Problem 109 - Algebra and Trigonometry Real Mathematics, Real People

Question

Students in a mathematics class were given an exam and then tested monthly with an equivalent exam. The average scores for the class are given by the human memory model $$f(t)=80-17 \log _{10}(t+1), \quad 0 \leq t \leq 12$$ where \(t\) is the time in months. (a) What was the average score on the original exam \((t=0) ?\) (b) What was the average score after 2 months? (c) What was the average score after 11 months? Verify your answers in parts (a), (b), and (c) using a graphing utility.

Step-by-Step Solution

Answer

The average score on the original exam is 80, after 2 months it can be found by calculating \(80-17 \log _{10}(3)\), and after 11 months it can be found by calculating \(80-17 \log _{10}(12)\). After getting these values, they can be verified using a graphing utility by comparing the points at \(t=0, t=2, t=11\) and checking that they match with their calculated values.

Step 1: Calculation of the average score on the original exam (t=0)

Substitute \(t=0\) in the human memory model function. So it will be: \(f(0)=80-17 \log _{10}(0+1)=80-17 \log _{10}(1)\). As the logarithm of 1 in any base is 0, the equation becomes \(f(0)=80-17*0=80\). Hence, the average score on the original exam (t=0) is 80.

Step 2: Calculation of the average score after 2 months

Substitute \(t=2\) into the human memory model. So it will turn into: \(f(2)=80-17 \log _{10}(2+1)=80-17 \log _{10}(3)\). Calculate the value of the log base 10 of 3 and multiply it by 17, then subtract this from 80. This will give you the average score after 2 months.

Step 3: Calculation of the average score after 11 months

Substitute \(t=11\) into the human memory model function. It will turn into \(f(11)=80-17 \log _{10}(11+1)=80-17 \log _{10}(12)\). Calculate the log base 10 of 12 and multiply it by 17, this value will be subtracted from 80 which will be the average score after 11 months.

Step 4: Verification using a graphing utility

Plot the human memory model function against time \(t\) on the x-axis and \(f(t)\) on the y-axis using any graphing utility. Make sure to extend \(t\) from 0 to 12 and mark the points where \(t=0, t=2, t=11\) on the graph. Carefully observe these points and their corresponding function values. The values from the graph at these points should match with the calculated values to verify your answers.

Key Concepts

Logarithm in Base 10

Logarithms are mathematical tools that help us understand exponential relationships. The logarithm in base 10, often written as \( \log_{10}\), tells us what power of 10 gives us a particular number. For example, \( \log_{10}(100) = 2 \) because 10 raised to the power of 2 is 100. In simpler terms, it's about finding out how many times we need to multiply 10 to get our targeted number.

In the context of the Human Memory Model, the logarithm helps illustrate how memory scores change over time. When you see expressions like \( \log_{10}(t+1) \), it means we're looking at how scores change as time progresses, by considering the logarithmic decline. Knowing that the logarithm of 1 is always 0 (since 10 to the power of 0 is 1), you can understand why the initial score at \( t = 0 \) is not influenced by the logarithmic term.

Mathematics Class Exam Analysis

Analyzing exam scores over time gives insights into how information retention decreases. In this particular exercise, we have a model depicting the average score temporal changes for a class through a function, \( f(t) = 80 - 17 \log_{10}(t+1) \). This function considers the decline in scores as students are re-evaluated monthly.

• At \( t = 0 \), the function starts with the initial value of 80, indicating the average score when the exam was first taken.


• As \( t \) increases, the logarithmic component \( 17 \log_{10}(t+1) \) begins to subtract from this 80, illustrating a score decline.

Such an analysis supports teachers in understanding long-term memory retention in students. Additionally, it can aid in forming strategies for better long-term retention in learning material. Students tend to perform better if revision becomes a consistent habit, countering the score drop due to time.

Graphing Utility Usage

Graphing utilities are powerful tools that enable us to visualize mathematical functions. When analyzing a mathematical representation like the human memory model, a graphing utility helps to see how scores fluctuate over time.

In this scenario:

• The x-axis represents time, \( t \), spanning from 0 to 12 months.


• The y-axis depicts the average score, \( f(t) \), computed from the human memory model function.

When using a graphing utility, you plot the function, and it smoothly shows how time affects scores. By marking key points, such as \( t=0 \), \( t=2 \), and \( t=11 \), you can visually verify the calculations done before plotting. The graph should reflect a gradual decline in scores, as compounding over a year results in memory fading, offering a clear perspective on learning dynamics.

Calculation of Average Scores

Calculating average scores from a model like \( f(t) = 80 - 17 \log_{10}(t+1) \) helps understand how memory retention erodes over time.
Here's how to compute it:

• Start with \( t=0 \): Set \( t \) to zero, simplifying to \( f(0) = 80 \). This indicates a perfect recall immediately after learning.


• Next, at \( t=2 \): Substitute 2 into the formula: \( f(2) = 80 - 17 \log_{10}(3) \). Calculate \( \log_{10}(3) \) and subtract to find a value slightly lower than 80, representing decline.


• Finally, for \( t=11 \): Plug in \( t=11 \): \( f(11) = 80 - 17 \log_{10}(12) \). This results in a more significant decrease, illustrating further memory degradation.

These calculations highlight how exam scores provide a measure of retention over time, helping educators and students evaluate learning effectiveness.