21M.1.SL.TZ1.7

pestleMathematicsAISLPaper 121M· sl-2-5-modelling-functionssource ↗

Professor Wei observed that students have difficulty remembering the information presented in his lectures.

He modelled the percentage of information retained, $R$ , by the function $R (t) = 100 e^{- p t}$ , $t \geq 0$ , where $t$ is the number of days after the lecture.

He found that $1$ day after a lecture, students had forgotten $50 %$ of the information presented.

Based on his model, Professor Wei believes that his students will always retain some information from his lecture.

Find the value of $p$ .

[2]

Use this model to find the percentage of information retained by his students $36$ hours after Professor Wei’s lecture.

[2]

State a mathematical reason why Professor Wei might believe this.

[1]

Write down one possible limitation of the domain of the model.

[1]

Markscheme / solution

EITHER

$50 = 100 e^{- 1 \times p}$ OR $0.5 = e^{- 1 \times p}$ (M1)

THEN

$0.693 (0.693147 \dots, \ln 2)$ A1

[2 marks]

$R (1.5) = 100 e^{- 0.693147 \dots \times 1.5}$ (M1)

$35.4 (%) (35.3553 \dots)$ A1

[2 marks]

$R (t) > 0$ OR $R (t)$ has a horizontal asymptote R1

[1 mark]

Award A1 for one reasonable limitation of the domain: A1

small values of $t$ produce unrealistic results

$R (0) = 100 %$

large values of $t$ are not possible

people do not live forever

model is not valid at small or large values of $t$

The reason should focus on the domain $t \geq 0$ . Do not accept answers such as:

recollection varies for different people

memories are discrete not continuous

the nature of the information will change how easily it is recalled

emotional/physical stress can affect recollection/concentration

Note: Do not accept $t \geq 0$ as this is a limitation that has been given in the question.

[1 mark]

Examiners’ report

[N/A]