Problem 106 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem involving rate of work. A winery has a vat to hold Merlot. An inlet pipe can fill the vat in 18 hours, and an outlet pipe can empty it in 24 hours. How long will it take to fill an empty vat if both the outlet pipe and the inlet pipe are open?

Step-by-Step Solution

Answer

72 hours.

Step 1: Understand the Problem

The problem involves finding how long it takes to fill a vat when both an inlet and an outlet pipe are open. The inlet pipe fills the vat in 18 hours, while the outlet pipe empties it in 24 hours.

Step 2: Determine the Rate of Work for Each Pipe

The inlet pipe can fill the vat in 18 hours, so its rate of work is \( \frac{1}{18} \) of the vat per hour. The outlet pipe can empty the vat in 24 hours, so its rate of work is \( \frac{1}{24} \) of the vat per hour.

Step 3: Calculate the Net Rate of Work

When both pipes are open, the combined rate of work is found by subtracting the outlet's rate from the inlet's rate: \( \frac{1}{18} - \frac{1}{24} \).

Step 4: Perform the Calculation

To find the net rate, find a common denominator (which is 72) and perform the subtraction:\[ \frac{1}{18} = \frac{4}{72}, \quad \frac{1}{24} = \frac{3}{72} \]Thus, the net rate is \( \frac{4}{72} - \frac{3}{72} = \frac{1}{72} \).

Step 5: Determine Time to Fill the Vat

Since the net rate is \( \frac{1}{72} \) of the vat per hour, it takes 72 hours to fill the vat.

Key Concepts

Inlet and Outlet Pipes

When solving rate of work problems in real-world scenarios, it's common to encounter situations involving inlet and outlet pipes. These pipes have opposite functions: one fills a container while the other empties it.
In our exercise, we have an inlet pipe that can fill the vat in 18 hours. This means if the inlet pipe works alone, it completes the task of filling the vat. Conversely, the outlet pipe is designed to empty the vat in 24 hours. Opening the outlet pipe alone would gradually remove all the contents of the vat over this time span.

• An effective approach is to think of each pipe working at a specific rate.
• The inlet pipe adds liquid, increasing the vat's content over time.
• The outlet pipe does the opposite, decreasing the vat's content.

Understanding the roles of these pipes sets the stage for calculating how their simultaneous operation affects the vat's fill rate.

Rate Calculation

Calculating the rate of work is the key step in solving work problems involving pipes. This is usually expressed as a fraction of the task completed in one unit of time, for instance, one hour.
For the inlet pipe:

• The rate is given as \( \frac{1}{18} \) of the vat per hour. This indicates the fraction of the vat that the inlet can fill in an hour when working alone.

For the outlet pipe:

• The rate is \( \frac{1}{24} \) of the vat per hour. This rate accounts for how much of the vat the outlet pipe would empty in one hour on its own.

The challenge comes in determining the net rate when both pipes are operational simultaneously. To do this, subtract the outlet's rate from the inlet's rate.
This subtraction gives us the net effect - how much the vat’s content increases per hour when both pipes are working.

Common Denominator Method

To find the combined effect of both pipes working at their respective rates, it's necessary to perform a subtraction.
This requires a common denominator, which is a shared multiple of the differing time periods, to combine the two rates.
In our given problem, the common denominator between 18 and 24 is 72, chosen because it is the smallest number that both 18 and 24 can divide evenly into:

• For the inlet pipe: \( \frac{1}{18} = \frac{4}{72} \)
• For the outlet pipe: \( \frac{1}{24} = \frac{3}{72} \)

With this common denominator, we can easily subtract the fractions: \( \frac{4}{72} - \frac{3}{72} = \frac{1}{72} \).
This result signifies that the net rate of work for both pipes is \( \frac{1}{72} \) of the vat per hour.
Hence, it will take 72 hours for the vat to be fully filled under these conditions.