We know that the volume of a sphere of radius r is given by

Now, we have been given the initial volume was 4,188.79 cubic centimeters. Let us find the radius at this time.
Substituting the value of volume in the above equation, we get

Now, the final volume is given by 14,137.167 cubic centimeters. Thus, we have

Now, let us find the surface areas for these two radii.
For 
Surface area is given by

Similarly, for

Therefore, the increase in the surface area is given by

Hence, the average rate at which the surface area is changing is given by

Therefore, the average rate at which the surface area is changing is given by 130.8 square centimeters per second.