# Thread: Maximizing the cuboid volume with fixed perimeter

1. ## 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 $\displaystyle C$. Which are the dimensions values that maximize the area?

A: One dimension is $\displaystyle x$, and thus the other is $\displaystyle C-x$. Therefore, the area is $\displaystyle x \cdot (C-x) = xC - x^2$. Then, setting the derivative equal to zero would provide us the answer ($\displaystyle C/2$ 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 $\displaystyle C$. Which are the dimensions values that maximize the volume?

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

2. ## Re: Maximizing the cuboid volume with fixed perimeter

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 $\displaystyle C$. Which are the dimensions values that maximize the area?

A: One dimension is $\displaystyle x$, and thus the other is $\displaystyle C-x$. Therefore, the area is $\displaystyle x \cdot (C-x) = xC - x^2$. Then, setting the derivative equal to zero would provide us the answer ($\displaystyle C/2$ 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 $\displaystyle C$. 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

$\displaystyle V(x,y,z)=xyz$

subject to the constraint that

$\displaystyle x+y+z=C \iff x+y+z-C=0 \implies F(x,y,z)=x+y+z-C$

Now we use the Lagrange multipliers to get

$\displaystyle \nabla V(x,y,z)=\lambda \nabla F(x,y,z)$

This gives

$\displaystyle yz \vec{i}+xz\vec{j}+xy\vec{k}=\lambda\vec{i}+\lambda \vec{j}+\lambda\vec{k}$

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

$\displaystyle yz=\lambda \\ xz=\lambda \\ xy=\lambda \\ x+y+z=C$

This gives that

$\displaystyle xyz=\lambda x =\lambda y =\lambda z$

or

$\displaystyle x=y=z= \frac{C}{3}$

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

3. ## Re: Maximizing the cuboid volume with fixed perimeter

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