In tank R, pipe A was opened for 9 hours and then closed. Pipes B and C were opened together for 3 hours and then both pipes were closed. After a total of 12 hours, approximately what volume of tank remained empty (in cubic meters), if both pipes A and C were opened together, the tank will always be empty?

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Accepted Answer

**Solution:**

1. Given that when pipes A and C are opened together, the tank will always be empty. This implies the efficiency of pipe A and pipe C are equal.

2. Time taken by pipe C alone to empty the tank = 18 hours.
   - Therefore, efficiency of pipe C = $ \frac{1}{18} $ tank per hour.

3. Time taken by pipe B alone to fill the tank = $ \frac{18}{3} = 6 $ hours.
   - Therefore, efficiency of pipe B = $ \frac{1}{6} $ tank per hour.

4. Let the capacity of tank R be 18 units (for simplicity).
   - Efficiency of pipe A = efficiency of pipe C = $ \frac{1}{18} = 1 $ unit/hour.
   - Efficiency of pipe B = 3 units/hour (since tank capacity is 18 units, $ \frac{18}{6} = 3 $ units/hour).

5. Calculate the volume filled in 12 hours:
   - Pipe A is opened for 9 hours: fills $ 9 \times 1 = 9 $ units.
   - Pipes B and C are opened together for 3 hours:
     - Combined efficiency = efficiency of B - efficiency of C = $ 3 - 1 = 2 $ units/hour (since C empties while B fills).
     - Volume filled in 3 hours = $ 3 \times 2 = 6 $ units.

6. Total volume filled = $ 9 + 6 = 15 $ units.

7. Remaining volume = Total capacity - filled volume = $ 18 - 15 = 3 $ units.

8. Given total tank capacity corresponds to 5390 m³, so 1 unit = $ \frac{5390}{18} \approx 299.44 $ m³.

9. Remaining volume in m³ = $ 3 \times 299.44 = 898.33 \approx 900 $ m³.

**Answer:** Approximately 900 m³

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