What are the dimensions of the largest-volume rectangular box, with a square base and an open top , that can be made from 10 000 cm^2 of materials ?
Here is the set up: we have a base and four sides. Each side has one dimension in common with the base and one dimension in common with each other (the height). Let's say that the base has dimensions $x\times x $ and the height is $y$. Then the surface area is $x^2+4xy$. You know that is equal to $10,000\text{ cm}^2$. You are trying to maximize volume: $V=x^2y$. So, solve for $y$ in the first equation, plug it into the volume formula, and then find the critical points for $x$.
If this problem is being done on a precalculus level (i.e. , you haven't learned about derivatives), then get volume in terms of $x$ by solving $x^2+4xy = 10000$ for $y$, then sub the result into $V = x^2y$ to get volume strictly in terms of $x$.
... determine the maximum by graphing the single-variable volume equation on a calculator.
If you do know to find critical values, then proceed with that method.
Hmmm...since a square provides maximum area
(a rectangle is a special square), wouldn't you
simply "stick together" 5 squares, each with
area=2000 sq.ft. ?
Or should I get out of the boxing business?