Problem 43 - Algebra and Trigonometry with Analytic Geometry

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 5 hours 20 minutes at LP speed and 40 minutes at SLP speed.

Step 1: Understand the Problem

We need to record a total of 6 hours using a tape that has different recording times and qualities at LP and SLP speeds. The challenge is to maximize the recording time at LP speed while fully capturing 6 hours of content.

Step 2: Define Variables

Let's denote the time in hours recorded at LP speed as \( x \) and at SLP speed as \( y \). We need to formulate the problem in terms of these variables.

Step 3: Set Up the Equation for Total Recording Time

We want to record 6 hours total, so the equation is:\[ x + y = 6 \]

Step 4: Determine Restrictions Due to Tape Capacity

The tape can hold 5 hours and 20 minutes at LP speed, which is \( \frac{80}{60} = \frac{4}{3} \) hours less than 6 hours, so: \[ x \leq 5.33 \] And 8 hours at SLP speed, but as \( y = 6 - x \), no extra conditional restriction is necessary beyond \( x \leq 5.33 \).

Step 5: Maximize LP Recording Time

To maximize the LP time, we simply take the maximum value possible that satisfies both the restriction and equation in steps 3 and 4. So set \( x = 5.33 \), then use the total time equation:\[ y = 6 - 5.33 = 0.67 \] hours.

Step 6: Convert from Hours to Minutes

Convert \( x \) and \( y \) to hours and minutes for final answer:\( x = 5.33 \) hours is 5 hours and 20 minutes, and \( y = 0.67 \) hours, equivalent to approximately 40 minutes.

Key Concepts

Linear Equations and Their Role in Problem Solving

Linear equations are fundamental in algebra and are used for solving a wide range of problems by representing relationships between variables. In the given problem, we have two critical variables: time recorded at LP speed \((x)\) and at SLP speed \((y)\). These variables allow us to express the total recording requirement and find a suitable solution. The equation \(x + y = 6\) represents the total recording time in hours, capturing the essence of our need to allocate resources - in this case, time - efficiently.

Linear equations like this one are straightforward because they represent constant rates of change. They facilitate a methodical approach to problem solving, allowing easy manipulation to explore all possible solutions. By balancing the equation with limits on each variable, we align with the constraints imposed by the problem, ensuring not only a solution but an optimal one.

The step-by-step reduction and adjustment of variables, as showcased in this problem, are typical of how linear equations help tackle optimization tasks in real-world scenarios.

Effective Time Management Through Algebra

Time management is crucial in various aspects of life, and algebra serves as an excellent tool to enhance it, even in everyday activities like recording a tennis tournament. The algebraic approach to managing the 6 hours of recording ensures all available time is used efficiently while respecting quality preferences.

Through the equation \(x + y = 6\), where \(x\) and \(y\) represent the LP and SLP recording times respectively, we allocate time resources wisely. This approach requires a meticulous breakdown of available time, set against the constraints defined by tape capacity. By focusing on time management within algebraic expressions, we not only organize our resources but also make informed decisions, maximizing the best usage at LP speed due to its higher quality.

Ultimately, using algebra in time management allows for a clear and structured method to maximize quality within given limitations, showcasing how such mathematical strategies can influence everyday decision-making positively.

Optimization: Maximizing Quality with Constraints

Optimization is about making the best possible decision within given limitations. In this problem, we sought to maximize high-quality recording time (LP speed) while ensuring the entire 6-hour tournament is covered. Constraints included the tape's holding limits at different recording speeds, and through the assessment of these values, we determined the optimal solution.

The tools of linear equations enabled us to model the constraints effectively. By setting \(x \leq 5.33\), where \(x\) is the hours recorded at LP speed, we catered for not just feasibility, but quality maximization. Selecting the highest possible value for \(x\) within this limitation resulted in \(x = 5.33\) hours, leaving the remainder \(y = 0.67\) hours for SLP speed.

Optimization using algebra ensures we balance between competing needs and practical limitations, providing a solution that prioritizes desired outcomes—in this case, better picture quality—within available resources. This strategic approach not only resolves immediate issues but also enhances one's ability to navigate complex decisions effectively in various contexts.