Final answer:
The next same-day pick-up date after September 2nd for both trash and recyclables by the sanitation department and recycling center respectively is September 14th, based on the least common multiple of their pick-up cycles.
Step-by-step explanation:
To find the next date when both the sanitation department and recycling center pick up on the same day, we need to determine the least common multiple (LCM) of their pick-up cycles, which are every 3 and 4 days, respectively. The LCM of 3 and 4 is 12, which means they will both pick up on the same day every 12 days. Since they both picked up on September 2nd, we add 12 days to this date to find the next same-day pick-up.
September 2nd + 12 days = September 14th.
Therefore, option D) September 12th is the correct answer. The answer provided in the question seems to have a typo, as it asks for an answer that is not logically consistent with the problem details. The correct answer determined through calculation is September 14th, assuming there were no typos in the problem statement regarding the pick-up dates.
Regarding the provided reference about children cleaning their rooms, it seems out of context, but to address it briefly, if the time taken by children to clean their rooms is uniformly distributed from one to 15 days, the expected wait time would be the average of these values, which is 8 days (1+15/2). So, the answer would be A) 8 days.