$3\mathrm{.}71$ Consider a plane composite wall that is composed of three materials $($materials A$,$ B$,$ and C are arranged left to right$)$ of thermal conductivities $k_A=0\mathrm{.}24\frac{W}{m}*K,k_B=0\mathrm{.}13\frac{W}{m}*K,$ and $k_C=0\mathrm{.}50\frac{W}{m}*K.$ The thicknesses of the three sections of the wall are $L_A=20mm,L_B=13mm,$ and $L_C=20mm.$ A contact resistance of $R_{t,c}^{\text{'}\text{'}}={10}^{-2}{m}^2*\frac{K}{W}$ exists at the interface between materials A and B$,$ as well as at the interface between materials B and C$.$ The left face of the composite wall is insulated, while the right face is exposed to convective conditions characterized by $h=10\frac{W}{m^2}*K,T_{-}=20C.$ For Case $1,$ thermal energy is generated within material A at the rate $q_A^{}=5000\frac{W}{m^3}.$ For Case $2,$ thermal energy is generated within material C at the rate $q_C^{}=5000\frac{W}{m^3}.$

$($a$)$ Determine the maximum temperature within the composite wall under steady$-$state conditions for Case $1.$

$($b$)$ Sketch the steady$-$state temperature distribution on $T-x$ coordinates for Case $1.$

$($c$)$ Sketch the steady$-$state temperature distribution for Case $2$ on the same $T-x$ coordinates used for Case $1.$

$($d$)($Extra question added by me$)$ Determine the maximum temperature within the composite wall under steady state conditions for Case $2.$