Write an equation for a ferris wheel problem

Question

Hi, I've been struggling with this question for weeks despite having adequate knowledge of the topic. If anyone could solve this for me and provide an explanation it would be greatly appreciated.

A ferris wheel has a diameter of 180m and the center of the wheel is 115m above ground. The wheel rotates continuously in a clock-wise direction at 1/26 rpm. Each visitor will get to enjoy 3 continuous rounds of the wheel.

At t = 0 (where t = time in minutes), Brandon gets into carriage A. The position of Brandon above the ground at the time t (in minutes) is given as y(A) (in meters).

You may use any of the following equations:

y(A) = a sin(bt + c) + d,

or y(A) = a cos(bt + c) + d

(a) Choose an equation that best describes the problem

(b) Find the amplitude, period, phase shift and vertical shift.

(c) Find the equation (a,b,c and d must have values)

(d) Determine the position of Brandon above the ground at t = 6 min

Answer 1

Answer 2

y(t)=a(sinbt+cj+d

because sine gives height, while cosine gives a horizontal distance, Y(t) represents height at time t

a=amplitude= half the diameter = 180/2 = 90 m = distance from midline to either max or min height

midline = 90+115=205

2bpi = period = 13/pi = 180/13 degrees = 13.85 degrees

1/26 rotations per minute = frequency = inverse of period =1/1/26=26 means 1 rotation every 26 minutes

sine curve repeats every 26 minutes, from t=0 to t=26. This is a very slow ferris wheel

the sine curve has a period of 2pi or 180 degrees if b=1

sinb(6.5)=1

6.5b = 90 degrees

b=90/6.5 = 18/1.3 = 13.85

c=phase shift = 16.128 degrees = 0.2815 radians = 0.0896 pi radians

Y(0)=115-90=25, since at t=0 ferris wheel is at minimum height.

90sin(0+c) =25 = 90sin(c)

sin(c)=25/90=5/18, c=16.128 degrees=0.2815 radians

d=vertical shift = 115m

y(t)=90(sin180t/13+16.128)+115

at t=6

Y(6)=90(sin6(180)/13+16.128+115

=90(sin83.077+16.128)+115

=90(sin99.205)+115

=0.9871(90)+115

=88.839 + 115 = 203.839 meters

at t=6.5 the ferris wheel is at its average height of 115+90=205 meters

at t=13 its at its maximum height = 115+180= 295

at t=0 it's at a minimum height of 115

a t=6, it should be slightly lower than its average height of 205

=

Y(26/4=6.5)=midline of the sine curve = 115+90= 205 meters= the height of the center of the ferris wheel

Y(6) should be a little less, <205