A composite wall consists of two layers with different materials in perfect thermal contact. The first layer $($Material A$)$ has thickness $L_1=0\mathrm{.}1mL$ and thermal conductivity $k_1=1\frac{W}{m}*K.$ The second layer $($Material B$)$ has thickness $L_2=0\mathrm{.}2mL$ and thermal conductivity $k_2=0\mathrm{.}5\frac{W}{m}*K.$ The inner surface of Material A is kept at $T_1=150C$ and the outer surface of Material B is exposed to air at $T_{}=30C$ with a convective heat transfer coefficient of $h=10\frac{W}{m^2}*K.$

a$)$ Find the temperature distribution $T(x)$ in each layer $($Material A and Material B$)$ under steady$-$state conditions.

b$)$ Calculate the temperature at the interface between the two materials.