Partial derivatives - Math Central

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Question from Meghan, a student:

Hi there,

I have a question I've been working at for a while with maxima/minima of partial derivatives.

"Postal rules require that the length + girth of a package (dimensions x, y, l) cannot exceed 84 inches in order to be mailed.
Find the dimensions of the rectangular package of greatest volume that can be mailed.
(84 = length + girth = l + 2x + 2y)"

I'm quite certain I'm supposed to substitute something, such as l = 84 - 2x - 2y.
However, I've been having trouble setting the partial derivatives to zero and solving for a certain variable.

Thank you so much for your help!
-Meghan

Hi Meghan,

You are to maximize the volume so you need an expression for the volume. The package is a "box" so the volume is given by V = l × x × y. You are correct in saying that l = 84 - 2x - 2y so substitute into the expression for the volume giving you the volume as a function of x and y.

Now try the partial derivatives. If you still have difficulties write back and tell us what you did.

Harley

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