A pump can be used for filling as well emptying a tank. The capacity of the tank is 2400 m^3 . The emptying tank capacity is 10 m^3 per minute higher than its filling capacity and the pump needs 8 minutes lesser to empty the tank than it needs to fill it. What is the filling capacity of the pump?

Solution

Let the filling capacity of the pump be \(x\) m³ per minute. Then, the emptying capacity of the pump is \(x + 10\) m³ per minute.

Given:
– The capacity of the tank is 2400 m³.
– The pump needs 8 minutes lesser to empty the tank than it needs to fill it.

Step 1: Express the time taken to fill and empty the tank

– Time taken to fill the tank: \(\frac{2400}{x}\) minutes
– Time taken to empty the tank: \(\frac{2400}{x + 10}\) minutes

Step 2: Set up the equation based on the given information

The pump needs 8 minutes lesser to empty the tank than to fill it:
\[ \frac{2400}{x} – \frac{2400}{x + 10} = 8 \]

Step 3: Solve the equation

\[ 2400(x + 10) – 2400x = 8x(x + 10) \]
\[ 24000 = 8x^2 + 80x \]
\[ x^2 + 10x – 3000 = 0 \]

Factoring the quadratic equation:
\[ (x + 60)(x – 50) = 0 \]

Since the filling capacity cannot be negative, we take the positive value:
\[ x = 50 \]

Conclusion

The filling capacity of the pump is 50 m³ per minute.