Open$-$Ended Problem $(25$ points$)$

You require a micro$-$fridge to keep in your dorm room. Your roommate proposes a design where heat is discarded from a Carnot heat pump at a rate of

$|Q_c^{}|=kA{T}_c^4$

where $k$ is a constant and $A$ is the area fridge.

$5\mathrm{.}1.$ For a constant power output $|W^{}|,$ find the ratio $\frac{T_C}{T_H}$ that minimizes the area needed. points$]$

$5\mathrm{.}2.$ A different roommate suggests that the current mini$-$fridge in your room could be repurposed as an air condition unit by simply propping the door open. Explain why or why not this would work in $20$ words or less. $(8$ points$).$