Problem 102 - College Algebra with Modeling and Visualization

Question

Pumping Water Suppose that a large pump can empty a swimming pool in 50 hours and a small pump can empty the pool in 80 hours. How long will it take to empty the pool if both pumps are used?

Step-by-Step Solution

Answer

Both pumps will empty the pool in approximately 30.77 hours.

Step 1: Define the Rates of Each Pump

First, determine the rate at which each pump empties the pool. The large pump empties the pool in 50 hours, so its rate is \( \frac{1}{50} \) pools per hour. Similarly, the small pump empties the pool in 80 hours, so its rate is \( \frac{1}{80} \) pools per hour.

Step 2: Find the Combined Rate of Both Pumps

To find the combined rate when both pumps are working together, add their individual rates:\[ \text{Combined Rate} = \frac{1}{50} + \frac{1}{80} \]This needs to be calculated by finding a common denominator.

Step 3: Calculate the Combined Rate

The least common denominator of 50 and 80 is 400. Convert the rates to have this common denominator:\[ \frac{1}{50} = \frac{8}{400} \quad \text{and} \quad \frac{1}{80} = \frac{5}{400} \]Add these rates:\[ \frac{8}{400} + \frac{5}{400} = \frac{13}{400} \]So, both pumps together empty the pool at a rate of \( \frac{13}{400} \) pools per hour.

Step 4: Calculate the Time to Empty the Pool with Both Pumps

To find how long it takes both pumps to empty the pool, take the reciprocal of the combined rate:\[ \text{Time} = \frac{400}{13} \approx 30.77 \text{ hours} \]This means it will take approximately 30.77 hours to empty the pool using both pumps.

Key Concepts

Work Rate

When dealing with rate problems, it's crucial to understand the concept of "work rate." This involves determining how quickly a task can be completed by a single worker or machine. In our example, we look at two pumps that empty a swimming pool. Each pump has its own work rate, defined as the fraction of the pool they can empty in one hour.

• The large pump has a work rate of \( \frac{1}{50} \), meaning it can empty \( \frac{1}{50} \) of the pool every hour.
• Similarly, the small pump has a work rate of \( \frac{1}{80} \).

Knowing the work rates helps us predict how the pumps will operate individually and in combination. Once you grasp this, other related calculations become much simpler.

Common Denominator

In order to combine two fractional rates, such as those of the two pumps, you need to find a common denominator. This is crucial because fractions can only be added when they share the same denominator. For our example:

• The denominators are 50 and 80.
• To find the least common denominator (LCD), you look for the smallest number that 50 and 80 can both divide evenly. The LCD in this case is 400.

This allows you to adjust the fractions so that they can be easily added together.

Fraction Addition

Once you've identified the common denominator, the next step is to adjust each fraction so they have equivalent denominators, making them ready for addition. In our problem:

• The large pump's rate is adjusted from \( \frac{1}{50} \) to \( \frac{8}{400} \).
• The small pump's rate is adjusted from \( \frac{1}{80} \) to \( \frac{5}{400} \).
• Now, simply add these fractions: \( \frac{8}{400} + \frac{5}{400} = \frac{13}{400} \).

Combining the rates in this way totals the amount of work completed in one hour by both pumps working together.

Reciprocal Calculation

After combining the rates of the two pumps, the last step is to figure out the total time required to finish the task using both pumps. This involves reciprocal calculation. To do this:

• Take the combined work rate, which is \( \frac{13}{400} \, ext{pools per hour} \).
• Find its reciprocal to determine the total time. The reciprocal is \( \frac{400}{13} \), which gives us approximately 30.77 hours.

Understanding reciprocal calculation is essential in solving rate problems, as it transforms a rate back into a total time or quantity, completing the solution to the problem.