$($a$)$ Assume you have a concrete pipe $($CHW $=110)$ with the following characteristics: the upstream end is at an elevation $($Z$1)$ of $630$ meters and has a pressure $($P$1)$ of $573$ kPa; the downstream end is at an elevation $($Z$2)$ of $534$ meters; and the pipe length between points $1$ and $2($L$12)$ is $1$ kilometer. At point $2,$ you desire a discharge $($Q$)$ of $179$ L$/$s and a minimum pressure $($P$2)$ of $458$ kPa. What is the minimum diameter of concrete pipe required to meet the above conditions? Pipes are available in $50$mm increments. Use Bernoulli$$s equation and the Hazen$-$Williams equation to determine your answer.

$($b$)$ Now, the pipe is extended to a third point. The third point is at an elevation $($Z$3)$ of $479$ meters, and you desire a minimum pressure $($P$3)$ of $385$ kPa. Assume Q remains constant and that the slope $($Sf $)$ between points $2$ and $3$ is $0\mathrm{.}030$ m$/$m$.$ What is the pipe length $($L$23)$ between these two points?

$($c$)$ Does the minimum diameter you found in part $($a$)$ satisfy the conditions at point $3?$ Show your work.