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:
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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: .