$6\mathrm{.}33$ A normal$-$shock inlet operates in a Mach $1\mathrm{.}86$ stream with ambient static pressure of $p_0=30$kPa. Neglecting total pressure loss in the subsonic diffuser, calculate

a$.$ inlet total pressure recovery with the shock at the lip, i$.$e$.$ the best backpressure

b$.$ inlet total pressure recovery when the shock is inside the duct at $\frac{A_x}{A_t}=1\mathrm{.}2,$ i$.$e$.$ the supercritical mode

c$.$ inlet total pressure recovery in subcritical mode with $10\%$ spillage, i$.$e$.\frac{A_{\text{s}\text{p}\text{i}\text{l}\text{l}\text{a}\text{g}\text{e}}}{A_1}=0\mathrm{.}1$

d$.$ flight dynamic pressure $q_0$