Nucleophile's too weak a base for elimination, especially in acid solvent.
This problem is kind of a dumb one; it's like an instructor sat down and thought "Hmm, what are all the ways people use to determine whether a reaction is SN1 or SN2" and then thought of an example that didn't fall into either. In my opinion, this reaction would probably give a mixture of the SN2 product and a bit of the rearranged SN1 product (nucleophile attaches to the tertiary carbon).
The closest thing I have ever done is a SN2 of a substituted benzoic acid with 1-bromooctane in DMF. It worked brilliantly, but these conditions are crappy compared to that. Really, this looks like a silly problem. If I wanted to make that acetate, I would start with the secondary alcohol and acetylate it with AcCl or Ac2O.