QCE Mathematical Methods: Exam practice Q12 (2020, Paper 2)

Mathematical Methods — Question 12

QCAA 2020, Paper 2 · 7 marks

Q12 · 2020 · Technology-activeSimple familiar7 marks

QUESTION 12 (7 marks)

The rates of change in population for two cities are given by

\text{City A: }A'(t)=\frac{45}{t+1},

\text{City B: }B'(t)=105e^{0.03t},

where

t

is the number of years since 2018 and both

A'(t)

and

B'(t)

are measured in people per year. At the beginning of 2018, City A had a population of 5000, and City B had a population of 3500.

Determine the population models for both cities.

[3 marks]

Use the information in 12a) to predict the population of City B at the beginning of 2028.

[1 mark]

Use the information in 12a) to predict the year in which the population of both cities will be the same.

[3 marks]

Question linkOriginal paper

WORKED SOLUTION

QCAA guide · typeset solution

7 marks

ANSWER

A(t)=45\ln(t+1)+5000

B(t)=3500e^{0.03t}

; b) 4724; c) during 2030

Worked solution

a) Integrating gives

A(t)=45\ln(t+1)+c

and

B(t)=3500e^{0.03t}+c

. Using

A(0)=5000

and

B(0)=3500

gives

A(t)=45\ln(t+1)+5000,\qquad B(t)=3500e^{0.03t}.

B(10)=3500e^{0.3}=4724

c) Solve

45\ln(t+1)+5000=3500e^{0.03t}.

A GDC gives

t=12.66

, so the populations are the same during 2030.

correctly determines the general population function for City A

[1 mark]

correctly determines the general population function for City B

[1 mark]

determines population functions for the two cities

[1 mark]

determines population of City B in 2028

[1 mark]

uses an appropriate mathematical representation to communicate approach

[1 mark]

determines time

[1 mark]

determines the required year

[1 mark]

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Mathematical Methods — Question 12

QUESTION 12 (7 marks)

QCAA guide · typeset solution

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