We are given that two pipes are connected to a swimming pool, with the first pipe filling the pool and the second pipe draining it. The first pipe takes 2 hours longer to fill the pool than the second pipe takes to drain it. Initially, the pool is 40% full, and after both pipes are opened, the pool is empty after 9 hours and 36 minutes.

Let's define the variables:

• Let $t$ be the time (in hours) it takes for the second pipe to drain the pool.
• The first pipe fills the pool in $t + 2$ hours.

Rate of filling and draining:

• The rate of the first pipe filling the pool is $\frac{1}{t+2}$ pools per hour.
• The rate of the second pipe draining the pool is $\frac{1}{t}$ pools per hour.

Initial condition:

The pool is initially 40% full, which means 60% of the pool needs to be drained.

Combined effect of both pipes:

When both pipes are opened, the net rate at which the pool is drained is the difference between the draining rate and the filling rate: $\text{Net rate} = \frac{1}{t} - \frac{1}{t+2}$ This is the rate at which the pool is being emptied.

Time condition:

The pool becomes empty after 9 hours and 36 minutes, which is equivalent to 9.6 hours. The total volume of the pool that needs to be drained is 60% of the pool, or $0.6$ pools.

Equation setup:

The net rate of emptying the pool multiplied by the time taken should equal 0.6 (since we are draining 60% of the pool): $\left( \frac{1}{t} - \frac{1}{t+2} \right) \times 9.6 = 0.6$

Let's solve this equation step by step.

Simplification:

The net rate can be simplified as: $\frac{1}{t} - \frac{1}{t+2} = \frac{(t+2) - t}{t(t+2)} = \frac{2}{t(t+2)}$

Substitute this into the time equation: $\frac{2}{t(t+2)} \times 9.6 = 0.6$

Now, solve for $t$: $\frac{2 \times 9.6}{t(t+2)} = 0.6$ $\frac{19.2}{t(t+2)} = 0.6$ $19.2 = 0.6 t(t+2)$ $32 = t(t+2)$ $t^2 + 2t - 32 = 0$

Solving the quadratic equation:

The quadratic equation $t^2 + 2t - 32 = 0$ can be solved using the quadratic formula: $t = \frac{-2 \pm \sqrt{2^2 - 4(1)(-32)}}{2(1)}$ $t = \frac{-2 \pm \sqrt{4 + 128}}{2}$ $t = \frac{-2 \pm \sqrt{132}}{2}$ $t = \frac{-2 \pm 11.49}{2}$ Taking the positive root (since time must be positive): $t = \frac{-2 + 11.49}{2} = \frac{9.49}{2} \approx 4.745$

Thus, it takes approximately 4.75 hours for the second pipe to drain the pool.

Time for the first pipe to fill the pool:

Since the first pipe takes 2 hours longer, it will take: $t + 2 = 4.75 + 2 = 6.75 \text{ hours}$

Summary:

• The second pipe drains the pool in approximately 4.75 hours.
• The first pipe fills the pool in approximately 6.75 hours.

Let me know if you'd like more details or have any questions!

5 Related Questions:

1. How would the solution change if the pool was 50% full at the start?
2. What if the first pipe took only 1 hour longer than the