$4-$ A small diameter metal rod has a heater installed on its tip. The rate of heat transfer provided by

the heater is $Q_h$ and is transferred by conduction into the tip of the rod as shown below. The

base of the rod is maintained at $T_b.$ The cross$-$sectional area of the rod is $A_c$ with a perimeter of

$P.$ The rod is surrounded by a fluid at $T_{}$ and a convective coefficient of $h.$ The rod has a

conductivity of $k.$ Assume the extended surface approximation $($i$.$e$.,$ temperature is a function of

$x$ only$).(20$ points$)$

A$)$ Sketch the temperature distribution in the rod. Assume $T_b{T}_{}.$ Justify your sketch by

listing some of the characteristics that you expect to see.

B$)$ Solve the differential equation analytically using the correct boundary conditions and obtain

the temperature profile within the rod.

C$)$ Solve part B using numerical solution with $15$ nodes along the fin longest axis $($i$.$e$,$ in the $x-$

direction$).$