Problem $(1)$: Consider a fin of length $L,$ perimeter $p,$ and cross$-$sectional area $A_c.$ The fin is exposed to a fluid at temperature $T_{}$ with convective heat transfer coefficient $h.$ One end of the fin is attached to a base at temperature $T_b$ while the other end $($tip$)$ is exposed to the same fluid at temperature $T_{}$ with convective heat transfer coefficient $h.$ Show that in this case the temperature profile $T(x)$ within the fin is given by

$\frac{T(x)-T_{}}{T_b-T_{}}=\frac{cosh[m(L-x)]+(\frac{h}{m}k)sinh[m(L-x)]}{cosh[mL]+(\frac{h}{m}k)sinh[mL]}$

and the rate of heat transfer is given by

$Q^{}=\sqrt[\phantom{2}]{hpk{A}_c}(T_b-T_{})\frac{sinh[mL]+(\frac{h}{m}k)cosh[mL]}{cosh[mL]+(\frac{h}{m}k)sinh[mL]}$

where $k$ is the thermal conductivity of the fin material and $m^2=\frac{hp}{k{A}_c}.$