Actually this is one of the first problems in the calculus of variations. It's named "the Newton problem" after Sir Isaac Newton, who studied it in 1685. It arises from the determination of the optimal profile for the motion of bodies (projectiles, ships, etc), that is, giving the minimal aerodynamic or hydrodynamic resistance. Here you are assuming axial symmetry, but the problem is even studied under more general assumptions.
So you can find a lot of material from the keywords "Newton problem" and "Calculus of variations". Here is a nice survey paper by Giuseppe Buttazzo (see in particular section 2 and the reference therein for the radial case).