A long 35-cm-diameter cylindrical shaft made of stainless steel 304 ($k = 14.9 \text{ W/m}^\circ\text{C}$, $\rho = 7900 \text{ kg/m}^3$, $C_p = 477 \text{ J/kg}^\circ\text{C}$, and $\alpha = 3.95 \times 10^{-6} \text{ m}^2/\text{s}$) comes out of an oven at a uniform temperature of 400$\circ$C. The shaft is then allowed to cool slowly in a chamber at 150$\circ$C with an average convection heat transfer coefficient of $h = 60 \text{ W/m}^2\circ\text{C}$. (a) Determine the temperature at $r = r_0/3$ from the center of the shaft, where $r_0$ is the radius of the shaft 20 min after the start of the cooling process. Solve the problem using analytical \textit{one-term approximation} method (not the Heisler charts). (b) Now, use the Lumped Capacitance method to estimate the temperature of the shaft, and compare it with the result obtained in (a)
