$5($a$)\_\backslash$#$3$: A heat pump must supply $\backslash (50,000\backslash$mathrm$\{$Btu$\}/\backslash$mathrm$\{$h$\}\backslash )$ on a $\backslash (30^\{\backslash$circ$\}\backslash$mathrm$\{$F$\}\backslash )$ day to maintain the temperature of a house at $\backslash (72\{\}^\{\backslash$circ$\}\backslash$mathrm$\{$F$\}\backslash ).$

a$)$ Determine the maximum coefficient of performance for this heat pump.

b$)$ What is the minimum power input, hp$,$ for this heat pump? $(1\mathrm{.}55$ hp $)$

c$)$ If the electricity cost is $8\mathrm{.}5$ center per kWh $,$ calculate the minimum cost to run this heat pump for one week. $(\backslash (\backslash$$ $16\mathrm{.}52/\backslash )$ month$)$

d$)$ Assuming that the power input that is available is in $1$ hp increments, determine actual COP and the corresponding increase in cost for one week for the closest available size. $(28\mathrm{.}9\backslash \%)$