$4.(60$ points$)$ A stationary normal shock stands in the diverging section of a convergingdiverging nozzle. The Mach number ahead of the shock is $3\mathrm{.}2.$ The nozzle area at the shock is $\backslash (550\backslash$mathrm$\{$~mm$\}^\{2\}\backslash ).$ The nozzle is fed from a large tank where the pressure is $1100$ kPa $($gage$)$ and the temperature is $420$ K $.$

$($a$)$ Find the Mach number, stagnation pressure, and static pressure after the shock.

$($b$)$ Calculate the nozzle throat area.

$($c$)$ If the nozzle exit area is $\backslash (620\backslash$mathrm$\{$~mm$\}^\{2\}\backslash ),$ estimate the exit Mach number.