optimize volume of box

Question

A rectangular box with a square base is made of 48 square meters.

What dimensions should have the box to have the maximum volume?

Is that correct my equation that i must maximize?

Answer 1

Hint

Your equations are correct but you did not answer the question : what is the value of $h$ which maximizes this volume $V(h)$.

To what corresponds the maximum of a function ?

If you answer this question, I am sure that you will easily finish your problem.

Added later to the answer

You could make the problem more general considering that the dimensions of the rectangular base are $x$ and $y$. So, as you wrote, the volume of the box is given by $$V(h)=h (x-2h)(y-2h)$$ As you did, you get $$V'(h)=12 h^2-4 (x+y) h+x y$$ which cancels for $$h=\frac{1}{6} \left(x+y -\sqrt{x^2-x y+y^2}\right)$$