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Exam-Style Question on Circular functions

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Question id: 461. This question is similar to one that appeared on an A-Level paper (specimen) for 2017. The use of a calculator is allowed.

The height above the ground, H metres, of a passenger on a Ferris wheel t minutes after the wheel starts turning, is modelled by the following equation:

$$H = k – 8\cos (60t)° + 5\sin (60t)°$$

where k is a constant.

(a) Express \(H\) in the form \(H = k - R \cos(60t + a)° \) where \(R\) and \(a\) are constants to be found (\( 0° \lt a \lt 90° \)).

(b) Given that the initial height of the passenger above the ground is 2 metres, find a complete equation for the model.

(c) Hence find the maximum height of the passenger above the ground.

(d) Find the time taken for the passenger to reach the maximum height on the fifth cycle. (Solutions based entirely on graphical or numerical methods are not acceptable.)

(e) It is decided that, to increase profits, the speed of the wheel is to be increased. How would you adapt the equation of the model to reflect this increase in speed?

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