fuel pump problem?

If your engine will stop after 10 min running, it sounds to me the problem is not in the battery. One way to see if your battery is drained at that time is once engine stopped by itself, turn your head light on to see if it is still bright. Bright light indicate you still have enough electric power in your battery. If the lack of electric power in the battery causes engine to stop, it has too be totally drained, and it should not even run your radio which only needs very little power. A failed fuel pump can cause all the symptoms you described. Now you cannot even to jump start the engine, this is good time to have the fuel pump checked.