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A Ferris wheel is 20 meters in diameter and completes 1 full revolution in 8 minutes.



A round Ferris wheel

A Ferris wheel is 20 meters in diameter and boarded from a platform that is 1 meter above the ground. The six o’clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 8 minutes. The function h(t) gives a person’s height in meters above the ground t minutes after the wheel begins to turn.



a. Find the amplitude, midline, and period of h(t).



Enter the exact answers.



Amplitude: A= meters

Midline: h= meters

Period: P= minutes



b. Assume that a person has just boarded the Ferris wheel from the platform and that the Ferris wheel starts spinning at time t=0. Find a formula for the height function h(t).



Hints:

What is the value of h(0)?
Is this the maximum value of h(t), the minimum value of h(t), or a value between the two?
The function sin(t) has a value between its maximum and minimum at t=0 , so can h(t) be a straight sine function?
The function cos(t) has its maximum at t=0, so can h(t) be a straight cosine function?


c. If the Ferris wheel continues to turn, how high off the ground is a person after 30 minutes?

Sagot :

Given that the Ferris diameter is 20 meters, with a rate of rotation of 1 turn in 8 minutes, we have;

a. Amplitude: A = 10 meters

Midline: h = 11 meters

Period, P = 8 minutes

b. h(t) = 10•cos((π/4)•t + π) + 11

c. 11 meters

How can the Ferris wheel be evaluated?

The amplitude is the same as the radius of the Ferris wheel,

The radius of the Ferris wheel = 20 ÷ 2 = 10

Therefore;

  • Amplitude: A = 10 meters

[tex]the \: midline \: = \frac{max \: height \: + min \: height}{2} [/tex]

Therefore;

[tex]midline \: = \frac{10 + 10 + 1 + 1}{2} = 11[/tex]

  • Midline: h = 11 meters

The period is the time to complete one rotation, therefore;

  • Period, P = 8 minutes

b. h(t) = A•cos(B•t + C) + h

Where;

B = 2•π/P

When t = 0, h(t) = 1

Which gives;

h(0) = 1 = 10 × cos(B×0 + C) + 11

-10/10 = -1 = cos(C)

C = arcos(-1) = π

Therefore;

  • h(t) = 10•cos((π/4)•t + π) + 11

h(0) = 1

h(0) is the minimum value of h(t)

  • h(t) cannot be a straight sine function because of the vertical shift

  • h(t) cannot be a straight cosine function because at t = 0, is the minimum point

c. After 30 minutes, we have;

h(30) = 10•cos((π/4)×30 + π) + 11 = 11

  • The height of a person after 30 minutes is 11 meters

Learn more about the Ferris wheel here:

https://brainly.com/question/86214

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