Problem 132 - College Algebra

Question

Students in a mathematics class took a final examination. They took equivalent forms of the exam in monthly intervals thereafter. The average score, \(f(t)\), for the group after \(t\) months was modeled by the human memory function \(f(t)=75-10 \log (t+1),\) where \(0 \leq t \leq 12\). Use a graphing utility to graph the function. Then determine how many months elapsed before the average score fell below 65.

Step-by-Step Solution

Answer

The number of months before the average score falls below 65 can be determined using the graph of the given function or by solving the mathematical inequality.

Step 1: Understand the function

The given function is \(f(t) = 75 - 10 \log (t + 1)\). This function describes the average score of the group after \(t\) months. Reading a log function, it can be noted that as \(t\) increases, the log value also increases. Hence, the subtraction part \(10 \log (t + 1)\) will lead to a decrease in the total score over time.

Step 2: Graph the function

Next, graph the function \(f(t) = 75 - 10 \log (t + 1)\) using any graphing software or utility. Apply the domain \(0 \leq t \leq 12\) while generating this graph. This will produce a curve showing the variation in the average score over twelve months.

Step 3: Analyze the graph

Analyze the plot for the point when the function value, that is average score, falls below 65. This point represents the number of months after which the average score drops to 65. Visually, it will be the point where the graph of falls below the line \(y = 65\).

Step 4: Determine the time period

From the graph, determine the value of \(t\) when \(f(t) < 65\). If the graph doesn't illustrate it precisely, using a mathematical approach could be beneficial. To do that, solve the inequality \(75 - 10 \log (t + 1) < 65\). The solution will give the time period after which the average score fell below 65.

Key Concepts

Average Score Model

The average score model presented in this exercise is defined by the function \(f(t) = 75 - 10 \log(t+1)\). This function is designed to represent how the average score of a group of students changes over a period of time. It's guided by the human memory function, which typically shows a pattern of decay.

• Initially, at \(t = 0\), the average score starts at 75.
• As time progresses, measured in months, the score decreases due to the factor \(10 \log(t+1)\).

The formula illustrates that as \(t\) increases, the effect of the logarithmic term becomes more significant, pulling the average score down. This model aligns with the natural tendency of memory retention, where recall abilities decline over time without reinforcement.

Logarithmic Function

The function \(f(t) = 75 - 10 \log(t+1)\) incorporates a logarithmic component. A logarithmic function, such as \(\log(t+1)\), exhibits slow growth. This means that as \(t\) gets larger, the rate of increase of \(\log(t+1)\) diminishes.Logarithmic functions are useful for modeling phenomena that experience rapid changes initially that then slow over time. In this problem:

• For small values of \(t\), the function \(\log(t+1)\) increases quickly, contributing to rapid decreases in the score.
• As \(t\) becomes larger, \(\log(t+1)\) increases at a slower rate, showing a slower decline in the average score.

These characteristics make logarithms fitting for modeling memory decay, as initial changes are pronounced, but stability increases over time.

Graphing Utility

Using a graphing utility is a practical way to visualize mathematical functions and understand their behavior over a range of data—in this case, the months elapsed. When you input the function \(f(t) = 75 - 10 \log(t+1)\) into a graphing tool, it generates a curve reflecting the average score against time. To graph this function:

• Ensure that your graphing utility is set to handle the logarithmic function.
• Apply the domain \(0 \leq t \leq 12\) because the context of the problem specifies a 12-month period.
• Analyze the graph to detect how the average score diminishes with time.

By observing the graph, you can easily pinpoint the moment when the score falls below 65, providing a visual method to address the problem at hand.

Mathematical Inequality

Solving mathematical inequalities helps determine specific values in real-world contexts. In this problem, you want to know when the average score falls below 65.To find this, set up the inequality:\[ 75 - 10 \log(t+1) < 65 \]This simplifies to:\[ 10 \log(t+1) > 10 \]\[ \log(t+1) > 1 \]To solve this, use properties of logarithms:

• Attempt to solve \(\log(t+1) = 1\), equating the log component to a known value.
• Recalling that \(\log_{10}(10) = 1\), you find that \(t+1 = 10\).
• Solve for \(t\): \(t = 9\).

Thus, the score falls below 65 after 9 months. This algebraic approach complements any visual findings from graphing the function.