Problem 74 - Precalculus: Functions and Graphs

Question

Filling a bin An extruder can fill an empty bin in 2 hours, and a packaging crew can empty a full bin in 5 hours. If a bin is half full when an extruder begins to fill it and a crew begins to empty it, how long will it take to fill the bin?

Step-by-Step Solution

Answer

It takes \( \frac{5}{3} \) hours to fill the bin.

Step 1: Determine Extruder and Crew Rates

The extruder fills 1 bin in 2 hours, so it fills \( \frac{1}{2} \) of a bin per hour. The packaging crew empties 1 bin in 5 hours, so it empties \( \frac{1}{5} \) of a bin per hour.

Step 2: Calculate Net Filling Rate

The net filling rate of the bin, when both the extruder fills and the crew empties it, is the difference of their rates: \( \frac{1}{2} - \frac{1}{5} = \frac{5}{10} - \frac{2}{10} = \frac{3}{10} \) of a bin per hour. This means the bin is effectively being filled at a rate of \( \frac{3}{10} \) of a bin per hour.

Step 3: Determine Bin Requirement

Since the bin is initially half full, the remaining amount to be filled is \( 1 - \frac{1}{2} = \frac{1}{2} \) of a bin.

Step 4: Compute Time to Fill Half Bin

To fill the remaining \( \frac{1}{2} \) of the bin at a rate of \( \frac{3}{10} \) of a bin per hour, we use the formula: \( \text{Time} = \frac{\text{Amount to Fill}}{\text{Rate}} = \frac{\frac{1}{2}}{\frac{3}{10}} = \frac{1}{2} \times \frac{10}{3} = \frac{10}{6} = \frac{5}{3} \) hours.

Key Concepts

Extruder Rate

Understanding how fast an extruder can fill a bin is crucial in solving work rate problems. The extruder in this scenario has a rate of filling 1 bin every 2 hours. This translates to filling half (\(\frac{1}{2}\)) of a bin in just one hour. When thinking about this concept, consider the reverse as well: knowing how long it takes to complete one whole unit (the bin). This concept can be applied to many practical situations where a machine or process works at a consistent pace. Always break down complex schedules into simple rates per hour for clarity.

Crew Rate

The packaging crew works at a rate of emptying one full bin in 5 hours. This translates to emptying \(\frac{1}{5}\) of a bin per hour. In many work rate problems, you might come across scenarios involving removing or draining something at a particular rate. This is similar to our example, where the crew removes a certain portion of the bin contents per hour. To solve such problems, converting the full-time task into a rate-per-hour figure is handy for calculations, just as we did here.

Net Filling Rate

When you have two opposing processes, such as filling and emptying, calculating the net filling rate is vital. The extruder adds content at \(\frac{1}{2}\) a bin per hour, while the crew removes at \(\frac{1}{5}\) a bin per hour. The net effect of these rates involves subtracting the crew's emptying rate from the extruder's filling rate. This gives a final rate of \(\frac{3}{10}\) of a bin being filled per hour. Understanding how to combine rates like this enables you to tackle many real-world problems where multiple workflows overlap.

Time Calculation

Calculating time needed to complete a task requires knowing how much of the task remains and dividing that by the net completion rate. In our situation, the bin starts half full, so only \(\frac{1}{2}\) a bin needs to be completed. Dividing the remaining task (\(\frac{1}{2}\) of a bin) by the net filling rate (\(\frac{3}{10}\) of a bin per hour) determines the time required. Using the formula \( \text{Time} = \frac{\text{Amount to Fill}}{\text{Rate}} \) shows us the task takes \( \frac{5}{3} \) hours to finish. Breaking tasks into smaller, more manageable steps always makes problem-solving more straightforward.