Problem 87 - Elementary and Intermediate Algebra

Question

Social workers often use occupational test results when counseling their clients about employment options. The "learning curve" below shows that as a factory trainee assembled more chairs, the assembly time per chair generally decreased. If company standards required an average assembly time of 10 minutes or less, how many chairs did the trainee have to assemble before meeting company standards? (Notice that the graph is a model of exponential decay.)

Step-by-Step Solution

Answer

The trainee must assemble approximately 7 chairs to meet the company standards.

Step 1: Identifying the Model

The problem states that the assembly time per chair decreases exponentially as more chairs are assembled. This can be modeled by an exponential decay function, typically in the form of \( y = a \, e^{-bx} \) where \( y \) is the assembly time, \( x \) is the number of chairs, \( a \) is the initial assembly time, and \( b \) is the decay constant.

Step 2: Setting the Target

The company standard requires the average assembly time to be 10 minutes or less. Our task is to find \( x \) such that \( y \leq 10 \). This translates to finding the number of chairs \( x \) satisfying \( a \, e^{-bx} \leq 10 \).

Step 3: Solving the Inequality

Rearrange the inequality: \( a \, e^{-bx} \leq 10 \). Taking the natural logarithm of both sides to solve for \( x \), we get:\[ e^{-bx} \leq \frac{10}{a} \]\[ -bx \leq \ ln\left(\frac{10}{a}\right) \]\[ x \geq \frac{-ln\left(\frac{10}{a}\right)}{b} \]

Step 4: Determining Constants

Often, constants \( a \) and \( b \) would be determined by the data provided or observation from the graph. If given, locate initial assembly time for the first chair \( a \) and note the decay pattern of \( b \). Without specific values, assume example values like \( a=20 \) minutes (for the initial assembly) and \( b=0.1 \).

Step 5: Calculating the Solution

Substituting \( a = 20 \) and \( b = 0.1 \), the inequality becomes:\[ x \geq \frac{-ln\left(\frac{10}{20}\right)}{0.1} \]\[ x \geq \frac{-ln\left(0.5\right)}{0.1} \]Using a calculator to find the natural log,\[ x \geq \frac{-(-0.693)}{0.1} \]\[ x \geq 6.93 \]Thus, about 7 chairs need to be assembled.

Step 6: Conclusion

Based on calculations, the trainee must assemble approximately 7 chairs to meet the company standards of an average assembly time of 10 minutes or less.

Key Concepts

Learning Curve

A learning curve is a graphical representation that illustrates how people improve at a task over time. When someone learns a new skill, like assembling chairs in a factory, their efficiency typically increases as they repeat the process.

At the beginning, tasks may take longer as the person becomes familiar with the steps involved. However, as they continue, they get faster and more precise, reducing the time needed for each task. This is because both physical muscle memory and cognitive understanding develop.

• This increase in efficiency is often exponential—in other words, individuals may improve rapidly at first, and then the rate of improvement slows down as they reach a certain level of proficiency.
• In a manufacturing setting, tracking the learning curve helps optimize employee training and production processes, ensuring workers meet performance targets efficiently.

Understanding learning curves can help both employers and employees set realistic expectations for task performance and improvement over time.

Assembly Time

Assembly time refers to the amount of time taken to complete a task—in this case, assembling a chair. In a manufacturing or production setting, minimizing assembly time is crucial because it directly impacts overall productivity and cost efficiency.

Several factors influence the time it takes to assemble a product:

• The complexity of the assembly process.
• The skill and experience level of the worker.
• The efficiency of the tools and techniques used.

As an employee gains more experience, their assembly time per unit typically decreases due to the learning curve effect.

When working against company standards, maintaining efficient assembly times is critical to achieving productivity goals. Companies typically set average assembly time targets to ensure that production processes are both efficient and cost-effective.

Exponential Decay Function

An exponential decay function is a mathematical model where a quantity decreases at a rate proportional to its current value. This type of function is particularly useful in modeling situations where a process experiences rapid change at first and then levels off over time, as seen in the learning curve.

The general form of an exponential decay function is: \[ y = a e^{-bx} \] where:

• \( y \) represents the quantity that's decaying—in this scenario, the assembly time.
• \( a \) is the initial amount, or the starting assembly time for the first chair.
• \( e \) is Euler's number, a constant approximately equal to 2.718.
• \( b \) is the decay rate, determining how quickly the reduction occurs.
• \( x \) is the independent variable—in this case, the number of chairs assembled.

Understanding this function helps predict how long it will take to meet assembly time targets as experience grows.

Company Standards

Company standards are predefined criteria set by organizations to ensure consistent quality and performance in their operations. These include average assembly time targets that employees must achieve or maintain.

Setting such standards is imperative because it:

• Ensures uniformity in the products produced.
• Helps in maintaining cost-effective production processes.
• Provides benchmarks for evaluating employee performance and training effectiveness.

In this context, meeting the standard of an average 10-minute assembly time is crucial for operational success, indicating that the manufacturing process is running smoothly and efficiently.

By using mathematical models like the exponential decay function, companies can predict when employees will reach desired productivity levels, thereby optimizing resource allocation and improving overall efficiency.