Pipes P and Q can fill a storage tank in full with water in 10 and 6 minutes, respectively. Pipe R draws the water out from the storage tank at a rate of 34 litres per minute. P, Q and R operate at a constant rate.
If it takes one hour to completely empty a full storage tank with all the pipes operating simultaneously, what is the capacity of the storage tank (in litres)?
Calculating Storage Tank Capacity
This problem involves calculating the capacity of a storage tank based on the filling and emptying rates of three pipes operating simultaneously.
Pipe Rates Analysis
- Pipe P fills the tank in 10 minutes. Its filling rate is $ \frac{1}{10} $ tank per minute.
- Pipe Q fills the tank in 6 minutes. Its filling rate is $ \frac{1}{6} $ tank per minute.
- Pipe R empties the tank at a rate of 34 litres per minute.
- The combined operation of P, Q, and R empties a full tank in 1 hour (60 minutes).
Net Rate Calculation
Let the capacity of the tank be C litres.
- Rate of P (filling) = $ \frac{C}{10} $ litres/minute.
- Rate of Q (filling) = $ \frac{C}{6} $ litres/minute.
- Rate of R (emptying) = 34 litres/minute.
When all pipes operate together, the net rate is the sum of filling rates minus the emptying rate:
Net Rate = (Rate of P + Rate of Q) - Rate of R
Net Rate = $ \left( \frac{C}{10} + \frac{C}{6} \right) - 34 $ litres/minute.
Determining Tank Capacity
Since the tank empties completely in 60 minutes with all pipes operating, the net rate must be negative, representing the rate at which the tank is emptying.
The rate of emptying the full tank (capacity C) in 60 minutes is $ \frac{C}{60} $ litres/minute.
Therefore, the net rate can also be expressed as $ -\frac{C}{60} $ litres/minute.
Equating the two expressions for the net rate:
$ \left( \frac{C}{10} + \frac{C}{6} \right) - 34 = -\frac{C}{60} $To solve for C, find a common denominator, which is 60:
$ \left( \frac{6C}{60} + \frac{10C}{60} \right) - 34 = -\frac{C}{60} $ $ \frac{16C}{60} - 34 = -\frac{C}{60} $Add $ \frac{C}{60} $ to both sides and add 34 to both sides:
$ \frac{16C}{60} + \frac{C}{60} = 34 $ $ \frac{17C}{60} = 34 $Now, solve for C:
$ 17C = 34 \times 60 $ $ C = \frac{34 \times 60}{17} $ $ C = 2 \times 60 $ $ C = 120 $The capacity of the storage tank is 120 litres.
Important Questions from Pipe and Cistern
A pipe can fill a cistern in 20 minutes where as the cistern when full can be emptied by a leak in 28 minutes. When both are opened, The time taken to fill the cistern is:
‘A’ pipe can empty a tank in 20 minutes. The second pipe ‘B’ has a diameter twice as that of ‘A’. If both A & B pipe are attached to the tank how much time will be required to empty the tank?
Pipes A and B can empty a full tank in 16 hours and 24 hours, respectively. Pipe C alone can fill the empty tank in 4 hours. If A, B and C are opened together, the tank will be 35% full after :
Pipes A and B can fill a tank in 36 minutes and 45 minutes, respectively. Both these pipes were opened simultaneously. After 20 minutes, a leak at the bottom of the tank was spotted which was immediately sealed. The tank was full in another 15 minutes. The leak alone can empty the full tank in:
A cistern has a leak which would empty it in 6 hours. A tap is turned on which admits 10 litres of water per minute into the cistern. When it is full it is now emptied in 10 hours. What is the capacity (in litres) of the cistern?
