Vectors and matrices — Question 19

QCAA 2024, Paper 2 · 6 marks

Q19 · 2024 · Technology-activeComplex unfamiliar6 marks

QUESTION 19 (6 marks)

An experiment researching the population changes of a certain species of insect was conducted over a four-week period. The insect has two distinct stages in its two-week lifespan. Each stage is approximately one week in length. A constant proportion of females survive from stage 1 into stage 2. The ratio of the reproduction rate for females in stage 2 to females in stage 1 is 2:1. All offspring are born into stage 1. The number of females in each stage was measured initially and then again after two weeks as shown.

Female population	Stage 1	Stage 2
Initially	48	32
After two weeks	25	21

Use a matrix approach to estimate the total number of females after four weeks.

Question linkOriginal paper

WORKED SOLUTION

QCAA guide · typeset solution

6 marks

ANSWER

Approximately 26 females.

Worked solution

Let the birth rate for stage 1 females be

x

and the survival rate from stage 1 to stage 2 be

y

. The Leslie matrix is

L=\begin{bmatrix}x&2x\\y&0\end{bmatrix}.

Let

P_n

represent the population after

n

weeks. Then

P_0=\begin{bmatrix}48\\32\end{bmatrix},\qquad P_2=L^2P_0=\begin{bmatrix}25\\21\end{bmatrix}.

This gives

112x^2+96xy=25,\qquad112xy=21.

Thus

xy=21/112

. Substitution gives

x=0.25

(rejecting the negative solution), and then

y=0.75

. Therefore

P_4=L^4P_0=\begin{bmatrix}0.25&0.5\\0.75&0\end{bmatrix}^{4}\begin{bmatrix}48\\32\end{bmatrix}\approx\begin{bmatrix}13.6\\12.6\end{bmatrix}.

The estimated total is about

13.6+12.6=26.2

, i.e. approximately 26 females.

correctly determines an appropriate Leslie matrix

[1 mark]

determines a matrix equation linking the initial population with the population after two weeks

[1 mark]

determines two simultaneous equations in terms of the relevant birth and survival rates

[1 mark]

determines appropriate values of

x

and

y

[1 mark]

determines the total number of females at the conclusion of the experiment

[1 mark]

shows logical organisation of a fully attempted solution, communicating key steps

[1 mark]

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Vectors and matrices — Question 19

QUESTION 19 (6 marks)

QCAA guide · typeset solution

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