A water tank doubles in volume every hour. If the tank is full at 11:00 AM, at what time was it a quarter full?

The amount of water in the tank doubles every hour.

At 11:00h, the tank was full. Let the full volume be $V$.
One hour before 11:00h, at 10:00h, the tank must have been half full, because it doubled to become full by 11:00h.
So, at 10:00h, the volume was $\frac{V}{2}$.

One hour before 10:00h, at 09:00h, the tank must have been a quarter full, because it doubled to become half full by 10:00h.
So, at 09:00h, the volume was $\frac{V}{4}$.

The tank was a quarter full at 09:00h.

Comparing this with the given options:
A 08:00h
B 09:00h
C 09:30h
D 10:00h
E 10:15h

The correct option is B.

The final answer is $\boxed{\text{09:00h}}$.