Question

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.

Answer 1

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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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