Hello, jlt1209!

This requires partial derivatives . . .

A cardboard box without a lid is to have a volume of 32,000 cm³.

Find the dimensions that minimize the amount of cardboard used. Code:

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

The length, width, height of the box are: . , respectively.

The volume is 32,000 cm³: . .[1]

The total surface area of the box is: . .[2]

Substitute [1] into [2]: .

. . and we have: .

Set the partial derivatives equal to 0.

. .

Substitute [3] into [4]: .

. .

. .

Substitute into [3]: .

Substitute into [1]: .

Therefore: .