Final answer:
The bell tolls 7 times between 11:45 am and 3:10 pm, starting at the next toll time of 12:00 pm and ending before the 3:30 pm toll.
Step-by-step explanation:
To calculate how many times a bell tolls between 11:45 am and 3:10 pm, we must consider that it rings every 30 minutes. We start counting from 12:00 pm, as it's the next toll after 11:45 am.
The tolls occur at 12 pm, 12:30 pm, 1:00 pm, 1:30 pm, 2:00 pm, 2:30 pm, and will not toll at 3:00 pm because we stop counting at 3:10 pm, which is before the 3:30 pm toll. In total, the bell tolls 7 times between 11:45 am and 3:10 pm.