Answer:
see attached for a graph
- period: 60 s
- amplitude: 20 m
- phase shift: 15 s
- vertical translation: 25 m
- equation: y = 25 +20sin(2π(x -15)/60)
- ending height: 45 m
Explanation:
You want a sine function equation and graph for the rider's height on a 40 m Ferris wheel centered 25 m above the ground. It makes 1 revolution per minute and a ride lasts 3.5 minutes.
a) Graph
A graph of the function is attached.
Since the average height if 5 m more than the radius, the low point at which the rider starts is 5 m above the ground.
The function repeats after one period of 60 seconds (1 revolution).
The peak is the radius above the midpoint: 25 m + 20 m = 45 m.
These values are sufficient for sketching a graph.
b) Parameters
The period is given as 60 seconds. The amplitude is the radius of the wheel, (40 m)/2 = 20 m. The function starts at the low point of the sine curve, which is 1/4 period before the first rising zero crossing. That is, the phase shift is 1/4 period, or 15 seconds.
The vertical translation is the height of the center of the wheel: 25 m.
c) Equation
The equation for the height is ...
h(x) = (vertical translation) + (amplitude)·sin(2π/(period)·(x -phase shift))
Using the above parameters, this is ...
h(x) = 25 +20·sin(2π/60·(x -15))
d) End height
Since the period is 1 minute, a 3.5 minute ride ends after a half-period, when the rider is at the top of the wheel.
At the end of the ride, the height of the rider is 45 m.
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Additional comment
When the start is at an extreme, we like to use the cosine function, which is at its extreme for x=0. That is the red curve in the attachment. The blue dots show the phase-shifted sine function gives the same curve.
3.5 minutes is 3.5·60 = 210 seconds.
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