QCE Mathematical Methods: Exam practice Q12 (2025, Paper 2)

Q12 · 2025 · Technology-activeSimple familiar7 marks

QUESTION 12 (7 marks)

A cockroach population is modelled by the function

P(t)=P_0e^{kt}

, where

P

is the population after

t

weeks and

k

is a population constant. Initially, 100 cockroaches were counted. After three weeks, there were 120.

Determine the constant

k

[2 marks]

Determine the function

P'(t)

[1 mark]

Determine when the rate of population growth would reach 10 cockroaches per week.

[1 mark]

To decrease the growth rate of the cockroach population, a pest control treatment was trialled. The new function to model the cockroach population

t

weeks after using the treatment is

I(t)=c\ln(t+8)+172,

where

c

is a constant. Three weeks after using the treatment, the rate of population growth was five cockroaches per week.

Determine the value of

c

[2 marks]

Determine the cockroach population when the treatment began.

[1 mark]

Question linkOriginal paper

WORKED SOLUTION

QCAA guide · typeset solution

7 marks

ANSWER

k=\frac13\ln(1.2)\approx0.06077

; b)

P'(t)=6.077e^{0.06077t}

; c)

t=8.1945

weeks; d)

c=55

; e) approximately 286 cockroaches.

Worked solution

a) Initially

P_0=100

. Since

P(3)=120

120=100e^{3k},

k=\frac13\ln(1.2)=0.06077.

Thus

P(t)=100e^{0.06077t}.

P'(t)=100(0.06077)e^{0.06077t}=6.077e^{0.06077t}.

c) Solving

10=P'(t)

gives

t=8.1945\text{ weeks}.

For the pest-control model,

I(t)=c\ln(t+8)+172,

I'(t)=\frac{c}{t+8}.

t=3

I'(3)=5

, hence

\frac{c}{11}=5,

c=55.

At the beginning of treatment,

I(0)=55\ln8+172=286.3693,

so there were approximately 286 cockroaches.

correctly identifies the value of

P_0

in the model

[1 mark]

determines the value of

k

[1 mark]

determines the derivative function

[1 mark]

determines the time when the rate of population growth reached 10 cockroaches per week

[1 mark]

correctly differentiates the pest control function

[1 mark]

determines the value of

c

[1 mark]

determines the population when the treatment began

[1 mark]

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Mathematical Methods — Question 12

QUESTION 12 (7 marks)

QCAA guide · typeset solution

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