Answer
a) The simplified value is $76$.
b) The simplified average score is $67,\text{ 63, 61, 59, and 56}$.
c) Shown below
Work Step by Step
(a)
For the first exam, the value of t is 0:
$\begin{align}
& f\left( 0 \right)=76-18\log \left( 0+1 \right) \\
& =76-18\log \left( 1 \right) \\
& =76-18\left( 0 \right) \\
& =76
\end{align}$
Thus, the average score for the first exam is 76.
(b)
We have the average score after the time period of 2 months:
$\begin{align}
& f\left( 2 \right)=76-18\log \left( 2+1 \right) \\
& =76-18\log \left( 3 \right) \\
& =76-18\left( 0.4771 \right) \\
& =67.4118
\end{align}$
Take the approximate value:
$ f\left( 2 \right)\approx 67$
And the average score after the time period of 4 months is:
$\begin{align}
& f\left( 4 \right)=76-18\log \left( 4+1 \right) \\
& =76-18\log \left( 5 \right) \\
& =76-18\left( 0.6990 \right) \\
& =63.4185
\end{align}$
Take the approximate value:
$ f\left( 4 \right)\approx 63$
And the average score after the time period of 6 months is:
$\begin{align}
& f\left( 6 \right)=76-18\log \left( 6+1 \right) \\
& =76-18\log \left( 7 \right) \\
& =76-18\left( 0.8451 \right) \\
& =60.7882
\end{align}$
Take the approximate value:
$ f\left( 6 \right)\approx 61$
The average score after the time period of 8 months is:
$\begin{align}
& f\left( 8 \right)=76-18\log \left( 8+1 \right) \\
& =76-18\log \left( 9 \right) \\
& =76-18\left( 0.9542 \right) \\
& =58.8236
\end{align}$
Take the approximate value:
$ f\left( 8 \right)\approx 59$
The average score after the time period of 1 year or 12 months is:
$\begin{align}
& f\left( 12 \right)=76-18\log \left( 12+1 \right) \\
& =76-18\log \left( 13 \right) \\
& =76-18\left( 1.1139 \right) \\
& =55.9490
\end{align}$
Take the approximate value:
$ f\left( 12 \right)\approx 56$
Thus, the average scores after 2 months, 4 months, 6 months, 8 months, and 1 year are $67,\text{ 63, 61, 59, and 56}$ respectively.
(c)
Draw the graph, formulate the table from values of t and f obtained in part(a) and part(b).
Thus, as the value of t increases, the value of $76-18\log \left( t+1 \right)$ decreases. We see that with an increase in time t, the retention of course content decreases for students.
