Question

If 1000 square centimeters of material is available to make a box with a square base and an open top, find the largest possible volume of the box.

Answer 1

\[1000=2x^2+4xy\]
This is your equation you use to solve for variables, you are maximizing volume which is
\[V=x^2y\]
and we solve for one variable in the first equation
\[\frac{ (500-x^2) }{ 2x}=y\]
Substitute
\[(\frac{ (500-x^2) }{ 2x})x^2\]
Which simplifies to
\[500x-x^3\]
Take the derivative of this to be able to discern min's and max's
\[\[500-3x^2=0\]\]
Set equal to zero
x=0 at 12.9099
Use the First derivative test to prove it is a max and you have your x value.
Plug the x's back in to find your y value

Answer 2

Refer to the attached solution using Mathematica 8 Home Edition.