Two pipes are used to carry water between two open tanks whose water surfaces are at different elevations. One pipe has twice the diameter of the other. If both pipes have the same friction factor, \(f\), all other sources of loss are negligible, and the volumetric flowrate is constant:

a. Prove that the head loss, \(h_{L}\) is proportional to \(1 / d^{5}\).

b. In the current situation, what is the ratio of flow rates in the two pipes in terms of the parameters given in the problem?