Problem 49 - Algebra A Combined Function

Question

Psychologists call the graph of the formula \(t=\frac{1}{c} \ln \left(\frac{A}{A-N}\right)\) the learning curve since the formula relates time t passed, in weeks, to a measure \(N\) of learning achieved, to a measure \(A\) of maximum learning possible, and to a measure c of an individual's learning style. Use this formula to answer. Round to the nearest whole number. Janine Jenkins is working on her dictation skills. She wants to take dictation at a rate of 150 words per minute and believes that the maximum rate she can hope for is 210 words per minute. How many weeks should it take her to achieve the 150 -word level if \(c\) is \(0.07 ?\)

Step-by-Step Solution

Answer

It will take approximately 18 weeks.

Step 1: Understand the Given Information

We are given the formula \( t=\frac{1}{c} \ln \left(\frac{A}{A-N}\right) \), where \( t \) is the time in weeks, \( A \) is the maximum learning possible, \( N \) is the learning achieved, and \( c \) is a constant representing the individual's learning style. We need to find \( t \) for Janine Jenkins when \( A = 210 \), \( N = 150 \), and \( c = 0.07 \).

Step 2: Substitute the Values

Substitute the given values into the formula: \( A = 210 \), \( N = 150 \), and \( c = 0.07 \). Thus, the equation becomes:\[ t = \frac{1}{0.07} \ln \left(\frac{210}{210-150}\right) \]

Step 3: Simplify the Denominator

Calculate \( A - N \):\[ 210 - 150 = 60 \]Thus, the equation simplifies to:\[ t = \frac{1}{0.07} \ln \left(\frac{210}{60}\right) \]

Step 4: Calculate the Fraction

Compute \( \frac{210}{60} \):\[ \frac{210}{60} = 3.5 \]Now, the equation is:\[ t = \frac{1}{0.07} \ln (3.5) \]

Step 5: Calculate the Natural Logarithm

Find \( \ln(3.5) \) using a calculator:\[ \ln(3.5) \approx 1.25276 \]Replace in the equation:\[ t = \frac{1}{0.07} \times 1.25276 \]

Step 6: Compute the Multiplication

Calculate \( \frac{1}{0.07} = 14.2857 \). Then multiply by the natural logarithm result:\[ t \approx 14.2857 \times 1.25276 \approx 17.868 \]

Step 7: Round to the Nearest Whole Number

Round \( t = 17.868 \) to the nearest whole number:\[ t \approx 18 \]

Step 8: Conclusion

Janine should expect it to take approximately 18 weeks to achieve her goal of 150 words per minute given the learning style constant \( c = 0.07 \).

Key Concepts

Logarithmic Functions

Logarithmic functions are fundamental in mathematics, providing a way to explore the inverse of exponential growth. The specific logarithmic function used in our learning curve formula is the natural logarithm, denoted as \( \ln \). This function relies on the constant \( e \), approximately 2.718, to compute its values. In simpler terms, the logarithm helps to determine the time it takes to reach a certain level of learning by compressing large scale differences into smaller, more manageable numbers.

• A primary property of logarithms is their ability to convert multiplication into addition, which is highly useful for simplifying equations.
• For example, \( \ln(3.5) \) represents the exponent to which \( e \) must be raised to yield a product of 3.5.

In the context of learning curves, logarithms are crucial because they model how learning typically starts quickly and then slows down over time as a person approaches their maximum potential. This characteristic is why studying logarithmic growth is practical in educational models.

Educational Psychology

Educational psychology explores the intriguing connection between teaching methods and how individuals learn. This field delves into learning styles, which represent different approaches individuals use to absorb information. These styles affect how quickly and effectively a person can learn a new skill or concept. Our exercise with Janine Jenkins highlights how a specific learning style, represented by the constant \( c \), influences learning speed.

• Janine's learning style constant is \( 0.07 \). This value affects the time it takes her to progress in her skills.
• Learning curves help educators and learners understand how different approaches can impact learning efficiency.

Overall, recognizing learning curve patterns can help identify optimal teaching techniques for enhancing educational outcomes, providing a bridge between theoretical models and practical applications.

Mathematical Modeling

Mathematical modeling is an essential tool in understanding complex phenomena such as learning. By creating formulas like the learning curve equation, we can quantify and predict outcomes based on initial parameters. The learning curve equation \( t = \frac{1}{c} \ln \left(\frac{A}{A-N}\right) \) encompasses several components:

• \( t \), the time in weeks, is the dependent variable we're solving for.
• Parameters \( A \), \( N \), and \( c \) impact the speed and extent of learning, encapsulating both personal and contextual factors.

This equation models learning as a process of diminishing returns, where initial progress is quicker and subsequent improvements take longer. Such modeling allows educators and psychologists to forecast how a learner will improve over time, and to customize learning strategies to fit individual needs and benchmarks. Hence, mathematical models in educational psychology are instrumental for evidence-based teaching methods.