A Ferris wheel with a radius of 20 metres rotates at a constant rate. The bottom of the wheel is 2 metres above the ground. A seat starts at the bottom of the wheel. The height, \(h\) metres, of the seat above the ground after \(t\) minutes is modelled by the function \(h(t) = a \cos(bt) + c\). The wheel completes one full revolution every 4 minutes.
(a) Find the value of \(b\). [2]
(b) Find the values of \(a\) and \(c\). [3]
(c) Find the height of the seat after 1.5 minutes. [1]
Question
Find the value of \(b\). [2]
Hint
Hint
Hint
Recall the period of a cosine function, \(T = 2\pi/|b|\), and set T = 4 minutes.
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\(b = \pi/2\)
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Question
Find the values of \(a\) and \(c\). [3]
Hint
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Hint
Compute the maximum height as 2×20 + 2 = 42 m and the minimum height as 2 m. Then find a = (max – min)/2 and c = (max + min)/2, adjusting the sign of a based on initial position.
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\(a = -20, \; c = 22\)
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Question
Find the height of the seat after 1.5 minutes. [1]
Hint
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Hint
Substitute t = 1.5 into the model \(h(t) = -20 \cos(\pi t/2) + 22\).
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\(h(1.5) = 10\sqrt{2} + 22 \approx 36.1 \text{ m}\)
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