Problem $1(4$ points$)$: A thin, long rod of total length $(L_{\text{i}\text{n}\text{s}}+L_0)mm,$ cross$-$section of diameter $25$ mm $,$ and thermal conductivity $50\frac{W}{m}-K,$ protrudes out of a furnace wall $($at $T_w=200C),$ which is insulated to a length of $L_{\text{i}\text{n}\text{s}}=200mm.$ Ambient temperature and convection coefficient are $25C$ and $15\frac{W}{m^2}-K,$ respectively. You are told that the maximum operating temperature of the exposed portion of the rod $($the portion $T_o$ where the insulation stops$)$ must be maintained at $92C.$ Determine whether the rod, with the parameters described in the problem, will meet the specified temperature requirement. In doing so$,$ consider two designs:

a$.$ Design $1$: Tip of the rod is insulated, and $L_o=200mm$

b$.$ Design $2$: Assume $m{L}_0=3.$ What case of fins does this fall under?

Hint: For each case, determine $T_o$ as if it is unknown, then compare it to the maximum temperature of $92C$ to see if either design would work.