We have a $10 \, {\rm m}^2$ metal plate and we want to construct a toroidal gas tank, as shown below.
Assuming that all metal is used, what must the values of inner radius $a$ and outer radius $b$ be to have maximal volume? What is the maximal volume?
I know that:
The surface area of torus is : $4\pi^2 ab$
The volume of torus is : $2 \pi^2 a^2 b$
I got $V = 5a$, how can I find the first derivative to test critical points?