A block coloured solid sphere of radius R and mass M is inside a cavity with vacuum inside .The walls of the cavity are maintained at temperature $\;T_{0}\;$ . The initial temperature of the sphere is $\;3T_{0}\;$.If the specific heat of the material of the sphere varies is $\;\alpha T^{3}\;$ per unit mass with the temperature T of the sphere , where $\;\alpha\;$ is a constant , then the time taken for the sphere to cool down to temperature $\;2T_{0}\;$ will be ($\;\sigma\; is stefan Boltzmann constant$

Question

$(a)\;\large\frac{M \alpha}{4 \pi R^{2} \sigma}\;ln(\large\frac{3}{2})\qquad(b)\;\large\frac{M \alpha}{4 \pi R^{2} \sigma}\;ln(\large\frac{16}{3})\qquad(c)\;\large\frac{M \alpha}{16 \pi R^{2} \sigma}\;ln(\large\frac{16}{3})\qquad(d)\;\large\frac{M \alpha}{16 \pi R^{2} \sigma}\;ln(\large\frac{3}{2})$