A $1$ km long pipe links two reservoirs whose difference in levels is maintained at $20$ m $.$ Calculate:

$($a$)$ the diameter of a "smooth" pipe that will convey a flow of $50\frac{L}{s}.$

$($b$)$ the diameter of a pipe with relative roughness $0\mathrm{.}002$ that will also convey $50\frac{L}{s}.$

$($c$)$ the flow rate that will be carried by a "smooth" $20$ cm diameter pipe.

$(17cm,19cm,76\frac{L}{s})$