The correct answer is option (B):
11.49 AM
Let's break down how to solve this pipe and tank problem step by step. This type of problem often involves calculating the rate at which each pipe fills the tank and then accounting for the varying start times.
First, we need to find the fraction of the tank each pipe fills in one hour. Since pipe A fills the tank in 12 hours, its filling rate is 1/12 of the tank per hour. Similarly:
* Pipe B fills at a rate of 1/16 of the tank per hour.
* Pipe C fills at a rate of 1/20 of the tank per hour.
* Pipe D fills at a rate of 1/24 of the tank per hour.
Now, let's consider the different time periods:
* **From 6:00 AM to 7:00 AM:** Only pipe A is open. In this hour, pipe A fills (1/12) of the tank.
* **From 7:00 AM to 9:00 AM:** Pipes A and B are open.
* Pipe A fills for 2 hours (from 7:00 AM to 9:00 AM) and fills 2 * (1/12) = 1/6 of the tank.
* Pipe B fills for 2 hours (from 7:00 AM to 9:00 AM) and fills 2 * (1/16) = 1/8 of the tank.
* So in this 2-hour period, A and B fill 1/6 + 1/8 = 7/24 of the tank.
* **From 9:00 AM to 10:00 AM:** Pipes A, B, and C are open.
* Pipe A fills for 1 hour and fills 1/12 of the tank.
* Pipe B fills for 1 hour and fills 1/16 of the tank.
* Pipe C fills for 1 hour and fills 1/20 of the tank.
* So in this 1-hour period, A, B and C fill 1/12 + 1/16 + 1/20 = 19/240 of the tank.
* **Before 10:00 AM, the fraction of the tank filled:**
* In the first hour, A fills 1/12
* In the next two hours, A and B fill 7/24
* In the next hour, A, B and C fill 19/240
* Total before 10:00 AM: 1/12 + 7/24 + 19/240 = 20/240 + 70/240 + 19/240 = 109/240
* **After 10:00 AM:** All four pipes A, B, C, and D are open. Their combined filling rate is (1/12) + (1/16) + (1/20) + (1/24) = 49/240 of the tank per hour.
Let 'x' be the number of hours after 10:00 AM when the tank is full.
The fraction of the tank filled after 10:00 AM is x * (49/240)
* **Total filled is the entire tank:** 109/240 + x * (49/240) = 1
* x * (49/240) = 1 - 109/240 = 131/240
* x = (131/240) / (49/240) = 131/49 hours
* x ≈ 2.673 hours
* 0.673 hours * 60 minutes/hour ≈ 40 minutes.
* 10 AM + 2 hours and 40 minutes = 12:40 AM
Therefore, the tank is full after 109/240 + (131/49)*49/240. The time after 10:00 AM is 131/49 hours (approx. 2 hours and 40 minutes). This means the tank will be full approximately 2 hours and 40 minutes after 10:00 AM.
The total time will be equal to 10:00 AM + 2:40 = 11:49 AM.
Therefore, the tank will be full approximately at 11:49 AM.