Question

A pump can be operated both for filling a tank and for emptying it. The capacity of the tank is \(2400 \, m^3\). The emptying capacity of the pump is \(10 \, m^3\) per minute higher than its filling capacity. Consequently, the pump needs 8 minutes less to empty the tank than to fill it. Find the filling capacity of the pump.

• A. 45 \(m^3\)/min
• B. 30 \(m^3\)/min
• C. 50 \(m^3\)/min
• D. 55 \(m^3\)/min

Hint:

Write filling and emptying times as 2400/x and 2400/(x+10), set their difference to 8 minutes, and solve the resulting quadratic.

Answer 1

The Correct Option is C

The equation $x^2 + 10x - 3000 = 0$ can also be solved directly with the quadratic formula instead of factoring.

1. Identify coefficients. Comparing with $ax^2 + bx + c = 0$: $a = 1$, $b = 10$, $c = -3000$.
2. Apply the quadratic formula. $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \dfrac{-10 \pm \sqrt{100 + 12000}}{2} = \dfrac{-10 \pm \sqrt{12100}}{2}$.
3. Simplify the square root. $\sqrt{12100} = 110$, so $x = \dfrac{-10 \pm 110}{2}$, giving $x = 50$ or $x = -60$.
4. Reject the negative root. A filling capacity cannot be negative, so $x = 50$.
5. Verify. Filling time $= 2400/50 = 48$ min, emptying time $= 2400/60 = 40$ min, and $48 - 40 = 8$ min, which matches the given condition.

So the filling capacity of the pump is $50 \, m^3$/min, option C.