A certain part of cast iron piping of a water distribution system involves a parallel section. Both parallel pipes have a diameter of $20cm.$ One of the branches $($pipe A$)$ is $1500m$ long while the other branch $($pipe B$)$ is $2500m$ long. If the flow rate through pipe A is $0\mathrm{.}5\frac{m^3}{s}.$ Disregard minor losses and assume the water temperature to be $15C.$

$($a$)$ Determine the flow rate through pipe B$.$

$($b$)$ Show that the flow is fully rough, and thus the friction factor is independent of Reynolds number.

$($c$)$ What is the system is used to distribute oil instead of water. Given that oil temperature is $50C.$