Question

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 t≥0. complete parts (a) through (e) below.
s(t)=75 - 12 ln(t + 1), 0≤t≤80
b) what was the average score after 4 months?
the average score after 4 months was 55.7% (round to one decimal place as needed.)
c) what was the average score after 24 months?
the average score after 24 months was 36.4% (round to one decimal place as needed.)
d) find s(t)
s(t)=-\frac{12}{t + 1}
e) find s(4) and s(24), and interpret the meaning of these numbers
s(4)=, which means that after 4 months, the average score is by points a month and
s(24)=, which means that after 24 months, the average score is by points a month
(type integers or decimals.)

Explanation:

Step1: Recall the derivative formula

We are given $S(t)=75 - 12\ln(t + 1)$ and its derivative $S'(t)=-\frac{12}{t + 1}$.

Step2: Calculate $S'(4)$

Substitute $t = 4$ into $S'(t)$:
$S'(4)=-\frac{12}{4 + 1}=-\frac{12}{5}=- 2.4$
This means that after 4 months, the average score is decreasing by 2.4 points a month.

Step3: Calculate $S'(24)$

Substitute $t = 24$ into $S'(t)$:
$S'(24)=-\frac{12}{24+1}=-\frac{12}{25}=-0.48$
This means that after 24 months, the average score is decreasing by 0.48 points a month.

Answer:

$S'(4)=-2.4$, which means that after 4 months, the average score is decreasing by 2.4 points a month.
$S'(24)=-0.48$, which means that after 24 months, the average score is decreasing by 0.48 points a month.