$$ 1 \leftarrow \quad \begin{array}{l} ext { Students in a psychology class took a final examination As } \ ext { part of an experiment to see how much of the course content } \ ext { they remembered over time, they took equivalent forms of the } \ ext { exam in monthly intervals thereafter. The average score for } \ ext { the group } f(t) ext {, after } t ext { months was modeled by the function } \ f(t)=78-16 \ln (t+1), 0 \leq t \leq 12 \ ext { a. What was the average score on the original exam? } \ ext { b. What was the average score after } 8 ext { months? } \ ext { c. Sketch the graph of } f ext { (either by hand or with a } \ ext { graphing utiity) Describe what the graph indicates in terms of } \ ext { the material retained by the students. } \ ext { a. The average score on the original exam was } \ ext { (Type a whole number) }\end{array} $$

Question

$$ 1 \leftarrow \quad \begin{array}{l}	ext { Students in a psychology class took a final examination As } \ 	ext { part of an experiment to see how much of the course content } \ 	ext { they remembered over time, they took equivalent forms of the } \ 	ext { exam in monthly intervals thereafter. The average score for } \ 	ext { the group } f(t) 	ext {, after } t 	ext { months was modeled by the function } \ f(t)=78-16 \ln (t+1), 0 \leq t \leq 12 \ 	ext { a. What was the average score on the original exam? } \ 	ext { b. What was the average score after } 8 	ext { months? } \ 	ext { c. Sketch the graph of } f 	ext { (either by hand or with a } \ 	ext { graphing utiity) Describe what the graph indicates in terms of } \ 	ext { the material retained by the students. } \ 	ext { a. The average score on the original exam was } \ 	ext { (Type a whole number) }\end{array} $$

UpStudy AI · Accepted Answer

**Step 1. Identify the function and its domain**

We are given the function  
$$
f(t)=78-16 \ln(t+1)
$$  
where $$0\leq t\leq 12$$. This function models the average exam score for the students $$t$$ months after the original exam.

**Step 2. Evaluate the function at $$t=0$$ (the original exam)**

For the original exam, $$t=0$$:
$$
f(0)=78-16\ln(0+1)=78-16\ln(1)
$$
Since $$\ln(1)=0$$, we have:
$$
f(0)=78-16(0)=78
$$
Thus, the average score on the original exam was $$\boxed{78}$$.

**Step 3. Evaluate the function at $$t=8$$ (after 8 months)**

For $$t=8$$:
$$
f(8)=78-16\ln(8+1)=78-16\ln(9)
$$

Leaving the answer in exact form, the average score after 8 months is:
$$
f(8)=78-16\ln(9)
$$

For an approximate numerical value, note that $$\ln(9)\approx2.1972$$. Then:
$$
16\ln(9)\approx16\times2.1972\approx35.155
$$
$$
f(8)\approx78-35.155\approx42.845
$$
Thus, the average score after 8 months is approximately $$\boxed{42.85}$$.

**Step 4. Sketch and interpret the graph of $$f(t)=78-16\ln(t+1)$$**

- **Intercept:** At $$t=0$$, $$f(0)=78$$. So the graph starts at the point $$(0,78)$$.
- **Behavior:** As $$t$$ increases, the logarithmic function $$\ln(t+1)$$ increases. Hence, $$f(t)$$ decreases. This shows that exam scores tend to drop as time passes.
- **Concavity:**  
  The first derivative is
  $$
  f'(t)=-\frac{16}{t+1},
  $$
  which is negative for all $$t\geq0$$, confirming that the function is decreasing.  
  The second derivative is
  $$
  f''(t)=\frac{16}{(t+1)^2},
  $$
  which is positive for $$t\geq0$$. Therefore, the graph is concave upward.
- **Interpretation:**  
  The graph starts at $$78$$ and decreases rapidly for small $$t$$ (representing an initial quick loss of retention), then levels off as $$t$$ increases (indicating that the rate of forgetting slows down). This suggests that while students forget a significant portion of the material soon after the exam, the rate of loss diminishes over time.

Thus, the graph of $$f(t)$$ is a decreasing, concave-up curve that models the gradual loss of exam recall over 12 months.
**a. The average score on the original exam was 78.** **b. The average score after 8 months was approximately 42.85.** **c. The graph of $$ f(t) = 78 - 16 \ln(t+1) $$ is a decreasing, concave-up curve starting at 78 when $$ t = 0 $$. It shows that students initially forget a lot of the material quickly, but the rate of forgetting slows down over time.

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