Pipe Filling Problem: Time Calculation
This problem involves calculating the time taken by two pipes, A and B, to fill a tank, considering Pipe A operates alone initially.
Calculating Pipe Rates
First, determine the rate at which each pipe fills the tank.
- Pipe A's time = 1 hour. Rate of Pipe A (\(R_A\)) = \(\frac{1}{1} = 1\) tank per hour.
- Pipe B's time = \(1\frac{1}{2}\) hours = \(\frac{3}{2}\) hours. Rate of Pipe B (\(R_B\)) = \(\frac{1}{3/2} = \frac{2}{3}\) tank per hour.
Work Done by Pipe A Alone
Pipe A is opened for 15 minutes before Pipe B is opened. Convert 15 minutes to hours:
Time A runs alone = 15 minutes = \(\frac{15}{60}\) hours = \(\frac{1}{4}\) hours = 0.25 hours.
Work done by Pipe A in this time = \(R_A \times \text{Time} = 1 \times \frac{1}{4} = \frac{1}{4}\) of the tank.
Remaining Work Calculation
Calculate the portion of the tank remaining to be filled.
Remaining Work = Total Capacity - Work Done by A = \(1 - \frac{1}{4} = \frac{3}{4}\) of the tank.
Combined Rate of Pipes A and B
When both pipes are open, their rates add up.
Combined Rate (\(R_{A+B}\)) = \(R_A + R_B = 1 + \frac{2}{3} = \frac{3}{3} + \frac{2}{3} = \frac{5}{3}\) tanks per hour.
Time Taken Together
Calculate the time needed for both pipes working together to fill the remaining \(\frac{3}{4}\) of the tank.
Time = \(\frac{\text{Remaining Work}}{\text{Combined Rate}} = \frac{3/4}{5/3}\)
Time = \(\frac{3}{4} \times \frac{3}{5} = \frac{9}{20}\) hours.
Convert Time to Minutes
Convert the calculated time from hours to minutes.
Time in minutes = \(\frac{9}{20} \times 60 = 9 \times 3 = 27\) minutes.
Therefore, both pipes will take 27 minutes more to fill the tank.
Similar Questions
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