$($b$)$ A centrifugal pump has an impeller diameter of $1$ m $.$ It is to be design so that it will supply a head rise of $200$ m at a flow rate of $4\mathrm{.}1\frac{m^3}{s}$ of water when operating at a speed of $1200$ rpm $.$ In order to investigate the performance of the pump a $\frac{1}{5}$ geometrically similar model operated at the same speed will be testing in a laboratory setting. Determine the required flow rate and head rise assuming both the model and full scale pump operate with the same efficiency and flow coefficient.