Expert: Ahmed Salami - 2/23/2006

Question
Hello, I tried to answer this question by drawing it all out, but I am not able to fully answer the question.

The ferris wheel has a diameter of 100 ft. When it runs at full speed, the wheel completes one rotation in 30 seconds. The bottom of the wheel is 4 feet above the ground.

a.) Sketch a graph that represents a rider's height above the ground with respect to time (t). Begin when the rider is at the lowest point on the wheel.

b.) Find a trigonometric equation that models the situation. (I know that it deals with sin, cos, tan.)

c.) Determine the rider's height above the ground when t=42 seconds.


I hope you can help me with this question. Thank You in advance!


Answer
Hi AFB,
Sorry for the time it took.
I'm unable to give out a diagram depicting the situation
here, but i'm sure you'll still understand the solution
especially when you try a sketch on your own. Draw a
circle, and for different angle values starting from the
lowest point and moving anticlockwise(or in the assumed
manner in which the wheel is moving), measure the
corresponding height from the ground. Note that the
wheel is 4ft above the ground. The values you have can
be used to draw a graph.
For a theoretical solution,
Let the radius of the wheel be r, and its angular
velocity be w.
w = 'theta'/t
'theta' = wt
where 'theta' is the angle rotated through as at time t.
The wheel makes a complete revolution in 30 seconds, its
angular velocity is therefore
w = 2#/30 seconds   (# = pi)
 = (#/15)rad/s
or in degrees
w = 360 degrees/30 seconds
 = 12 deg/s
Measuring from the lowest point on the wheel, the height
h of the rider would be (r - rcos'theta')
h = r(1 - cos'theta')
Considering that the wheel is 4ft above the ground, the
height of the rider at any time t;
H = h + 4
 = r(1 - cos'theta') + 4
 = r[1 - cos(wt)] + 4
 = (r + 4) - rcos(wt)
 = (r + 4) - rcos(12t)
The graph of H against t would be the graph of rcos(wt)
inverted and then moved up (r+4) units.
At t = 42 seconds
H = (r + 4) - rcos(12x42)
 = (r + 4) - rcos(504)
 = (r + 4) - rcos(144)
 = (r + 4) - (-0.8090)r
 = r + 4 + (0.8090)r
 = (1.8090)r + 4
But d = 100ft, r = 50ft
H = 94.45 ft
(above the ground)

I hope i have helped. You can always get back to me.
Regards.