If 2100 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.
First find an equation to represent the surface area of the box. The surface area will be the surface area of the base plus the surface area of the sides:
$\displaystyle S_t=S_b+S_s$
The surface area of the base will be the width multiplied by the length, which in this case is just the width squared:
S_b=w^2
The surface area of the sides will be 4 multiplied by the surface area of one of the sides. The surface area of one of the sides is the width multiplied by the height:
$\displaystyle
S_s=4wh$
$\displaystyle
S_t=w^2+4wh
$
We also know that the total surface area will be equal to 2100:
$\displaystyle 2100=w^2+4wh$
$\displaystyle h=\frac {2100-w^2}{4w}$
Volume is represented by the equation:
$\displaystyle
V=l\cdot w\cdot h$
In this case, we know that l=w and we have already solved for h in terms of w, so we have that:
$\displaystyle V(w)=w^2\cdot \frac {2100-w^2}{4w}$
Now that you have the volume as a function of one variable, all you have to do is find the derivative and find the critical points. Once you have found the critical points, plug them back into the original equation to find the highest possible volume.