I'm learning the use of derivatives and I have found a problem:
Supposing we want to build a box of $4000\, \textrm{cm}^3$ of volume without top and a square base. Which are the measures so we can use less material?
I have no idea where to start :C, Any hint would be appreciated, thank you.
Since the base is square, there are two important dimensions: the length of each side of the base, which I’ll call $x$, and the height, which I’ll call $y$. The volume of the box is $x^2y$, and we know that this is $4000$, so we have the equation $x^2y=4000$. That equation completely determines the relationship between $x$ and $y$: if we know one, we can use the equation to find the other.
The amount of material used is the area of the base together with the combined areas of the four upright sides; that’s $x^2+4xy$, and we want to choose $x$ and $y$ to minimize this. At this point we want to get rid of one of the variables $x$ and $y$, so that we have the area in terms of just one of them and can apply standard elementary calculus techniques to find the minimum. It’s easier to solve $x^2y=4000$ for $y$ than to solve it for $x$, so we’ll do that: $y=\dfrac{4000}{x^2}$. Now we can write the area (amount of material) as a function of $x$ alone:
$$A(x)=x^2+4xy=x^2+\frac{4x\cdot4000}{x^2}=x^2+\frac{16000}x\;.$$
At this point you can apply the usual technique of finding a critical point of $A(x)=x^2+\dfrac{16000}x$ and making sure that it’s a minimum. That will give you the right value of $x$, and from that you can get the associated value of $y$.