Problem $3$: A heat pump operating at steady state maintains a dwelling at $22C$ on a day when the outside temperature is $-5C.$ Energy is loss by heat transfer from the dwelling at a rate of $20000\frac{J}{s}$ while the heat pump's power input is $8$ kW $.$

a$)$ Determine the coefficient of performance of the heat pump.

b$)$ Determine the power input required if it was a Carnot heat pump.

Sketch a well$-$labelled diagram $($indicating thermal reservoirs, temperatures, heat flow, work, etc.$)$