Okay, first of all:
Here, it is smart to express the surface S in terms of the volume V:
S=6V^{\frac{2}{3}}
Thus, the differential equation for the rate of change of the volume is:
\frac{dV}{dt}=-6kV^{\frac{2}{3}}
This is a separable equation:
\frac{dV}{V^{\frac{2}{3}}}=-6kdt
or, integrating both sides from t=0 and and t=T:
3(V(T)^{\frac{1}{3}}-V(0)^{\frac{1}{3}})=-6kT
or simply, for arbitrary T:
V(T)=(V(0)^{\frac{1}{3}}-2kT)^{3}
Now you should be able to do the last steps on your own!