$1)$ A heat pump with a coefficient of performance of $4\mathrm{.}5$ receives a work input at a rate of $300\backslash (\backslash$mathrm$\{$Btu$\}/\backslash$mathrm$\{$hr$\}\backslash )$ while extracting heat from the surroundings at $\backslash (40^\{\backslash$circ$\}\backslash$mathrm$\{$F$\}\backslash ).$ Find the rate at which heat is extracted from the surroundings $($Btu$/$hr$)$ and provided to a dwelling $($Btu$/$hr$)$ to maintain an interior temperature of $\backslash (75^\{\backslash$circ$\}\backslash$mathrm$\{$F$\}\backslash ).$ What is the maximum coefficient of performance for the heat pump? How much electricity can be saved if the heat pump operated ideally? $(8$ pts$)$