$3.$ A rocket engine is designed to operate at a pressure ratio $($reservoir pressure to back pressure$)$ of $37.$ Determine:

a$.$ The exit$-$to$-$throat area ratio required for the supersonic exhaust to be properly expanded.

b$.$ The Mach number of the exit flow under correctly expanded conditions.

c$.$ The minimum pressure ratio $\backslash (\backslash$left$($p$\_\{0\}/$ p$\_\{$b$\}\backslash$right$)\backslash )$ at which the nozzle becomes choked.

d$.$ The pressure ratio $\backslash (\backslash$left$($p$\_\{0\}/$ p$\_\{$b$\}\backslash$right$)\backslash )$ at which a normal shock wave occurs at the nozzle exit. Assume the specific heat ratio of the gas is $1\mathrm{.}4.$