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