# Math

The largest Ferris Wheel in the world is the London Eye in England. The height (in metres) of a rider on the London Eye after t minutes can be described by the function h(t) = 65 sin[12(t − 7.5)] + 70.

• What is the diameter of this Ferris wheel?
• Where is the rider at t = 0? Explain the significance of this value.
• How high off the ground is the rider at the top of the wheel?
• At what time(s) will the rider be at a height of 100 m?
• How long does it take for the Ferris wheel to go through one rotation?
• What is the minimum value of this function? Explain the significance of this value. We are given `h(t)=65sin(12(t-7.5))+70` where t is time in minutes and h(t) is the height in meters at time t.

The general form for a sinusoid is `y=asin(b(x-h))+k` where:

a: a is the amplitude (if a<0 the graph of teh base function is reflected over the horizontal axis)

b: the period is found by `p=360/b` ;

h: h is a horizontal translation called the phase shift

k: k is a vertical translation. The line y=k is the midline

So for this function the amplitude is 65, the midline is y=70, the period is `p=360/12=30` , and there is a phase shift of 7.5 units right.

(1) The diameter of the Ferris wheel is 130m. (The amplitude is the maximum distance from the midline; the highest point will be 70+65=135m and the lowest point will be 70-65=5m. 135-5=130m or twice the amplitude.)

(2) h(0)=5. The rider begins the ride 5m above the ground; this is the lowest point on the ride.

(3) As stated above, the highest point is 70+65=135m above the ground.'

(4) Solve h(t)=100:

100=65sin(12(t-7.5))+70

30=65sin(12(t-7.5))

sin(12(t-7.5))=6/13

`12(t-7.5)=sin^(-1)(6/13)~~27.486`

`t-7.5~~2.291`

`t~~9.791`

` ` The sin will also be approximately 6/13 at 152.514 so we also get `t~~20.210`

Each of these times will repeat every 30 minutes. (The period is 30 minutes)

The rider will be at 100m at approximately 9.8 minutes and 20.2 minutes after the beginning of the ride.

(5) The period is 30 minutes.

(6) The minimum value is 5, as stated above. This is the lowest point for the rider; the beginning and presumably the ending of each ride.

The graph: