A rod of diameter \(25 \mathrm{~mm}\) and thermal conductivity \(k=40 \mathrm{~W} / \mathrm{m} \mathrm{K}\) is part of a handle that protrudes from a furnace wall. The wall temperature is \(T_{w}=200^{\circ} \mathrm{C}\) and the first portion of the rod is covered by a thick blanket of insulation. To be useful as a handle, the end of the rod must be no more than \(50^{\circ} \mathrm{C}\). The ambient air temperature is \(T_{\infty}=25^{\circ} \mathrm{C}\) and the heat transfer coefficient to the environment is \(h=40 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}\).

a. Derive an expression for the temperature of the rod. Assume the exposed portion of the rod acts as an infinite fin.

b. How long must the insulated portion of the rod be to meet specifications?