5.07 Sinusoidal models

Sounds around us create pressure waves. Our ears interpret the amplitude and frequency of these waves to make sense of the sounds.

A speaker is set to create a single tone, and the graph below shows how the pressure intensity (I) of the tone, relative to atmospheric pressure, changes over t seconds:

Marge has asthma, so she is having some breathing tests done to monitor her lung capacity V measured in millilitres \left(\text{mL}\right). The respirologist models Marge's breathing patterns according to the model V = a \sin \left( b \left(t + c\right)\right) + d , where the quantity b \left(t + c\right) is in degrees. The graph of the function is shown below:

A circular Ferris wheel that is 40 \text{ m} in diameter contains several carriages. Hannah and Michael enter a carriage at the bottom of the Ferris wheel, and get off 6 minutes later after having gone around completely 3 times. When a carriage is at the bottom of the wheel, it is 1 \text{ m} above the ground.

The height above the ground of a rider on a Ferris wheel can be modelled by the function:h \left( t \right) = 25 \cos \left( 3 \left(t - 60\right)\right) + 30where h \left( t \right) is the height in metres, t is the time in seconds, and the quantity \left( 3 \left(t - 60\right)\right) is in degrees.

The height in metres of the tide above mean sea level t hours after midnight is given by: h = 4 \sin \left(30 \left(t - 2\right)\right)

The tide rises and falls in a periodic manner which can be modelled by a trigonometric function. Brad charts tide levels in order to determine when he can sail his ship into a bay and takes the following measurements:

• Low tide occured at 8 \text{ am}, when the bay was 7 \text{ m} deep.
• High tide occured at 2 \text{ pm}, when the bay was 15 \text{ m} deep.

The height of water in metres on a beach wall t hours after midnight is given by:h \left( t \right) = 6 + 4 \cos \left( 30 t - 60\right)

In a tidal river, the time between high tides is 12 hours. The average depth of water at a point in the river is 5 \text{ m}, and at high tide the depth is 8 \text{ m}. The depth of water at this point, t hours after noon, is given by: h \left( t \right) = a \sin \left( b t + c\right) + dwhere a \gt 0, c is the least positive number, the quantity \left(b t + c\right) is in degrees.

Tobias is jumping on a trampoline. Victoria watches him bounce at a regular rate and wants to try to model his height over time. When Victoria starts her stopwatch, Tobias is at a minimum height of 30 \text{ cm} below the trampoline frame. A moment later Victoria records Tobias reaching a maximum height of 50 \text{ cm} above the trampoline frame. To model the situation Victoria uses the function:H \left( t \right) = a \sin \left( 360 \left(t - c\right)\right) + dwhere H is the height in centimetres above the trampoline frame and t is the time in seconds and 0 < c < 1.

Three objects, X, Y and Z are placed in a magnetic field such that:

• X is 2 cm from object Y.
• X is 4 cm from object Z.
• As object X is moved closer to line YZ, objects Y and Z move in such a way that the lengths XY and XZ remain fixed.

Let \theta = \angle ZXY, and let the area of triangle XYZ be represented by A.

a

Find an equation for A in terms of \theta.

b

State the domain of the function of A in this context.

c

Sketch a graph of the function A.

d

Find the acute angle which will form an area that is exactly half of the maximum possible area.

A metronome is a device used to help keep the beat consistent when playing a musical instrument. It swings back and forth between its end points, just like a pendulum.

For a particular speed, the given graph represents the metronome's distance, x \text{ cm}, from the centre of its swing, t seconds after it starts swinging. Negative values of x represent swinging to the left, and positive values of x represent swinging to the right of the centre: