Problem 48 - 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. An experiment of teaching chimpanzees sign language shows that a typical chimp can master a maximum of 65 signs. How many weeks should it take a chimpanzee to master 30 signs if \(c\) is 0.03 ?

Step-by-Step Solution

Answer

21 weeks

Step 1: Identify Given Values

From the problem, we know that the maximum learning possible, \( A \), is 65 signs. The number of signs mastered, \( N \), is 30 signs, and the learning style constant, \( c \), is 0.03.

Step 2: Substitute Values into the Formula

Insert the given values \( A = 65 \), \( N = 30 \), and \( c = 0.03 \) into the learning curve formula: \[t = \frac{1}{0.03} \ln\left(\frac{65}{65-30}\right)\]

Step 3: Calculate the Denominator Part

Calculate the denominator part of the fraction inside the logarithm function:\( 65 - 30 = 35 \).

Step 4: Compute the Fraction Inside the Logarithm

Compute the fraction inside the logarithm function with the calculated denominator:\( \frac{65}{35} \approx 1.857 \).

Step 5: Apply the Natural Logarithm

Calculate the natural logarithm of 1.857:\( \ln(1.857) \approx 0.619 \).

Step 6: Divide by the Learning Style Constant

Divide the natural logarithm result by the learning style constant \( c = 0.03 \):\[t = \frac{0.619}{0.03} \approx 20.63\]

Step 7: Round to the Nearest Whole Number

Round the calculated time, \( t \), to the nearest whole number:\( t \approx 21 \text{ weeks} \).

Key Concepts

Psychological Graphs

In psychology, graphs are invaluable tools used to illustrate abstract concepts in a more tangible way. Among these, the learning curve is particularly significant as it visually represents the progression of learning over time. The learning curve provides a graphical depiction of how time and effort are invested in acquiring new knowledge or skills. By plotting the formula \(t=\frac{1}{c} \ln \left(\frac{A}{A-N}\right)\), where \(t\) represents time, psychologists can examine learning patterns. This curve typically starts steep, indicating fast learning, and gradually flattens as one reaches the maximum capacity, \(A\).
Such psychological graphs help educators and psychologists:

• Track learning efficiency.
• Identify optimal learning techniques.
• Evaluate factors affecting learning speed.

Through these insights, educators can tailor learning experiences to better suit individual learning styles, making education more effective and efficient.

Exponential Functions in Learning

Exponential functions play a crucial role in modeling real-world phenomena, including learning processes. The learning curve equation is a prime example of how exponential functions are integral to understanding learning dynamics.
Exponential growth and decay principles can help explain how quickly or slowly one acquires new skills, driven by variables like the learning style constant \( c \). In the equation \(t=\frac{1}{c} \ln\left(\frac{A}{A-N}\right)\), the logarithmic and exponential components reveal the non-linear characteristics of learning.
These characteristics include:

• Initial fast learning, where the learner picks up skills rapidly.
• Slowed progress as the learner approaches their capacity for knowledge.

Exploring exponential functions allows us to better predict learning outcomes and tailor instructional methods to individual needs by adjusting variables that influence learning speed and efficiency.

Chimpanzee Sign Language Learning

Teaching sign language to chimpanzees offers fascinating insights into both learning curves and exponential functions. In experiments, researchers have observed that chimpanzees follow similar learning patterns to humans when mastering sign language. They experience rapid learning initially, which gradually plateaus as they approach the maximum number of signs they can learn, \(A\), in this case, 65 signs. These findings reflect the general shape of the learning curve graph for both humans and animals.
Due to a chimpanzee's specific learning style, characterized by a constant \(c = 0.03\), predicting how long it takes to learn a certain number of signs, like 30 in this example, becomes feasible by applying the learning curve formula. Observations from such studies show:

• Chimps learn new signs quickly when they start.
• Progress slows down as they near their known limit.
• Chimpanzee communication capacity offers valuable data for understanding general learning processes.

By analyzing these patterns, researchers can not only better grasp animal cognition but also apply such learning principles to enhance educational practices for humans.