Consider steady two$-$dimensional heat transfer in a long solid body whose cross section shown in Figure $1.$ The temperatures at the selected nodes and the thermal conditions at the boundaries are also presented on Figure $1.$ The thermal conductivity of the body is $k=14W{m}^{-1}K,$ and heat is generated in the body uniformly at a rate of $e^{}=2\mathrm{.}3{10}^{6}W{m}^{-3}.$ Using the finite difference method with a mesh size of $x=1\mathrm{.}5cm$ and $y=0\mathrm{.}75cm,$ determine:

a$.$ The temperatures at nodes $1,2,$ and $3$

b$.$ The rate of heat loss from the bottom surface through a $1-$m$-$long $($in the $z-$direction$)$ section of the body.

Figure $1$: Schematic of the solid body cross section with temperatures and boundary conditions