Question

Students in a psychology class took a final examination. As part of an experiment to see how much of the course content they remembered over time, they took equivalent forms of the exam in monthly intervals thereafter. The average score for the group, f(t), after t months was modeled by the function

F (t) = 88-15 In(t + 1), 0 ≤ t ≤ 12
Answer the following questions and show all work to obtaining that answer.
1. What was the average score on the original exam?
2. What was the average score, to the nearest tenth, after 2 months? 4 months? 6 months? 1 year?
3. Sketch the graph of f (either by hand or with a graphing utility). Describe what the graph indicates in terms of the material retained by the students. You do not need to include a picture of the graph in your assignment.
Note: ln represents the natural logarithm in the function.

Answer 1

a) 88

b) F ( 2 ) = 71.5 , F ( 4 ) = 63.9 , F ( 6 ) = 58.8 , F ( 12 ) = 49.5

c) See explanation

Explanation:

Solution:-

- Students in a psychology class took a final examination. As part of an experiment to see how much of the course content they remembered over time, they took equivalent forms of the exam in monthly intervals thereafter.

- The average score was modeled by a function f(t), given:

f ( t ) = 88 - 15*Ln ( t + 1 )

Interval: 0 ≤ t ≤ 12 , t : Months

- The average score obtained by the group on the original exam can be determined from the given relation at t = 0 months. Plug the value of t = 0 in the given relation to determine the average score on original exam:

f ( 0 ) = 88 - 15*Ln ( 0 + 1 )

f ( 0 ) = 88 - 15*Ln ( 1 ) = 88 - 15*0

f ( 0 ) = 88

Answer: The average score on the original exam was 88 .

- The average score obtained by the group on the exam after two months can be determined from the given relation at t = 2 months. Plug the value of t = 2 in the given relation to determine the average score on the exam:

f ( 2 ) = 88 - 15*Ln ( 2 + 1 )

f ( 2 ) = 88 - 15*Ln ( 3 ) = 88 - 16.47918

f ( 2 ) = 71.520

Answer: The average score on the exam after two months was 71.5

- The average score obtained by the group on the exam after four months can be determined from the given relation at t = 4 months. Plug the value of t = 4 in the given relation to determine the average score on the exam:

f ( 4 ) = 88 - 15*Ln ( 4 + 1 )

f ( 4 ) = 88 - 15*Ln ( 5 ) = 88 - 24.14156

f ( 4 ) = 63.858

Answer: The average score on the exam after four months was 63.9

- The average score obtained by the group on the exam after six months can be determined from the given relation at t = 6 months. Plug the value of t = 6 in the given relation to determine the average score on the exam:

f ( 6 ) = 88 - 15*Ln ( 6 + 1 )

f ( 6 ) = 88 - 15*Ln ( 7 ) = 88 - 29.18865

f ( 6 ) = 58.811

Answer: The average score on the exam after six months was 58.8

- The average score obtained by the group on the exam after a year can be determined from the given relation at t = 12 months. Plug the value of t = 12 in the given relation to determine the average score on the exam:

f ( 12 ) = 88 - 15*Ln ( 12 + 1 )

f ( 12 ) = 88 - 15*Ln ( 13 ) = 88 - 38.47424

f ( 12 ) = 49.525

Answer: The average score on the exam after a year was 49.5

- The function F ( t ) i.e the average score of the group for the equivalent exams they took over the course of months is shown by a decreasing curve.

- The average score decreases as time passes by. This hints towards the material retention of the students decreases with time because they score less after every successive exam.