SPS SPS SM 2021 November — Question 9 4 marks

Exam BoardSPS
ModuleSPS SM (SPS SM)
Year2021
SessionNovember
Marks4
TopicExponential Functions
TypeCritique single model appropriateness
DifficultyEasy -1.3 This is a straightforward exponential modelling question requiring only basic substitution (parts a-b), simple logarithm manipulation (part b), and solving an inequality (part d). Part (c) asks for critique of a model which is standard A-level literacy. All techniques are routine with no novel problem-solving required, making it easier than average.
Spec1.02z Models in context: use functions in modelling1.06a Exponential function: a^x and e^x graphs and properties1.06g Equations with exponentials: solve a^x = b1.06i Exponential growth/decay: in modelling context
9. David has been investigating the population of rabbits on an island during a three-year period. Based on data that he has collected, David decides to model the population of rabbits, \(R\), by the formula $$R = 50 \mathrm { e } ^ { 0.5 t }$$ where \(t\) is the time in years after 1 January 2016.
  1. Using David's model:
    1. (i) state the population of rabbits on the island on 1 January 2016;
    2. (ii) predict the population of rabbits on 1 January 2021.
    3. Use David's model to find the value of \(t\) when \(R = 150\), giving your answer to three significant figures.
    4. Give one reason why David's model may not be appropriate.
      [0pt] [1 mark]
    5. On the same island, the population of crickets, \(C\), can be modelled by the formula $$C = 1000 \mathrm { e } ^ { 0.1 t }$$ where \(t\) is the time in years after 1 January 2016.
      Using the two models, find the year during which the population of rabbits first exceeds the population of crickets.
      [0pt] [3 marks]
9.

David has been investigating the population of rabbits on an island during a three-year period.

Based on data that he has collected, David decides to model the population of rabbits, $R$, by the formula

$$R = 50 \mathrm { e } ^ { 0.5 t }$$

where $t$ is the time in years after 1 January 2016.
\begin{enumerate}[label=(\alph*)]
\item Using David's model:\\
(a) (i) state the population of rabbits on the island on 1 January 2016;\\

(a) (ii) predict the population of rabbits on 1 January 2021.
\item Use David's model to find the value of $t$ when $R = 150$, giving your answer to three significant figures.
\item Give one reason why David's model may not be appropriate.\\[0pt]
[1 mark]
\item On the same island, the population of crickets, $C$, can be modelled by the formula

$$C = 1000 \mathrm { e } ^ { 0.1 t }$$

where $t$ is the time in years after 1 January 2016.\\
Using the two models, find the year during which the population of rabbits first exceeds the population of crickets.\\[0pt]
[3 marks]
\end{enumerate}

\hfill \mbox{\textit{SPS SPS SM 2021 Q9 [4]}}
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