Problem 41 - Precalculus: Functions and Graphs

Question

An avid tennis watcher wants to record 6 hours of a major tournament on a single tape. Her tape can hold 5 hours and 20 minutes at the LP speed and 8 hours at the slower SLP speed. The LP speed produces a better quality picture, so she wishes to maximize the time recorded at the LP speed. Find the amount of time to be recorded at each speed.

Step-by-Step Solution

Answer

Record 320 minutes at LP speed and 40 minutes at SLP speed.

Step 1: Convert Recording Time into Minutes

First, convert the recording time from hours and minutes into total minutes. The goal is to record 6 hours of a tournament. Since 1 hour equals 60 minutes, 6 hours is equivalent to 6 * 60 = 360 minutes. Also, convert the LP speed capacity: 5 hours and 20 minutes equals 5 * 60 + 20 = 320 minutes. The SLP speed capacity is 8 hours, so it's 8 * 60 = 480 minutes.

Step 2: Maximize LP Speed Time

To maximize the recording time at the LP speed, we should use as much of the LP capacity as possible. The maximum LP capacity is 320 minutes. Check if this capacity fits within the goal of recording 360 minutes. Since 320 < 360, it is possible to use the full 320 minutes at LP speed and still have minutes left to record.

Step 3: Calculate Remaining Time for SLP Speed

Subtract the time recorded at LP speed from the total recording goal. Total minutes needed are 360 minutes, and 320 minutes has been allocated to LP speed. Therefore, the remaining time to recorded is 360 - 320 = 40 minutes.

Step 4: Conclusion

The watcher should record 320 minutes at LP speed and the remaining 40 minutes at SLP speed to achieve a total of 360 minutes of recording.

Key Concepts

Time Conversion

Understanding time conversion is crucial when dealing with problems involving durations across different units such as hours and minutes. In our example, the watcher wants to record 6 hours of a tennis tournament. To solve the problem effectively, we begin by converting this time into minutes to maintain consistency in calculations.

• 1 hour is equal to 60 minutes.
• Thus, 6 hours translate into 6 * 60 = 360 minutes.

Similarly, we convert the recording capacities of the tape at different speeds into minutes:

• The LP speed offers 5 hours and 20 minutes, which is calculated as 5 * 60 + 20 = 320 minutes.
• The SLP speed provides 8 hours, equating to 8 * 60 = 480 minutes.

These conversions help handle all quantities in the same denomination, simplifying comparisons and calculations.

Optimization

Optimization in this context refers to making the best or most effective use of available resources. Our goal is to maximize the recording quality, which is best at LP speed. With the total recording requirement at 360 minutes, the challenge lies in determining how much of this can be recorded at the superior LP speed. Maximization involves using as much of the resource - in this case, LP speed time - before moving to the next available option. Since the LP capacity is 320 minutes and fits within the 360-minute requirement, it’s optimal to use all 320 minutes available at this speed. Optimizing ensures that:

• The highest quality recording time (LP) is maximized.
• The constraints of total duration and recording capacity are respected.

This approach effectively maximizes the viewer's satisfaction given the constraints of tape capacity and desired total recording time.

Mixed Use of Resources

The efficient use of available resources involves understanding and combining different capacities to meet a target objective. In scenarios involving resources like tapes with multiple recording speeds, mixed-use strategies allow one to capitalize on the strengths of each speed. Once it's determined that 320 minutes can be recorded at the LP speed, we need to address the remaining required 40 minutes to meet the total recording goal of 360 minutes. This is achieved by switching to the SLP speed, which offers sufficient additional capacity (480 minutes available). By using two different speeds:

• We achieve the desired total recording time of 360 minutes effectively.
• We maximize the high-quality LP recording while utilizing the excess capacity of SLP for the remainder.

This way, by employing a mixed strategy, the best possible outcome is attained using the available resources, ensuring a satisfying recording of the tennis tournament.