$(40$ points$)$ A farmer bought a used pump in really good condition for cheap! The pump is rated at $8$ hp $($shaft$)$ and has an $87\%$ efficiency. She plans to use it to pump water from a nearby river, with water at $40F$ moving at speed $6\frac{ft}{s},$ to a tank at an elevation of $260$ ft $.$ If she desires to pump water at a rate of $80\frac{\text{g}\text{a}\text{l}}{\text{m}\text{i}\text{n}},$ and she knows that she will need $1640$ ft of plastic piping, six flanged $90$ smooth bends and one flanged $180$ smooth bend, determine the minimum pipe diameter she should choose so she is able to use her pump. The pipe inlet will be aligned facing the coming water stream and there will no be losses due to the water inlet. Take the kinematic correction factor $=1.$