Water to run a Pelton wheel is supplied by a penstock of length \(\ell\) and diameter \(D\) with a friction factor \(f\). If the only losses associated with the flow in the penstock are due to pipe friction, show that the maximum power output of the turbine occurs when the nozzle diameter, \(D_{1}\), is given by \(D_{1}=D /(2 f \ell / D)^{1 / 4}\).