Problem 83 - College Algebra and Calculus: An Applied Approach

Question

Students in a mathematics class were given an exam and then retested monthly with an equivalent exam. The average score \(g\) for the class can be approximated by the human memory model \(g(t)=78-14 \log _{10}(t+1), \quad 0 \leq t \leq 12\) where \(t\) is the time (in months). (a) What was the average score on the original exam? (b) What was the average score after 4 months? (c) When did the average score drop below 70 ?

Step-by-Step Solution

Answer

The average score on the original exam was 78. The score after four months was approximately 68.27. The average score dropped below 70 around the 3rd month.

Step 1: Find the original exam score

To find the original exam score, we need to substitute \(t = 0\) into the function \(g(t) = 78 - 14 \log_{10}(t + 1)\). Therefore, the original exam score is \(g(0) = 78 - 14 \log_{10}(0 + 1)\). Hence, \(g(t) = 78 - 14 \times 0 = 78\). The original average score was 78.

Step 2: Find the score after 4 months

To find the score after four months, substitute \(t = 4\) into the function. This gives \(g(4) = 78 - 14 \log_{10}(4 + 1) = 78 - 14 \log_{10}(5)\). Using a calculator to evaluate this gives approximately \(g(4) = 78 - 14 \times 0.69897 = 68.27442\). Therefore, the average score after four months was approximately 68.27.

Step 3: Finding when the average score dropped below 70

To solve the inequality \(g(t) < 70\), we can initially set \(g(t) = 70\) and solve for \(t\). This gives us the equation \(70 = 78 - 14 \log_{10}(t + 1)\). Therefore, \(8 = 14 \log_{10}(t + 1)\), and \(\log_{10}(t + 1) = \frac{8}{14} = 0.57143\). Using properties of logarithms, this can be rewritten as \(10^{0.57143} = t + 1\). Solving this gives \(t = 10^{0.57143} - 1\), and using a calculator gives this as approximately \(t = 2.68\) months. Therefore, the average score dropped below 70 around the 3rd month.

Key Concepts

Logarithmic Function

The logarithmic function used in student exam score predictions is part of the human memory model. This function helps describe how the ability to recall information decreases over time. In our exercise, the function given is \( g(t) = 78 - 14 \log_{10}(t + 1) \). By using the logarithmic term, \( \log_{10}(t + 1) \), it expresses the rate of memory loss. Here’s why the logarithmic function is effective:

• Non-linear decay: A logarithmic function decreases incrementally slower over time. This models how human memory retention often diminishes most drastically immediately after learning, then tapers off.
• Base 10 log: This specific log base (10) simplifies the mathematical operations and aligns well with common analytic tools like calculators.
• Modeling capability: It captures complex cognitive effects simply and is easier to understand and compute.

By understanding logarithmic functions, we appreciate how models can predict phenomena such as memory retention effectively.

Average Score

The average score in this context is a statistical representation of the class's collective performance. It provides a convenient summary of student performance at various points in time using the function \( g(t) = 78 - 14 \log_{10}(t + 1) \).In the original exercise, the average score is calculated as:

• On original exam: At \( t = 0 \), the average score is \( g(0) = 78 \). This tells us the initial collective performance of the students when they’re first tested.


• After 4 months: At \( t = 4 \), the average score was computed as approximately \( 68.27 \) using the function. This means that on average, students' scores dropped over time.

The average score helps educators gauge the overall effectiveness of teaching methods and materials over time and make informed decisions on curriculum adjustments.

Time-Dependent Function

A time-dependent function, such as \( g(t) = 78 - 14 \log_{10}(t + 1) \), represents relationships where an element (the score) changes based on another factor (time in months). In this exercise, understanding the time dependency is crucial since it provides insight into how the students' memory retention evolves.This time-dependent function helps analyze:

• Memory Recollection: How well students can recall what they have learned over different periods.


• Performance trends: By solving \( g(t) < 70 \), we discovered that scores drop below a certain threshold around 3 months, illustrating the point at which retention sharply declines.


• Predictive Modeling: Educators can predict future performance and determine when more revision or reinforcement may be necessary to better student outcomes.

Understanding this function's behavior over time is vital for making strategic educational decisions, guiding curriculum adjustments, and enhancing overall learning.