Radioactive wastes are packed in a thin-walled spherical container. The wastes generate thermal energy nonuniformly according to the relation \(\dot{q}=\dot{q}_{o}\left[1-\left(r / r_{o}\right)^{2}\right]\) where \(\dot{q}\) is the local rate of energy generation per unit volume, \(\dot{q}\) is a constant, and \(r_{o}\) is the radius of the container. Steady-state conditions are maintained by submerging the container in a liquid that is at \(T_{\infty}\) and provides a uniform convection coefficient \(h\).

Determine the temperature distribution, \(T(r)\), in the container. Express your result in terms of \(\dot{q}_{o}, r_{o}, T_{\infty}, h\), and the thermal conductivity \(k\) of the radioactive wastes.

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Coolant Th -=[1-(rir)]