Hello, ikickyouindanuts!

Find the dimensions of an open-top rectangular box with a volume of 32 ft³,

using the least amount of material. You need to be familiar with partial derivatives . . . Code:

*-----------*
/| /|
/ | / |
/ | / |H
*-----------* |
| | |
| | *
H | | /
| | /W
| |/
*-----------*
L

The volume is 32 ft³: .

The base has area: .

The front and back has area: .

The sides have area: .

The total area is: .

Substitute [1] into [2]: .

. . and we have: .

Take partial dereivatives and equate to zero:

. .

Equate [3] and [4]: .

Substitute into [3]: .

Substitute into [1]: .

. . Therefore: .