A $2mm$ diameter and $10m$ long electric wire $($copper; $k_c=300\frac{W}{m*K})$ is tightly wrapped with a $1mm$ thick plastic cover whose thermal conductivity is $k_{\text{i}\text{n}\text{s}}=0\mathrm{.}15\frac{W}{m*K}.$ Electrical measurements indicate that a current of $10$A passes through the wire and that there is a voltage drop of $8V$ along the wire. $($For those that are pre$-$Physics II$,$ power in electricity $=$ current times voltage $P=IV,$ so if the current is constant, then you can figure out the difference in power at the beginning and end of the wire, and then assume that the power went to$....)$

A$)$ If the insulated wire is exposed to a medium at $T_{}=30C$ with a heat transfer coefficient $h=24\frac{W}{m^2*K},$ calculate the heat flux through the outside of the insulation layer at steady state.

B$)$ Determine the steady state temperatures at the interface between the plastic cover and the wire and at the outside of the insulation layer.

C$)$ What happens to the values found in A and B if the thickness of the insulation layer is doubled to $2mm?$ Explain your results.