b
(b) Rate of cooling \((R) = \frac{{\Delta \theta }}{t} = \frac{{A \in \sigma ({T^4} - T_0^4)}}{{mc}}\)

==> \(R \propto \frac{A}{m} \propto \frac{{{\rm{Area}}}}{{{\rm{volume}}}} \propto \frac{{{r^2}}}{{{r^3}}} \propto \frac{1}{r}\)

==> Rate \((R) \propto \frac{1}{r} \propto \frac{1}{{{m^{1/3}}}}\)  \(\left[ {\because \,\;m = \rho  \times \frac{4}{3}\pi {r^3} \Rightarrow r \propto {m^{1/3}}} \right]\)

==> \(\frac{{{R_1}}}{{{R_2}}} = {\left( {\frac{{{m_2}}}{{{m_1}}}} \right)^{1/3}} = {\left( {\frac{1}{3}} \right)^{1/3}}\)