Find the triangle with perimeter $2p$ given that, when we rotate it around one of its sides, the solid obtained have the maximum volume.
Well, suppose that we have a triangle of sides $a,b,c$ such that $a+b+c=2p$ and, without loss of generality, when we rotate it around the support line of the side with size c, we want to get the solid with maximum volume. We can clearly see that this solid is formed by 2 cones and, to find the volume of both of them, we need the radius of its bases that is given by the relative height to the side with size $c$ . But, I don't see now to construct the function of my volume without adding new variables (using sine and consine, for example). Maybe we can construct another restriction for my volume function (using the intern angles of my triangle) and use Lagrange multipliers, but I am not sure that this is the better form to do this exercise.
Edit: Probably another way to find my height is to use that $Area = \sqrt{p(p-a)(p-b)(p-c)}$ = $\frac{c.h}{2}$
Thanks for your help!