$.($a$)$ Water flows from a $\backslash (10-\backslash$mathrm$\{$cm$\}\backslash )$ diameter pipe to a $\backslash (5-\backslash$mathrm$\{$cm$\}\backslash )$ diameter pipe through a transition section. Calculate the discharge $($in $\backslash (\backslash$mathrm$\{$m$\}^\{3\}/\backslash$mathrm$\{$s$\}\backslash ))$ through the pipe when the water level difference in the two piezometers is $30$ cm $.$ The piezometric height in the $\backslash (5-\backslash$mathrm$\{$cm$\}\backslash )$ pipe is $80$ cm $.$

$(10$ marks$)$

Figure. $5($a$)$