Consider a large plane wall of thickness $L=0\mathrm{.}4m,$ thermal conductivity $k=2\mathrm{.}3\frac{W}{m}*C,$ and surface area $A=20m^2.$ The left side of the wall is maintained at a constant temperature of $T_1=80C$ while the right side loses heat by convection to the surrounding air at $T_2=15C$ with a heat transfer coefficient of $h=24\frac{W}{m^2}*C.$ Assuming constant thermal conductivity and no heat generation in the wall, with expressing the differential equation and the boundary conditions for steady $1$D heat conduction through the wall, $($a$)$ find a relation for the variation of temperature in the wall by solving the differential equation, and $($b$)$ evaluate the rate of heat transfer through the wall. $($Solution of Steady One$-$Dimensional Heat Conduction Problems$)$

$($Answers: $($a$)T(x)=80-131\mathrm{.}1x,($b$)6\mathrm{.}03kW)$