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
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