$10$ points$)$ A heat pump operating at steady state in the winter requires a power input of $1\mathrm{.}2$

kW to maintain a house at $23$C$.$ If heat transfer through the walls and roof is $0\mathrm{.}43$ kW per

degree temperature difference between the inside and outside temperatures,

a$.$ Determine the lowest theoretical

outside temperature, in $$C$,$ for which

the heating requirements can be met.

b$.$ If the actual coefficient of

performance is $9\mathrm{.}5,$ what is the

minimum outside temperature

possible to maintain the home at

$23$C