2. One pipe can fill a pool 1.5 times faster than a second pipe. If both pipes are open, the pool

can be filled in 6 hours. If only the slower pipe is open, how long would it take to fill the pool?

Let's denote the time it takes for the slower pipe to fill the pool by itself as \( t \) hours.

Since the first pipe can fill the pool 1.5 times faster than the second pipe, the time it takes for the faster pipe to fill the pool is \( \frac{t}{1.5} \) or \( \frac{2t}{3} \) hours.

The rates at which the pipes fill the pool are:
- Slower pipe: \( \frac{1}{t} \) pools per hour
- Faster pipe: \( \frac{1}{\frac{2t}{3}} = \frac{3}{2t} \) pools per hour

When both pipes are open, their combined rate is the sum of their individual rates:
\[ \frac{1}{t} + \frac{3}{2t} = \frac{2}{2t} + \frac{3}{2t} = \frac{5}{2t} \]

We know that together, they can fill the pool in 6 hours, so their combined rate is:
\[ \frac{1}{6} \text{ pools per hour} \]

Equating the combined rate to the rate when both pipes are open, we get:
\[ \frac{5}{2t} = \frac{1}{6} \]

Solving for \( t \):
\[ \frac{5}{2t} = \frac{1}{6} \]
\[ 5 \cdot 6 = 2t \]
\[ 30 = 2t \]
\[ t = 15 \]

Thus, if only the slower pipe is open, it would take 15 hours to fill the pool.

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. How long would it take to fill the pool if only the faster pipe is open?
2. If the slower pipe were 2 times slower instead of 1.5 times, how would that affect the filling time?
3. What would be the combined rate of the pipes if the slower pipe were twice as slow?
4. How would the total filling time change if a third pipe, filling at the same rate as the faster pipe, were added?
5. How would you determine the filling time if the slower pipe had an unknown leak reducing its efficiency by 10%?
6. What if the pool had to be emptied as well, at a rate of 1 pool per 24 hours; how would this affect the filling time?
7. If the pool's volume were doubled, how would this impact the filling time using both pipes?
8. What assumptions are made in these calculations, and how could real-world factors alter the outcome?

**Tip:** When dealing with combined rates, remember to add the rates (work per unit time) rather than the times to complete the work.

Explore how to solve a combined rates problem involving two pipes filling a pool. Learn how to calculate filling times when both pipes are open or when only the slower pipe operates. Suitable for high school students studying rates and linear equations.

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