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 $\displaystyle x^2$ mē.

. . At $4/mē, its cost is: .$\displaystyle 4x^2$ dollars.

The four sides have an area of $\displaystyle 4xy$ mē.

. . At $3/mē, their cost is: .$\displaystyle 12xy$ dollars.

The total cost is: .$\displaystyle 4x^2 + 12xy$ which is limited to $48.

. . There is our constraint: .$\displaystyle 4x^2 + 12xy \:=\:48\quad\Rightarrow\quad y \:=\:\frac{48 - 4x^2}{12x} $ .[1]

The volume of the box is: .$\displaystyle V \;=\;x^2y$ .[2]

Substitute [1] into [2]: . $\displaystyle V \;=\;x^2\left(\frac{48-4x^2}{12x}\right)$

. . And we have: .$\displaystyle V \;=\;\frac{1}{3}(12x - x^3)$

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