Problem $1(50$ points$)$

A refrigerator $($shown in the figure below$)$ operates at steady state with a coefficient of performance of $5\mathrm{.}0$ within a kitchen at $23C.$ The refrigerator rejects $4\mathrm{.}8$ kW by heat transfer to its surroundings from metal coils located on its exterior. Determine:

$($a$)$ the power input, in $kW.$

$($b$)$ the lowest theoretical temperature inside the refrigerator, in $[$K$].$

Refrigerator

$=5\mathrm{.}0$

Surroundings, $23C$