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Exam-Style Question on Periodic Functions

A mathematics exam-style question with a worked solution that can be revealed gradually

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Question id: 680. This question is similar to one that appeared on an IB AA Higher paper in 2023. The use of a calculator is allowed.

A carriage attached to a tall vertical pole in an amusement park ride whisks customers up and down. The height, \(H\) metres, of the base of the carriage above the ground can be modelled by the function \(H(t) = a\cos(0.4t) + b\), for \(a, b \in \mathbb{R}\) and \(t\) is the time in seconds after the ride starts.

(a) Find the period of the function.

Amusement Park

The ride begins when its base is at a minimum height of 1 metre above the ground, and it reaches a maximum height of 31 metres above the ground.

(b) Find the values of a and b.

(c) Find the number of times that the carriage reaches its maximum height in the first minute of its motion.

(d) Find the first time that the base of the carriage reaches a height of 15 metres.

A camera is set to take a picture of the ride at a random time during the first 15 seconds of its motion.

(e) Find the probability that the height of the base of the carriage is greater than 10 metres at the time the picture is taken.

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