Just bought 2011 THREE I'm afraid battery is faulty. Please help

If you are able, return the car
Out of curiosity, was the battery behavior present during your test drive ?

 

By the way the battery behavior noted is similar to another thread we have going right now. The other case is low MPG and close to warranty so the guy is wondering if Toyota owes him a battery, but of course his battery hasn't died yet either.

But Jimbo is correct this car has warranty left.

 

Two possibilities come to mind:
1. You're hearing one of the many systems that runs after you turn the car off, such as checking that the fuel tank is not leaking vapors to the atmosphere, or the brake pump, or a variety of other things.
2. If it's actually the engine, are you sure you shut the car off at night? The READY light and everything else on the dashboard is dark? MANY of the people on this forum have forgotten at one point or another. Sometimes, the car beeping reminds us, or it starts moving as you get out (;-)); other times, people don't notice until they hear the engine running much later. That would explain all the symptoms adequately; the battery will discharge most of the way, then the engine will run periodically to keep it slightly charged. Running for a while overnight is enough to drop your MPG pretty significantly. (The average MPH on the trip meter should be pretty low too, in that case.)

I'd bet my money that your car is probably fine. All three of your symptoms are common worries about the car, especially from people who are new to it.
1. The car can't accurately track how charged the battery is while it's running - it counts how much electricity goes in and how much comes out, but over time it can get off. The next time you turn the car on, it can recalibrate since it has been resting for a while. This is why it won't necessarily show the same charge level when you turn it back on as it had when you turned it off.
2. If you're driving at low speed, or in the left half of the HSI, this is par for the course. Full battery to empty can happen in just a half mile or so.
3. Reset the trip meter. Lots of things can drag it down, like getting stuck in a traffic jam with the A/C blasting, or if the dealer left the car on for a long time (as noted above). Do some "normal" driving; more than 5 miles or so to give the engine time to warm up. Pay attention to if the engine shuts off when you are stopped. If the battery is having trouble, the engine will stay running - but there are also lots of times that the engine stays running while stopped even if your battery is just fine.