Three pipes A, B and C can fill a tank in 12 hours, 18 hours and 24 hours, respectively. A leak at the bottom can empty the full tank in 36 hours. If all the pipes are opened together, in how many hours will the tank be filled? (Round off your answer to two decimal places.)
6.55 hours
To find out how long it will take to fill the tank with all the pipes and the leak working together, we need to determine the net rate at which the tank fills. Let's calculate this step by step:
- The rate at which pipe A fills the tank is \( \frac{1}{12} \) of the tank per hour because it can fill the entire tank in 12 hours.
- Similarly, the rate at which pipe B fills the tank is \( \frac{1}{18} \) of the tank per hour.
- And pipe C fills the tank at a rate of \( \frac{1}{24} \) of the tank per hour.
- The leak, which empties the tank, works at a rate of \( \frac{1}{36} \) of the tank per hour.
- To find the net rate of filling the tank, we add the rates of pipes A, B, and C and subtract the rate at which the leak empties the tank. The equation is given by: \(\text{Net rate} = \left(\frac{1}{12} + \frac{1}{18} + \frac{1}{24}\right) - \frac{1}{36}\)
- Calculating each part:
- \( \frac{1}{12} = 0.0833 \) (approx.)
- \( \frac{1}{18} = 0.0556 \) (approx.)
- \( \frac{1}{24} = 0.0417 \) (approx.)
- \( \frac{1}{36} = 0.0278 \) (approx.)
- Add these rates: \(0.0833 + 0.0556 + 0.0417 - 0.0278 = 0.1528\)
- Therefore, the net rate at which the tank is filled is \( 0.1528 \) of the tank per hour.
- To find the total time required to fill one tank, we calculate the reciprocal of the net rate: \(\text{Time} = \frac{1}{0.1528} \approx 6.55 \text{ hours}\)
Hence, the time taken to fill the tank with all the pipes working along with the leak is approximately 6.55 hours. Therefore, the correct answer is 6.55 hours.
Similar Questions
Two pipes A and B can fill an empty cistern in 32 and 48 hours, respectively. Pipe C can drain the entire cistern in 64 hours when no other pipe is in operation. Initially, when the cistern was empty Pipe A and Pipe C were turned on. After a few hours, Pipe A was turned off and Pipe B was turned on instantly. In all it took 112 hours to fill the cistern. For how many hours was Pipe B turned on?
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