Let the total capacity of the cistern be $x$ liters.

Step 1: Rate of filling by pipes A and B

• Pipe A can fill the cistern in 15 minutes, so its rate of filling is $\frac{x}{15}$ liters per minute.
• Pipe B can fill the cistern in 10 minutes, so its rate of filling is $\frac{x}{10}$ liters per minute.

Step 2: Rate of emptying by pipe C

• Pipe C removes water at a constant rate of 7 liters per minute.

Step 3: Net rate of filling when all pipes are open

When all three pipes are open, the net rate of filling is the sum of the filling rates of pipes A and B minus the emptying rate of pipe C. This is:

$\text{Net rate} = \left( \frac{x}{15} + \frac{x}{10} - 7 \right) \text{ liters per minute}.$

Step 4: Given information

When all pipes are open, the cistern is emptied in 2 hours, i.e., 120 minutes. This means that the net rate of water flow is negative, and in 120 minutes, the entire cistern is emptied. Therefore, the total volume $x$ is emptied in 120 minutes, implying the following equation:

$\text{Net rate} \times 120 = -x.$

Step 5: Solve for $x$

The net rate can be simplified by finding a common denominator for the fractions:

$\frac{x}{15} + \frac{x}{10} = \frac{2x}{30} + \frac{3x}{30} = \frac{5x}{30} = \frac{x}{6}.$

Thus, the net rate of water flow is:

$\frac{x}{6} - 7 \text{ liters per minute}.$

Now, using the fact that the cistern is emptied in 120 minutes:

$\left( \frac{x}{6} - 7 \right) \times 120 = -x.$

Expanding the equation:

$\frac{120x}{6} - 7 \times 120 = -x,$ $20x - 840 = -x.$

Now, solve for $x$:

$20x + x = 840,$ $21x = 840,$ $x = \frac{840}{21} = 40 \text{ liters}.$

Final Answer:

The cistern holds 40 liters.

Would you like more details or have any questions?

Here are 5 relative questions for further practice:

1. What would be the net rate of filling if the waste pipe C was not opened?
2. How long would it take to fill the cistern if pipe B was closed?
3. If the waste pipe C increased its rate to 10 liters per minute, how would that affect the time to empty the cistern?
4. How much time would it take for pipes A and C to fill the cistern together?
5. What is the efficiency of each pipe in terms of liters filled per minute?

Tip: When solving problems with multiple rates, always ensure to carefully consider the net effect of all rates combined (filling vs. emptying).