April 18, 2016

Trig question involving a ferris wheel?

A large ferris wheel has a diameter of 135 m and passengers get on at the bottom 4m above the ground once every three minutes.

a) How do I graph to represent the height of a passenger in metres as a function of time?
b) Determine the equation that expresses your height as a function of time?
c) how high is a passenger 5 minutes after the wheel starts rotating?
d) How many seconds after the wheel starts rotating is a passenger 85m above the ground for the first time?

Comments

oweemytoe

maths nerd…

netsfan4life

a.
When t is zero, we have an initial height of 4m, so h = 4.
When t is a non-integer multiple of 1.5 we have a height of 139m, so h = 139.
When t is a multiple of 3 we have a height of 4m, so h = 4.
You should be able to graph this and you will get a sine wave.

b.
We know from the graph that the function is a sine wave. We assume that it will take the form of h = A + Bsin[C(t – D)], and we then find the value of each constant.

A is the vertical shift, the average of the maximum and minimum values:
a = (139 + 4) / 2
a = 143 / 2

B is the amplitude, half of the difference between the maximum and minimum values.
B = (139 – 4) / 2
B = 135 / 2

C is the frequency, and we need to use this to obtain the period. Since after 3 minutes the graph is at the same point, we want the period to be 3 minutes. The period is given by 2? / C, and we can now solve for C:
2? / C = 3
C = 2? / 3

D is the phase shift, we want the minimum to be at t = 0 but it is at t = -0.75, so we want to shift the graph 0.75 to the right, so D = -0.75.

Then the height as a function of time is h(t) = 143 / 2 + 135sin[2?(t – 0.75) / 3] / 2.

c.
Plug in this value for time into the equation for height:
h(5) = 143 / 2 + 135sin[2?(5 – 0.75) / 3] / 2
h(5) = 105.25
After 5 minutes, a passenger will be at a height of 105.25m.

d.
Set the equation for height equal to the value and solve:
h(t) = 85
143 / 2 + 135sin[2?(t – 0.75) / 3] / 2 = 85
135sin[2?(t – 0.75) / 3] / 2 = 27 / 2
135sin[2?(t – 0.75) / 3] = 27
sin[2?(t – 0.75) / 3] = 27 / 135
2?(t – 0.75) / 3 = sin?¹(27 / 135)
2?(t – 0.75) = 3sin?¹(27 / 135)
t – 0.75 = 3sin?¹(27 / 135) / 2?
t = 0.75 + 3sin?¹(27 / 135) / 2?
t = 0.85 mins
A passenger will first reach a height of 85m after 0.85 minutes, or 50.77 seconds.

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