Problem $4.$ Fins. $(25$ pts$)$ Heat is uniformly generated at the rate of $\backslash (2\backslash$times $10^\{5\}\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$m$\}^\{3\}\backslash )$ in a wall of thermal conductivity $\backslash (25\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$m$\}\backslash$cdot $\backslash$mathrm$\{$K$\}\backslash )$ and thickness $60$ mm $.$ The wall is exposed to convection on both sides, with different heat transfer coefficients and temperatures as shown. There are straight rectangular fins on the right$-$hand side of the wall, with dimensions as shown and thermal conductivity of $\backslash (250\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$m$\}\backslash$cdot $\backslash$mathrm$\{$K$\}\backslash ).$

a$)$ List all the assumptions $(\backslash (\backslash$mathbf$\{5\}\backslash$mathrm$\{$pts$\}\backslash ))$

b$)$ Find the expressions of the temperatures at the two ends of the wall $(8$ pts $)$

c$)$ Find the expression of the maximum temperature that will occur in the wall in terms of temperatures at the two ends of the wall as well? $(8$ pts $)$

d$)$ Intuitively, rank the $3$ temperatures in terms of order of increasing. $(4$ pts$)$