Consider a system where the work generated by a Carnot heat engine is used to drive a Carnot refrigerator. The Carnot heat engine operates with a source at $\backslash (900^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ and a sink at $\backslash (27^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$ The Carnot refrigerator removes heat from the cooled space at $\backslash (0^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ at a rate of $30$ kW and rejects heat to the same environment at $\backslash (27^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$ Determine $($a$)$ the rate of heat supplied to the heat engine and $($b$)$ the rate of heat rejected by the heat engine to the environment.