# Math Grade 11, Applications, Please answer fully. thank you.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...

Math Grade 11, Applications, Please answer fully. thank you.

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.

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### 1 Answer

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: