These Ten-Year-Series (TYS) worked solutions with video explanations for 2022 A Level H2 Mathematics Paper 1 Question 12 are suggested by Mr Gan. For any comments or suggestions please contact us at support@timganmath.edu.sg.
2022 A Level H2 Math Paper 1 Question 12
Scientists are interested in the population of a particular species. They attempt to model the population $P$ at time $t$ days using a differential equation. Initially the population is observed to be $50$ and after $10$ days the population is $100$.
The first model the scientists use assumes that the rate of change of the population is proportional to the population.
(a)
Write down a differential equation for this model and solve it for $P$ in terms of $t$.
[5]
(a) Write down a differential equation for this model and solve it for $P$ in terms of $t$.
[5]
To allow for constraints on population growth, the model is refined to
$\frac{\text{d}P}{\text{d}t}=\lambda P\left( 500-P \right)$
where $\lambda $ is a constant.
(b)
Solve this differential equation to find $P$ in terms of $t$.
[6]
(b) Solve this differential equation to find $P$ in terms of $t$.
[6]
(c)
Using the refined model, state the population of this species in the long term. Comment on how this value suggests the refined model is an improvement on the first model.
[2]
(c) Using the refined model, state the population of this species in the long term. Comment on how this value suggests the refined model is an improvement on the first model.
[2]
Suggested Video Solutions
Suggested Handwritten Solutions
In the long run, the population of this species will increase from $50$ and stabalise at $500$ in the refined model whereas the first model would suggest that the population will increase indefinitely. Thus, the refined model is better as in real life a population growth of a species will be limited by external factors such as death rate and competition to survive.
In the long run, the population of this species will increase from $50$ and stabalise at $500$ in the refined model whereas the first model would suggest that the population will increase indefinitely. Thus, the refined model is better as in real life a population growth of a species will be limited by external factors such as death rate and competition to survive.
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