A booster pump can be used both for filling and for emptying a tank of capacity 2400 cubic metres. The emptying rate of the pump is 10 cubic metres per minute higher than its filling rate, and the pump needs 8 minutes less time to empty the full tank than to fill it. What is the filling capacity of the pump in cubic metres per minute?

Introduction / Context:
This problem involves a single pump that can operate in two modes: filling and emptying. We are given the total capacity of the tank, the difference between the two rates, and the difference in times taken in the two modes. The goal is to find the filling rate of the pump. This is solved by setting up an algebraic equation based on time equals volume divided by rate.

Given Data / Assumptions:

Tank capacity = 2400 cubic metres.

Emptying rate is 10 cubic metres per minute greater than filling rate.

Time taken to empty the tank is 8 minutes less than the time taken to fill it.

Concept / Approach:
Let the filling rate be F cubic metres per minute. Then the emptying rate is F + 10 cubic metres per minute. The time to fill the tank is 2400 / F minutes. The time to empty it is 2400 / (F + 10) minutes. We are told that emptying is 8 minutes faster, so we can set up the equation 2400 / F - 2400 / (F + 10) = 8 and solve for F.

Step-by-Step Solution:
Step 1: Let filling rate = F m^3 per minute.Step 2: Then emptying rate = F + 10 m^3 per minute.Step 3: Time to fill the tank = 2400 / F minutes.Step 4: Time to empty the tank = 2400 / (F + 10) minutes.Step 5: Given that emptying is 8 minutes faster, we write 2400 / F - 2400 / (F + 10) = 8.Step 6: Divide both sides by 8 to simplify: 300 / F - 300 / (F + 10) = 1.Step 7: Multiply through by F(F + 10): 300(F + 10) - 300F = F(F + 10).Step 8: Left side simplifies to 300F + 3000 - 300F = 3000.Step 9: So F(F + 10) = 3000, giving the quadratic equation F^2 + 10F - 3000 = 0.Step 10: Solve this equation: F^2 + 10F - 3000 = 0 has a positive root F = 50.Step 11: Therefore, filling rate of the pump = 50 cubic metres per minute.

Verification / Alternative check:
If F = 50, then filling time = 2400 / 50 = 48 minutes. Emptying rate = 60 m^3 per minute, so emptying time = 2400 / 60 = 40 minutes. The difference is 48 - 40 = 8 minutes, which matches the problem statement. Hence our value for F is correct.

Why Other Options Are Wrong:
60 m^3 per minute would make emptying rate 70, and the time difference would not equal 8 minutes.

72 m^3 per minute or 40 m^3 per minute similarly fail to satisfy the time difference when used in the equation and thus do not fit all the constraints.

Common Pitfalls:
One common mistake is to take 2400 / (F + 10) - 2400 / F = 8, reversing the sign of the time difference. Another is to cancel 2400 incorrectly or to forget that rates and times are reciprocals. Always express each time as volume divided by rate and then use the given information about the difference in times to set up a precise algebraic equation.

Final Answer:
The filling capacity of the pump is 50 m^3 per minute.