Problem 132 - Algebra and Trigonometry

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 \(f(t)=75-10 \log (t+1), \quad\) where \(\quad 0 \leq t \leq 12 .\) 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 average score falls below 65 after 9 months.

Step 1: Understand the Function

The function \(f(t) = 75 - 10 \log (t + 1)\) represents the average score of the students after \(t\) months. To figure out when the average score falls below 65, we need to find the value of \(t\) when \(f(t) < 65\). This translates to solving for \(t\) in the inequality \(75 - 10 \log (t + 1) < 65\).

Step 2: Solve the Inequality

Subtract 75 from both sides of the inequality to isolate the logarithm term, resulting in \(-10 \log (t+1) < -10. Divide through by -10 to get \(\log (t+1) > 1\). We know the base of the logarithm here is 10 as it's not stipulated, hence can write it as \( \log_{10}{(t+1)} > 1\).

Step 3: Find the Value of t

The inequality \( \log_{10}{(t+1)} > 1\) means that \(t+1\) is greater than 10. This can be solved as \(t > 10 - 1\) or \(t > 9\). This means the average score fell below 65 after 9 months.

Step 4: Verification using a Graphing Utility

Plotting the function \(f(t) = 75 - 10 \log (t + 1)\) on a graphing utility, you will observe that the function falls below 65 around 9 months, corroborating the solution obtained in step 3.

Key Concepts

Logarithmic Function

A logarithmic function is a mathematical expression involving a logarithm, which is the inverse operation of exponentiation. It allows us to determine the power to which a number (called a base) must be raised to obtain another number. In our exercise, the function is of the form \[f(t) = 75 - 10 \log (t + 1)\]Here, the logarithm is a base 10 logarithm since no other base is specified. This base 10 logarithm, also known as the common logarithm, is widely used in scientific calculations.

• The term \(75\) represents an initial value, which in this context is the starting average exam score.
• The term \(-10 \log (t + 1)\) signifies how the average score decreases over time.
• The variable \(t\) stands for time in months, making the equation a model for exponential decay in memory retention.

Understanding how this function works helps in interpreting the memory loss over time.

Inequality Solving

Inequality solving is a crucial skill in algebra and helps find ranges of possible solutions rather than a single exact answer. In this exercise, we need to find when the average score falls below a specific value, 65.

Starting from the inequality \[75 - 10 \log (t + 1) < 65\]we solve for \(t\):

• First, subtract 75 from both sides to focus on the logarithmic part: \(-10 \log (t+1) < -10\).
• By dividing each side by \(-10\), and flipping the inequality sign (as per rules of inequalities when dividing by a negative), we get: \(\log (t+1) > 1\).
• This implies \(t+1 > 10\), yielding \(t > 9\).

Hence, the average score falls below 65 after month 9. Solving inequalities often involves reversing operations and careful attention to the direction of inequality signs.

Graphing Utility

Graphing utilities, like graphing calculators or software, visualize mathematical functions and their behavior over a domain. For our function: \[f(t) = 75 - 10 \log (t + 1)\]a graphing utility can illustrate the decrease in the average score over months. Using a graphing tool is often recommended for:

• Understanding complex behaviors in functions, such as points of intersection or maximum/minimum values.
• Verifying algebraic solutions by visual inspection of where the graph falls below defined values.
• Exploring changes over time, especially with models involving decay or growth.

In this task, plotting helps to confirm our algebraic solution by showing the score dipping below 65 at around 9 months.

Mathematical Modeling

Mathematical modeling is the formation of mathematical formulas or functions that describe real-world phenomena. In this exercise, \[f(t) = 75 - 10 \log (t + 1)\]is a model that simulates human memory retention over months.

• The initial score of 75 represents the starting point of retention, reflecting the immediate post-exam score.
• The logarithmic term \(-10 \log (t+1)\) signifies decreasing memory retention, a natural process observed to be logarithmic in nature.
• The time variable \(t\) shows that as time progresses, the remembered score diminishes.

Mathematical models like this help us predict future outcomes and understand the dynamics at play in various scientific and social phenomena.