I have no idea how to do this problem at all.
A cylindrical can without a top is made to contain V cm^3 of liquid. Find the dimensions that will minimize the cost of the metal to make the can.
Since no specific volume is given the smallest amount of metal for the can would be zero, which would held zero cm^3 of liquid. How is this wrong? It is not possible to make a cylinder out of a negative amount of metal.
Best Answer
In the cylinder without top, the volume $V$ is given by:
$$V=\pi R^2h$$ the surface, $$S=2\pi Rh+\pi R^2$$
Solving the first eq. respect to $R$, you find:
$$h=\frac{V}{\pi R^2}$$ Putting this into the equation of the surface, you obtaine: $$S=2\frac{V}{R}+\pi R^2$$ deriving this expression respect to $R$ and putting the result to zero in order to find the minimum, you have:
$$R=\sqrt[3]\frac{V}{\pi}$$