theme park math question

Question

A popular ride at a theme park consists of a large wheel that starts in a horizontal position and

as the ride picks up its rotational speed the wheel tilts up to a moderate acute angle. A Sine

Function may model the vertical position of a rider, H, in metres, above the ground after time,

t, in seconds. The maximum height above the ground is 7.5 metres and the minimum height is

150 cm. The wheel completes one turn in 5 s. This model predicts the rider’s highest point at

t =0 seconds. Determine a sine function model for, H, in metres, in terms of time, t, in seconds.

The function may be modelled as: H = a sin [k (t – d)] + c, where the exact values of a, k, d,

and c can be calculated.

Answer 1

a is the amplitude: 1/2 (highest -lowest height)

c is the height at the midpoint 1/2(highest + lowest height)

k is 2pi/period so that when change in t = period the sine function will repeat its value

The ride does not start with the lowest point (sin(kt) = 0 and H=c) so we need to displace the function in time (d is nonzero). It's starting at the highpoint which is 1/4 period to the right: d= -T/4

I believe that's everything. There are other ways to solve this.