Hello, kdogg121!

Your professor's advice is correct . . .

A carpenter has been asked to build an open box with a square base.

The sides of the box will cost $3/mē, and the base will cost $4/mē.

What are the dimensions of the box of greatest volume

that can be constructed for $48?

The answer is supposed to be: 2 by 2 by 4/3 meters Code:

*-------*
/| /|
/ | / |
*-------* |y
| | |
| | |
y| | *
| | /x
| |/
*-------*
x

The base has an area of mē.

. . At $4/mē, its cost is: . dollars.

The four sides have an area of mē.

. . At $3/mē, their cost is: . dollars.

The total cost is: . which is limited to $48.

. . There is our constraint: . .[1]

The volume of the box is: . .[2]

Substitute [1] into [2]: .

. . And we have: .

And **that** is the function we must maximize . . .