Minimizing the Cost of a Box

Date: 11/14/95 at 17:53:11
From: Anonymous
Subject: Business Calculus

A closed box with a square base is to contain 252 cubic feet.
The bottom costs $5 per square foot, the top costs $2 per square foot,
and the sides cost $3 per square foot.  Find the dimensions that will
minimize the cost.

Date: 12/7/95 at 6:3:3
From: Doctor Jonathan
Subject: Re: Business Calculus

When you want to minimize a function, you find the points where that 
function's first derivative is zero.  This will either be a maximum
or a minimum.  To determine which, you need only look at the sign of the
second derivative at that point; if it's positive you have a minimum, if
negative, a maximum. I have a feeling your problem will require the 
minimization of a function of two variables, but the same principles 
apply (replace deriv. with gradient). If you need further help, please 
feel free to write back.

-Doctor Jonathan,  The Geometry Forum