Consider a cylindrical pipe transporting ethanol vapor with a length of $L=12m.$ The pipe has an imer radius of $r_1=15cm,$ an outer radius of $r_2=20cm,$ and a thermal conductivity of $k=8\mathrm{.}5\frac{W}{m}*C.$ The ethanol vapor flowing inside has an average temperature of $260C,$ and the heat transfer coefficient on the inner surface is given as $h=15\frac{W}{m^2}C.$ The outer surface of the pipe was maintained at an average temperature of $170C.$

$($a$)$ Formulate the differential equation governing steady$-$state one$-$dimensional heat conduction through the pipe, along with the corresponding boundary conditions.

$($b$)$ Derive the temperature distribution inside the pipe by solving a differential equation.

$($c$)$ Calculate the total heat loss from ethanol vapor to the surroundings through the pipe wall.