A hydraulic ram uses the kinetic energy of flowing fluid to raise part of that fluid to a higher derivation. The principle of operation is as follows: When water flowing down a supply tube is abruptly stopped, the pressure at the bottom of the pipe surges, and this allows some of the water to be raised above the feed level. With proper valving this pulsing action repeats $15200$ times$/$min$,$ with pumping efficiencies as high as $90\%.$ This type of pump requires no motor and is convenient to use in faraway places. $($a$)$ Develop an expression relating the fraction of feed which can be delivered to the higher elevation to the pertinent variables shown above. State the assumptions you make if any $($b$)$ A stream flows past my home, and I want to use it for my domestic water supply. I do not want to buy electricity to pump the water up to my house, so I've decided to use a hydraulic ram. If the fall of the stream is $3$ m$,$ the usable flow rate in the feed pipe is $2$ lit$/$s$,$ and the height from the ram to my storage tank is $8$ m$,$ what delivery rate can I expect at $50\%$ efficiency if the efficiency is defined as actual delivery rate divided by the theoretical?