$10\mathrm{.}2$ Consider an academic problem $($i$.$e$.$ not likely to occur is a practical design situation$)$ where temperatures for a real $($i$.$e$.$ work is not isentropic$),$ ideal gas refrigeration cycle are given at the turbomachinery exit states $(\backslash ($ T$\_\{2\}\backslash )$ and $\backslash ($ T$\_\{4\}\backslash )),$ but temperatures at turbomachinery inlet states $(\backslash ($ T$\_\{1\}\backslash )$ and $\backslash ($ T$\_\{3\}\backslash ))$ are unknowns. Derive two accurate engineering formulas, one for the compressor and one for the turbine, that will permit the inlet state temperatures to be computed using thermodynamic formulas and given values for isentropic efficiencies $\backslash (\backslash$eta$\_\{$t$\}\backslash )$ and $\backslash (\backslash$eta$\_\{$c$\}\backslash ).$