- On a roller coaster ride, passengers travel in carriages around a track.
On the ride, carriages complete multiple circuits of the track such that
- the maximum vertical height of a carriage above the ground is 60 m
- a carriage starts a circuit at a vertical height of 2 m above the ground
- the ground is horizontal
The vertical height, \(H \mathrm {~m}\), of a carriage above the ground, \(t\) seconds after the carriage starts the first circuit, is modelled by the equation
$$H = a - b ( t - 20 ) ^ { 2 }$$
where \(a\) and \(b\) are positive constants.
- Find a complete equation for the model.
- Use the model to determine the height of the carriage above the ground when \(t = 40\)
In an alternative model, the vertical height, \(H \mathrm {~m}\), of a carriage above the ground, \(t\) seconds after the carriage starts the first circuit, is given by
$$H = 29 \cos ( 9 t + \alpha ) ^ { \circ } + \beta \quad 0 \leqslant \alpha < 360 ^ { \circ }$$
where \(\alpha\) and \(\beta\) are constants.
- Find a complete equation for the alternative model.
Given that the carriage moves continuously for 2 minutes,
- give a reason why the alternative model would be more appropriate.