Problem 132 - College Algebra Essentials

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 \(\quad 0 \leq t \leq 12 . \quad\) Use \(\quad\) 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 resulting time will depend on the accuracy of the graphing used. The number of months can be approximated but to get an accurate number, solving equation \(75-10 \log (t+1) = 65\) could be necessary, which would involve more complex algebraic manipulations and use of logarithm properties.

Step 1: Understanding and graphing the function

The function to be graphed is \(f(t)=75-10 \log (t+1)\), where \(t\) represents time in months. We will graph this function for \(0 \leq t \leq 12\). Most graphing utilities would require entering the function as it is, specifying the range of \(t\) and then generating the graph.

Step 2: Analyzing the graph

Once a graph is constructed, we can see how the function \(f(t)\) evolves over time. Specifically, we want to check how many months it takes for the average score to fall under 65, that is when \(f(t) < 65\). This can be done by following the line on the graph from starting point and see where it crosses the \(y=65\) horizontal line.

Step 3: Finding the exact time

Make use of the graphing utility's feature to find the exact value of \(t\) when \(f(t) = 65\). This would be the solution to the problem as this is the time it takes for the score to fall below 65.