QCAA Maths Methods Paper 2 Integral Calculus Mini Test 6
External Assessment Paper 2 — Technology-active
Number of marks: 11
Perusal time: 1 minute
Writing time: 15 minutes
Section 2
Instructions
• Write using black or blue pen.
• Questions worth more than one mark require mathematical reasoning and/or working to be shown to support answers.
• If you need more space for a response, use the additional pages at the back of this book.
– On the additional pages, write the question number you are responding to.
– Cancel any incorrect response by ruling a single diagonal line through your work.
– Write the page number of your alternative/additional response, i.e. See page …
– If you do not do this, your original response will be marked.
• This section has nine questions and is worth 45 marks.
The rates of change in population for two cities are given by
City A: \( A'(t) = \frac{45}{t+1} \)
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.
a) Determine the population models for both cities. [3 marks]
b) Use the information in 12a) to predict the population of City B at the beginning of 2028. [1 mark]
c) Use the information in 12a) to predict the year in which the population of both cities will be the same. [3 marks]
Flying foxes enter and leave a fruit-growing region every evening. The rate at which the flying foxes enter the region is modelled by the function \[ A(t) = 42 \sin\left(0.03t - \frac{\pi}{3}\right) + 71, \quad 0 \le t \le 240 \]
The rate at which the flying foxes leave the region is modelled by the function \[ L(t) = 42 \sin\left(0.04t - \frac{\pi}{3}\right) + 42, \quad 0 \le t \le 240 \]
Both \(A(t)\) and \(L(t)\) are measured in animals per minute and \(t\) is measured in minutes after 7 pm.
There are 100 flying foxes in the region at 7 pm.
Determine the maximum number of flying foxes in the region and the time that this occurs.
END OF PAPER
