Answer: Period = 2 seconds
Graph is shown below
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Explanation:
1 minute = 60 seconds
We have 30 revolutions every 60 seconds, so we can write the ratio
30 rev: 60 sec
Divide both parts by 30 to turn that "30 rev" into "1 rev"
30 rev: 60 sec
30/30 rev: 60/30 sec
1 rev : 2 sec
This tells us that each full revolution takes 2 seconds. This is the period of the ferris wheel. The period is the length of each cycle.
The cyclic or periodic nature of this motion heavily implies we'll be dealing with a sine or cosine function of some kind. Let's go with sine.
Side note: cosine is really a sine function in disguise (it's just a phase shifted version of it).
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The general sine curve equation is
y = A*sin(B(x-C)) + D
where
- |A| = amplitude
- B = 2pi/T with T being the period
- C handles phase shifts (left and right shifting)
- D handles vertical shifting, and y = D is the equation of the midline
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The wheel is 10 feet in diameter, so 10/2 = 5 is the radius. This is also the amplitude because it is the distance from the midline to either the highest point (10 ft) or lowest point (0 ft). So A = 5.
Since T = 2 is the period, this means,
B = 2pi/T
B = 2pi/2
B = pi
We don't have any phase shifts to worry about so C = 0.
The midline is D = 5 since this is halfway up
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Putting all the pieces together, we get this sine equation
y = A*sin(B(x-C)) + D
y = 5*sin(pi*(x-0)) + 5
y = 5*sin(pi*x) + 5
The graph is shown below.
x = time in seconds, y = height of the seat
