Trigonometry Application Task Mini Test 1
Number of marks: 10
Reading time: 2 minutes
Writing time: 15 minutes
Section B – Calculator Allowed
Instructions
• Answer all questions in the spaces provided.
• Write your responses in English.
• In questions where a numerical answer is required, an exact value must be given unless otherwise specified.
• In questions where more than one mark is available, appropriate working must be shown.
• Unless otherwise indicated, the diagrams in this book are not drawn to scale.
On a remote island, there are only two species of animals: foxes and rabbits. The foxes are the predators and the rabbits are their prey.
The populations of foxes and rabbits increase and decrease in a periodic pattern, with the period of both populations being the same, as shown in the graph below, for all \( t \geq 0 \), where time \( t \) is measured in weeks.
One point of minimum fox population, \( (20, 700) \), and one point of maximum fox population, \( (100, 2500) \), are also shown on the graph.
The graph has been drawn to scale.
a. i. State the initial population of rabbits. 1 mark
ii. State the minimum and maximum population of rabbits. 1 mark
iii. State the number of weeks between maximum populations of rabbits. 1 mark
The population of foxes can be modelled by the rule \( f(t) = a \sin\big(b(t - 60)\big) + 1600 \).
b. Show that \( a = 900 \) and \( b = \frac{\pi}{80} \). 2 marks
An area of parkland has a river running through it, as shown below. The river is shown shaded.
The north bank of the river is modelled by the function \(f_1: [0, 200] \to \mathbb{R}\), \(f_1(x) = 20\cos\left(\frac{\pi x}{100}\right) + 40\).
The south bank of the river is modelled by the function \(f_2: [0, 200] \to \mathbb{R}\), \(f_2(x) = 20\cos\left(\frac{\pi x}{100}\right) + 30\).
The horizontal axis points east and the vertical axis points north.
All distances are measured in metres.
A swimmer always starts at point \(P\), which has coordinates \((50, 30)\).
Assume that no movement of water in the river affects the motion or path of the swimmer, which is always a straight line.
a. The swimmer swims north from point \(P\).
Find the distance, in metres, that the swimmer needs to swim to get to the north bank of the river. 1 mark
b. The swimmer swims east from point \(P\).
Find the distance, in metres, that the swimmer needs to swim to get to the north bank of the river. 2 marks
c. On another occasion, the swimmer swims the minimum distance from point \(P\) to the north bank of the river.
Find this minimum distance. Give your answer in metres, correct to one decimal place. 2 marks
End of examination questions
VCE is a registered trademark of the VCAA. The VCAA does not endorse or make any warranties regarding this study resource. Past VCE exams and related content can be accessed directly at www.vcaa.vic.edu.au
