all i want to add to this is:

AA here IS freerolling with their money. If you don't understand this think harder. it is impossible for aa to lose money on this hand. if you don't understand that either, think even harder. That doesn't mean its correct to throw away 50BB if you think you might be behind, but in this situation, it probably is.

It couldn't be more wrong to say you shouldn't base your actions on how much you've already put in the pot. This is the driving force behind the entire game.