Question

In: Advanced Math

The largest Ferris Wheel in the world is the London Eye in England. The height (in...

The largest Ferris Wheel in the world is the London Eye in England. The height (in meters) 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. (15)

  1. How long does it take for the Ferris wheel to go through one rotation?
  2. What is the minimum value of this function? Explain the significance of this value.

Solutions

Expert Solution

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Answer:

Explanation:

A) period=2*180/12

=30 minutes

B)

You need to evaluate the minimum value of the given function [h(t), ] hence, you need to solve for t the equation [h'(t) = 0] .

You need to differentiate the given function with respect to t, using the chain rule, such that:

[h'(t) = (65 sin(12(t- 7.5)) + 70)']

[h'(t) = 65*cos(12(t - 7.5))*(12(t - 7.5))' + 0]

[h'(t) = 65*12*cos(12(t - 7.5))]

You need to solve for t the equation [h'(t) = 0] , such that:

[65*12*cos(12(t - 7.5)) = 0]

Dividing by [65*12] yields:

[cos(12(t - 7.5)) = 0 => 12(t - 7.5) = cos^(-1)(0) + 2n*pi]

Substituting [pi/2] for [cos^(-1)(0)] yields:

[12(t - 7.5) = pi/2 + 2n*pi => t - 7.5 = pi/24 + n*pi/6]

[t = pi/24 + n*pi/6 + 7.5]

Substituting [(3pi)/2] for [cos^(-1)(0)] yields:

[12(t - 7.5) = (3pi)/2 + 2n*pi => t - 7.5 = pi/8 + n*pi/6]

[t = pi/8 + n*pi/6 + 7.5]

Hence, evaluating the points where the function [h(t) ] reaches its minimum point yields that [t = pi/8 + n*pi/6 + 7.5.]

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