2021 HSC Mathematics Extension 1
Question 14(b):
Differential equation
In a certain country, the population of deer was estimated in to be . The population growth is given by the logistic equation , where is the number of years after and is the carrying capacity.
In the year , the population of deer was estimated to be .
Use the fact that to show that the carrying capacity is approximately .
How to recognise this question
Question type: differential equations
- The command and notation point to differential equations.
- The requested response is a show that.
How to handle it: separate or integrate the differential equation, include the constant, then use the initial condition.
Watch out: Do not select a formula until its domain, interval, sign and units match the question.
Step-by-step answer
- Separate: .
- Using the given decomposition: .
- .
- At , : .
- At , : .
- Take reciprocals: .
- .
- .
- (to the accuracy required).
- NESA: 4 marks full; 3 for two equations in and ; 2 for correct integration; 1 for separating variables.
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 4 total marks for this group. Where the official marking guide provides useful partial-credit criteria, those criteria are stated in the steps above.
Try your hand at it again by answering these questions
Question 1
Given with , express in terms of .
Question 2
Same DE and . If , set up the equation for (do not simplify numerically).
Question 3
Evaluate to the nearest thousand.