Question

A ferris wheel has a radius of 13 m. It rotates once every 24 seconds. A passenger gets on at the bottom
of the wheel from a ramp which is one metre above ground level.

a) If the height of the passenger is measured from the ground, determine an equation for the height of the
passenger as a function of time in the form of h(t)= acos(bt)+d
b) To the nearest metre, find the height of the passenger after 55 seconds.

Answer 1

2π/k = period
2π/k = 24
k = 2π/24 = π/12 <---- that's the b of your equation.
also we know a = 13

so we start with h(t) = 13 cos (π/12 t)
the minimum of this graph is -13, we want the min to be +1, so we have to raise it 14 units

h(t) = 13 cos (π/12 t) + 14

I would have started with a sine function, rather than a cosine function, since the sine starts at 0 when t = 0 and would be increasing.
The cosine curve would start at 1 when t = 0
So we have to move our cosine function horizontally to achieve this.
Let's see what we have so far

http://www.wolframalpha.com/input/?i=h(t)+%3D+13+cos+(%CF%80%2F12+t)+%2B+14

if we translate our curve 12 units to the right, we get:
http://www.wolframalpha.com/input/?i=h(t)+%3D+13+cos+((%CF%80%2F12)(t+-+12)))+%2B+14

so h(t) = 13cos ( (π/12)(t-12) ) + 14

checking:
when t = 0 , h(t) = 13 cos (-π) + 14 = 1 --> bottom
when t = 6 , h(6) = 13cos(-π/2)+14 = 14 --> half-way up
when t = 12, h(12)=13cos(0)+14 = 27 --> the top
when t= 18, h(18) =13cos(π/2)+14 = 14
when t=24, h(24) = 1

equation is good,

b) when t = 55
wheel has gone 55/24 = 2 + 7/24 periods
so it would be in the same position as it would be at 7 seconds

h(7) = 13cos (-5π/12) + 14
= appr 17.4 m high