Hello, davesface!

We usually needCalculusto maximize a quantity . . .

The length of the body diagonal of a rectangular prism is

What is the largest possible volume enclosed by the prism?

Let the dimensions be

Then: . .[1]

The volume is: . .[2]

Substitute [1] into [2]: .

Equate the partial derivatives to zero ... and solve:

. . . . .[3]

. . . . .[4]

. . And we find that: . . . . The rectangular prism is a.cube

Hence: .

. . And the maximum volume is: .