To solve this problem, I need to relate the number of rubber bands to the diameter of the ball. I can do this by using the fact that the volume of the ball is proportional to the cube of its diameter. Let V be the volume of the ball, D be its diameter, and N be the number of rubber bands. Then I can write V = kD^3, where k is some constant of proportionality. I also know that V = Nv, where v is the volume of one rubber band. So I have Nv = kD^3. Now I can plug in the given values for N, D, and v, and find k. I have 54v = k(3)^3, so k = 54v/27 = 2v. Now I can use this value of k to find the number of rubber bands when the diameter is 4 cm. I have Nv = k(4)^3, so Nv = 2v(64), and N = 128. This means Alana needs to add 128 - 54 = 74 rubber bands to increase the diameter by 1 cm. 
Alana is making a ball out of rubber bands.  When the ball has 54 rubber bands, it has a diameter of 3 cm.  How many rubber bands should Alana add to the ball to increase its diameter by 1 cm?  Assume that all of Alana's rubber bands have the same volume.