VCE Methods Trigonometry Application Task 2
Number of marks: 11
Reading time: 2 minutes
Writing time: 16 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.
Let \(f: [0, 8\pi] \to R, f(x) = 2\cos\left(\frac{x}{2}\right) + \pi\).
a. Find the period and range of \(f\). 2 marks
The following diagram represents an observation wheel, with its centre at point P. Passengers are seated in pods, which are carried around as the wheel turns. The wheel moves anticlockwise with constant speed and completes one full rotation every 30 minutes. When a pod is at the lowest point of the wheel (point A), it is 15 metres above the ground. The wheel has a radius of 60 metres.
Consider the function \( h(t) = -60 \cos(bt) + c \) for some \( b, c \in \mathbb{R} \), which models the height above the ground of a pod originally situated at point A, after time \( t \) minutes.
a. Show that \( b = \frac{\pi}{15} \) and \( c = 75 \). 2 marks
Let \( g(x) = f(\sin(2x)) \).
The following table gives values for \( f(x) \):
| \( x \) | \( \frac{1}{2} \) | \( \frac{\sqrt{2}}{2} \) | \( \frac{\sqrt{3}}{2} \) |
|---|---|---|---|
| \( f(x) \) | \( -2 \) | \( 5 \) | \( 3 \) |
a. Find the value of \( g\left( \frac{\pi}{6} \right) \). 1 mark
During a telephone call, a phone uses a dual-tone frequency electrical signal to communicate with the telephone exchange.
The strength, \(f\), of a simple dual-tone frequency signal is given by the function \(f(t) = \sin\left(\frac{\pi t}{3}\right) + \sin\left(\frac{\pi t}{6}\right)\) where \(t\) is a measure of time and \(t \ge 0\).
Part of the graph of \(y = f(t)\) is shown below.
a. State the period of the function. 1 mark
b. Find the values of \(t\) where \(f(t) = 0\) for the interval \(t \in [0, 6]\). 1 mark
A horizontal bridge positioned 5 m above level ground is 110 m in length. The bridge also touches the top of three arches. Each arch begins and ends at ground level. The arches are 5 m apart at the base, as shown in the diagram below.
Let \(x\) be the horizontal distance, in metres, from the left side of the bridge and let \(y\) be the height, in metres, above ground level.
Arch 1 can be modelled by the function \( h_1 : [5, 35] \rightarrow \mathbb{R}, h_1(x) = 5\sin\left(\frac{(x-5)\pi}{30}\right) \).
Arch 2 can be modelled by the function \( h_2 : [40, 70] \rightarrow \mathbb{R}, h_2(x) = 5\sin\left(\frac{(x-40)\pi}{30}\right) \).
Arch 3 can be modelled by the function \( h_3 : [a, 105] \rightarrow \mathbb{R}, h_3(x) = 5\sin\left(\frac{(x-a)\pi}{30}\right) \).
a. State the value of \(a\), where \(a \in \mathbb{R}\). 1 mark
b. Describe the transformation that maps the graph of \(y = h_2(x)\) to \(y = h_3(x)\). 1 mark
Sammy visits a giant Ferris wheel. Sammy enters a capsule on the Ferris wheel from a platform above the ground. The Ferris wheel is rotating anticlockwise. The capsule is attached to the Ferris wheel at point \(P\). The height of \(P\) above the ground, \(h\), is modelled by \(h(t) = 65 - 55\cos\left(\frac{\pi t}{15}\right)\), where \(t\) is the time in minutes after Sammy enters the capsule and \(h\) is measured in metres.
Sammy exits the capsule after one complete rotation of the Ferris wheel.
a. State the minimum and maximum heights of \(P\) above the ground. 1 mark
b. For how much time is Sammy in the capsule? 1 mark
End of examination questions
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