: side length > 0).

 

[member 392791]

Your saying for a box with a given surface area, it wouldn't have a minimum value for its volume?

 

[Mark44]

Your saying for a box with a given surface area, it wouldn't have a minimum value for its volume?

The volume can't go below 0, so that is effectively a minimum value. If you restrict your attention to actual, physical boxes for which all dimensions are positive, you can't get to a volume of 0.

 

[member 392791]

Ok if you are given that the surface area has a value that it cannot deviate from, I'm wondering how you guys are proposing a box of zero volume and finite surface area. I know there is a physical minimum volume for a given surface area, how do I go about finding it?

 

[AlephZero]

Make a box 1 inch long, 1 inch wide, and 0 inches deep.

Volume = 0, surface area = 2. The box still has a finite sized top and bottom, even though you can't put anything inside it.

 

[member 392791]

what would a box like this look like? Are you talking about a square?

 

[mfb]

It is like a square, indeed. If you want a "physical" box (with positive side lengths), there is no minimum, but you can make it as close to 0 as you like with an extremely tiny depth.

 

[symbolipoint]

Suppose you are given a problem to find the dimensions for the maximum volume of a cube given the surface area. These problems involve 2 equations, taking the derivative and setting it equal to zero (local minimum or maximum) and substituting the 2nd equation to find the parameters. However, suppose I wanted to know the dimensions with the minimum volume, how can I go about doing that? Clearly using the same method will result in a maximum? When doing these, since setting a derivative to zero gives a local max or min, how is one to know which one will be found?

You already set the volume at a constant when you assigned the surface area. You cannot find a maximum.