$13.$ Show all work $[12$ marks in total$]$

A fountain ejects water $(\backslash (\backslash$rho$=1000\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$m$\}^\{3\}\backslash )$ and $\backslash (\backslash$mu$=1\backslash$times $10^\{-3\}\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$m$\}\backslash$cdot $\backslash$mathrm$\{$s$\}\backslash ))$ upwards into the surrounding atmosphere through a nozzle with an inner diameter of $10$ mm $,$ as shown below. The inner diameter of the supply pipe is $60$ mm $.$ Water ejected from the nozzle reaches a height of $4$ m above the pipe's centerline, before falling back down. Assume steady, frictionless flow in performing the calculations required below.

a$)$ Estimate the volumetric flow rate of water through the pipe at section $\backslash ($ A $\backslash ).[6$ marks$]$

b$)$ Estimate the gauge pressure of water in the supply pipe at section $\backslash ($ A $\backslash ).[6$ marks$]$