Notice that it is asking you to "minimize" therefore this is an optimization problem, not related rates.
I too find this problem hard but I can assist you with what I know:
Notice that they ask you to minimize, thus you have to find c'(h), then set c'(h)=0 and when you have found that h, use c''(h) to see which solutions are minimum or maximum.
I don't think you have to relate h to l since l was given as a constant, and they also tell you that your answer for h will be in terms of l.
Again, I'm not completly sure but I hope I helped.
Oh. I was under the impression that was variable?
Also, don't we need to somehow use that information about the density, rho?
I do know about finding critical points of a function, but I'm certain that I first must eliminate one of the variables, since this is a single-variable calculus subject, and there are no partials here.