Question $1(20$ points$)$

A reversible refrigeration cycle operates between cold and hot reservoirs at temperatures $T_C$ and $T_H,$ respectively.

$($a$)$ If the coefficient of performance is $3\mathrm{.}5$ and $T_C=-48F,$ determine $T_H,$ in $F.$

$($b$)$ If $T_C=-38C$ and $T_H=38C,$ determine the coefficient of performance.

$($c$)$ If $Q_C=580$Btu,$Q_H=880$Btu, and $T_C=28F,$ determine $T_H,$ in $F.$

$($d$)$ If $T_C=38F$ and $T_H=180F,$ determine the coefficient of performance. $($e$)$ If the coefficient of performance is $8\mathrm{.}8$ and $T_C=-18C,$ find $T_H,C.$