Maximizing the cuboid volume with fixed perimeter

First, the *2D* case is easy...

**Q0**: In a rectangle, we know that the sum of the dimensions (lenght and width) is a fixed . Which are the dimensions values that maximize the area?

**A**: One dimension is , and thus the other is . Therefore, the area is . Then, setting the derivative equal to zero would provide us the answer ( on both dimensions)

Okay... now I want to generalize this to the *3D* case. More precisely:

**Q1**: In a cuboid (see en.wikipedia.org/wiki/Cuboid), we know that the sum of the dimensions (lenght, width and depth) is a fixed . Which are the dimensions values that maximize the volume?

Finally, is there a generalization of the answer to Q1 for *higher dimensions*?

Re: Maximizing the cuboid volume with fixed perimeter

Quote:

Originally Posted by

**andre.vignatti** First, the

*2D* case is easy...

**Q0**: In a rectangle, we know that the sum of the dimensions (lenght and width) is a fixed

. Which are the dimensions values that maximize the area?

**A**: One dimension is

, and thus the other is

. Therefore, the area is

. Then, setting the derivative equal to zero would provide us the answer (

on both dimensions)

Okay... now I want to generalize this to the

*3D* case. More precisely:

**Q1**: In a cuboid (see

en.wikipedia.org/wiki/Cuboid), we know that the sum of the dimensions (lenght, width and depth) is a fixed

. Which are the dimensions values that maximize the volume?

Finally, is there a generalization of the answer to Q1 for

*higher dimensions*?

The easiest way is to use Lagrange Multipliers. So the problem would be we want to maximize the volume

subject to the constraint that

Now we use the Lagrange multipliers to get

This gives

This gives a system of three equations with four unknowns. Using the original contraint gives the non linear system of equations

This gives that

or

You can use Lagrange multipliers in any dimention to solve this problem

Re: Maximizing the cuboid volume with fixed perimeter

Hummm, thank you TheEmptySet! Now I need to learn about the Lagrange multipliers :-)