$(80$ pts $)$ Air flows steadily from a pressurized upstream tank through a well$-$insulated, round, converging$-$diverging duct. The pressure at the nozzle exit is adjusted until there is a normal shock sitting at a location in the diverging part of the nozzle where the duct radius is $r=0\mathrm{.}2757m.$ Here are some values:

Throat radius is $0\mathrm{.}2154$ m

Stagnation pressure in the tank is $651$ kPa

Stagnation temperature in the tank is $467$ K

$($a$)$ Calculate Mach number just upstream of the normal shock. Note: You will need to solve implicitly for Mach number using any technique you wish.

$($b$)$ Calculate the pressure just upstream of the normal shock.

$($c$)$ Calculate Mach number just downstream of the normal shock.

$($d$)$ Calculate the pressure just downstream of the normal shock.