Since the radius of a circle is half its diameter, the radius of our Ferris wheel is [tex]r= \frac{220}{2} =110[/tex]ft
Next, we are going to convert from revolutions per minute to degrees per second.
We know that the wheel makes a complete turn every 2 minutes, so it makes a complete turn in 120 seconds. Since there are 360° in a complete turn, we can set up our conversion factor:
[tex] \frac{1*360}{120}=3 [/tex] degrees per second
Now, lets find the height:
We know that the passenger is at the lowest point on the wheel when t=0; since the wheel is 125 feet above the ground, at t=0 h=125. To find t at the top, we are going to take advantage of the fact that the wheel will turn 180° from the lowest point to the top and that it turns 3° every second:
[tex]t= \frac{180}{3} [/tex]
[tex]t=60[/tex]
Notice that the height at the top is the diameter of the wheel plus the height above the ground, so [tex]h=220+125=345[/tex].
To model the situation we are going to use the cosine function, but notice that [tex]cos (\alpha) [/tex] is 1 when [tex] \alpha =0[/tex] and -1 wen [tex] \alpha =180[/tex]. Since we want the opposite, we are going to use negative cosine.
Notice that we want [tex] \alpha =180[/tex] when [tex]t=60[/tex], so we are going to use [tex]-cos(3t)[/tex]. Next, we are going to multiply our cosine by the radius of our wheel: [tex]-110cos(3t)[/tex], and last but not least we are going to add the sum of the radius of the wheel plus the height above the ground:
[tex]h=110+125-110cos(3t)[/tex]
[tex]h=225-110cos(3t)[/tex]
Now that we have our height function lets check if everything is working:
the passenger is at the lowest point at t=0; we also know that the lowest point is 125 feet above the ground, so lets evaluate our function at t=0:
[tex]h=225-110cos(3t)[/tex]
[tex]h=225-110cos(3*0)[/tex]
[tex]h=125[/tex] feet
So far so good.
We also know that at t=60, our passenger is 345 feet above the ground, so lets evaluate our function at t=60 and check if coincides:
[tex]h=225-110cos(3t)[/tex]
[tex]h=225-110cos(3*60)[/tex]
[tex]h=225-110cos(180)[/tex]
[tex]h=345[/tex]feet
We can conclude that cosine function that express the height h (in feet) of a passenger on the wheel as a function of time t (in minutes) ) is: [tex]h=225-110cos(3t)[/tex]
