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?

maths nerd…

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.