Pipes A and B can fill a tank in 5 and 6 hours respectively. Pipe C can empty it in 12 hours. If all the three pipes are opened together, how long will it take to fill the tank from empty to full?
How much water flows in or out during 1 hour? If the result indicates that the tank will fill in x hours, then
1/5 + 1/6 - 1/12 = 1/x
Now just solve for x
To find the time it will take to fill the tank from empty to full when all three pipes are open, we need to calculate their combined rate of filling the tank.
Let's start by finding the rate at which each pipe fills or empties the tank per hour:
Pipe A fills the tank in 5 hours, so its filling rate is 1/5 of the tank per hour. (1 tank / 5 hours)
Pipe B fills the tank in 6 hours, so its filling rate is 1/6 of the tank per hour. (1 tank / 6 hours)
Pipe C empties the tank in 12 hours, so its emptying rate is -1/12 of the tank per hour. (1 tank / -12 hours)
Now, add up the rates of the three pipes to find their combined rate of filling the tank:
Combined rate = (1/5) + (1/6) + (-1/12)
To make the denominators the same, we multiply and divide by the least common multiple of 5, 6, and 12, which is 60:
Combined rate = (1/5) * (12/12) + (1/6) * (10/10) + (-1/12) * (5/5)
Simplifying:
Combined rate = 12/60 + 10/60 - 5/60
Combined rate = 17/60
So the combined rate at which the three pipes fill the tank is 17 tanks per 60 hours.
To find the time it will take to fill the tank, we can use the formula:
Time = Amount / Rate
Since we want to fill the tank completely, the amount is 1 tank:
Time = 1 tank / (17/60 tanks per hour)
Simplifying:
Time = 60/17
So, it will take approximately 3.53 hours (or 3 hours and 32 minutes) to fill the tank from empty to full when all three pipes are opened together.