Optimization of a rectangular box I am suppose to find the volume if 1200 cm^2 of material is available to make a box with a square base and an open top, find the largest possible volume of the box.
I think what I need to do is set up the formulas to be 
$4(lw) + w^2 = 1200$ for area
$lwh = v$ for volume
I know that if the base is a square than the rectangle will have the same dimensions and the only different variable would be the height so I can solve for length like so
$l=\frac{1200-w^2}{4w}$
Now that I have that I can put it in my formula
$lwh = v$ for volume
which I can rewrite as 
$l^2 * h = v$
I then take the derivative of this and I get some ridiculous answer that is wrong.
$\frac{1200w^2 - w^4}{4w}$
the derivative
$300 \frac{-3w^2}{4}$
Which gives me $+-20$ which is an incorrect answer.
 A: When you write down a formula, you must write down what the letters stand for. 
When you write down $4lw+w^2=1200$, you must add, "where $l$ is the length of the base of the box, and $w$ is the width of the base of the box". 
If you do that, you might see right away where you've messed up. You've written down a formula for the area which doesn't include the height of the box---that can't possibly be right, can it? 
In fact, the formula you have written down only makes sense if $l$ is the height of the box, right? 
So go back and identify the variables explicitly and then write down formulas that make sense. 
A: The formula $\rm V=lwh$ means "volume = length times width times height." The variable $\rm l$ is length, the variable $\rm w$ is width, and the variable $\rm h$ is height. Using these, the total area is actually
$$\rm 2(l\times h)+2(w\times h)+w^2=1200.$$
We know that $\rm l=w$ (because the base of the box is square), so this is $\rm 4wh+w^2=1200$. This allows us to solve for the height $\rm h$ in terms of width $\rm w$ as $\rm h=(1200-w^2)/(4w)$.
We have the formula for $\rm h$ in terms of $\rm w$, and know $\rm l=w$, so we have the volume function
$$\rm V=l\,wh=w^2\frac{1200-w^2}{4w}=300w-\frac{1}{4}w^3.$$
Now can you take the derivative of this, equate it with zero and solve for $\rm w$?
