How Long Will 12 Pipes Take to Fill the Same Pool If 4 Pipes Take 6 Hours?

In this article, we will explore the concept of work rates and how to apply it to solve a real-world problem involving pipes and pools. By understanding work rates, you can determine how different numbers of pipes will affect the time it takes to fill a pool.

Understanding Work Rates

Work rates are a measure of how much of a task can be completed in a given amount of time. In the context of pipes and pools, the work rate is the rate at which a pipe fills a pool. If 4 pipes can fill a pool in 6 hours, we can use work rates to determine how long it will take 12 pipes to fill the same pool.

Step-by-Step Solution

Step 1: Calculate the Work Done by 4 Pipes in 6 Hours

The total work done by 4 pipes in 6 hours can be expressed as the product of the number of pipes and the time they work:

Work Number of pipes times; Time

Therefore, the work done by 4 pipes in 6 hours is:

Work 4 pipes times; 6 hours 24 pipe-hours

Step 2: Determine the Work Rate of One Pipe

The work rate of 4 pipes is 1 pool per 6 hours. To find the work rate of one pipe, we divide the total work by the number of pipes:

Rate of 1 pipe (frac{1 text{ pool}}{24 text{ hours}} frac{1}{24} text{ pools per hour})

Step 3: Calculate the Work Rate of 12 Pipes

Now, to find the work rate of 12 pipes, we multiply the work rate of one pipe by 12:

Rate of 12 pipes 12 times; (frac{1}{24} text{ pools per hour} frac{12}{24} text{ pools per hour} frac{1}{2} text{ pools per hour})

Step 4: Determine the Time Taken by 12 Pipes to Fill the Pool

If 12 pipes can fill (frac{1}{2}) of the pool in 1 hour, then to fill the entire pool, it will take:

Time (frac{1 text{ pool}}{frac{1}{2} text{ pools per hour}} 2 text{ hours})

Therefore, 12 pipes will fill the pool in 2 hours.

Alternative Solution

Another approach to solving this problem involves setting up equations based on the volume of the pool:

Step 1: Define Variables

Let V denote the full volume of the given pool, and T denote the time in hours required for 12 pipes to fill the pool.

From the given information, we can create the following equations:

12 pipes times; (frac{V}{4 text{ pipes} times 6 text{ hours}} V div T)

This simplifies to:

(frac{12V}{24} frac{V}{T})

(T 2 text{ hours})

Hence, 12 pipes will take 2 hours to fill the same pool.

Conclusion

By applying the concept of work rates, we can easily determine how the number of pipes affects the time required to fill a pool. In this case, 12 pipes will fill the pool in 2 hours, assuming the pool filling rate remains constant.

Keywords: work rate, pipes, pool