Bert is on a Ferris wheel. The radius of the Ferris wheel is 6 metres. Its axle is 7 metres above the ground. It takes 100 seconds to complete one full revolution. Bert enters the Ferris wheel at its lowest height. The ride starts at 0 seconds.
Work or explanation must be provided for full marks. Answers involving radians will result in a mark of zero.
a) Sketch a height vs time graph of ONE cycle of Bert’s motion on the ride. Show at least 5 KEY POINTS and label them on your graph.
b) Determine an equation of the sinusoidal function h(t) to represent Bert’s journey, where h represents the height of the Ferris Wheel and t represents the time in seconds.
Choose the MOST appropriate sinusoidal function based on the information provided.

a) Please find attached the height vs time graph of ONE cycle of Bert's motion on the ride, showing 5 Key points, including; The period, amplitude, the vertical shift, horizontal shift, the neutral axis

b) The equation to represent Bert's journey is h(t) = 6·sin(3.6·(t - 25)) + 7

The reason the above equation are correct are:

The given parameters of the Ferris wheel are;

Radius, r = 6 meters

The height of the axle above the ground, h = 7 metres

The time it takes to complete one revolution, T = 100 seconds

The level at which Bert enters the Ferris wheel = The lowest level

The time at which the ride starts = 0 seconds

a) The graph of a Ferris wheel is a graph sinusoidal function with the following details

The general form of the sinusoidal function is, y = a·sin(b·(t - h)) + D

Amplitude = The radius = 6 meters

The vertical shift, D = The elevation of the axle above the ground = 7 meters

The period, T = 360/b

∴ b = 360/T = 360/100 = 3.6

At t = 0, sin(b·(t - h)) = -1

Given that sin(-90) = -1

Therefore; (3.6·(0 - h)) = -90

π/50 = -π/2

-h = -25

∴ The horizontal shift, h = 25

The function of the Ferris wheel is y = 6·sin(3.6·(t - 25)) + 7

The graph of the function is created on MS Excel, using the above sinusoidal equation of the Ferris wheel

The 5 key points included in the graph are;

The period, T

Horizontal shift, h

Amplitude, a

Neutral axis

Vertical shift, D

b) The appropriate equation of the sinusoidal function of the Ferris wheel is determined from the general sinusoidal function equation, y = a·sin(b·(t - h)) + D, as follows;

From part (a);

a = 6, b = 3.6, h = 25, and D = 7

The equation of the sinusoidal function h(t) to represent Bert's journey, is h(t) = a·sin(b·(t - h)) + D

Where;

h = The height of the Ferris Wheel

t = The time in seconds

a = The amplitude = 7

b = The 360/(The period) = 3.6

h = The horizontal shift = 25°

D = The vertical shift = 7 meters

The MOST appropriate equation of the sinusoidal function h(t) is therefore;

h(t) = 6·sin(3.6·(t - 25)) + 7

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