What are you trying to optimize?
Neither. It's one of
- If he has an unlimited number of tapes: Minimize the number of tapes needed (bin packing problem).
- If he has an limited number of tapes: Maximize the value of what will fit on the tapes he has (multiple knapsack problem).
Efficiently fill drives of fixed size? The latter is (as AppleFritter points out) the knapsack problem.
No, it's not, as I already pointed out. In the knapsack problem, you have one bin (tape), and you're trying the maximize what you can put on it. Stuff is left behind.