Students in a zoology class took a final exam. They took equivalent forms of the exam at monthly intervals thereafter. After $t$ months, the average score $S(t)$, as a percentage, was found to be given by the following equation, where $1 \geq 0$. Complete parts (a) through $(e)$ below.
\[
S(t)=77-14 \ln (t+1), 0 \leq 1 \leq 80
\]
a) What was the average score when they initially took the test?
The average score was $\%$. (Round to one decimal place as needed.)
Answer
The average score when they initially took the test was \(\boxed{77.0\%}\).
Step 1 ：The question is asking for the average score when the students initially took the test. This corresponds to the time \(t=0\). We can substitute \(t=0\) into the equation \(S(t)=77-14 \ln (t+1)\) to find the initial average score.
Step 2 ：Substitute \(t=0\) into the equation: \(S(0)=77-14 \ln (0+1)\)
Step 3 ：Calculate the value of \(S(0)\) to get the initial average score.
Step 4 ：The average score when they initially took the test was \(\boxed{77.0\%}\).
