The plane wall in Figure $3$ is constructed from two materials with thermal conductivities $k_A$ and $k_B$ and thicknesses $L_4$ and $L_8$ respectively. The left side of Material $A$ is maintained at a temperature $T_S$ while Material $B$ is next to a fluid at a temperature $T_{}$ with a convective heat transfer coefficient $h.$ The wall area is given b$,$ A$.)$ Assume an idealized smooth interface between Material A and Material B such that there is no contact resistance.

$($a$)$ Draw the equivalent resistance circuit, label the nodes and equivalent resistances $(15$ points$).$

$($b$)$ Find the algebraic expression denoting the heat transfer rate, $q,$ through the wall in terms of given symbolic parameters $(15$ points$).$

$($c$)$ Using the solution from part $($b$)$ Calculate the heat transfer rate through the wall if $k_A=0\mathrm{.}06\frac{W}{m}K,k_B=0\mathrm{.}04\frac{W}{m}$ K$,L_A=12mm,L_B=14mm,T_{}=20C,T_S=187C,h=13\frac{W}{m^2}*K,A=1m^2*$ Convert all temperatures to Kelvin and all dimensions including thicknesses to meters. Show detailed calculations $(10$ points$).$