Since the radius of a circle is half its diameter, the radius of our Ferris wheel is
ft
Next, we are going to convert from revolutions per minute to degrees per second.
We know that t<span>he 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:
</span>
degrees per second
<span>
Now, lets find the height:
</span>We know that <span>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:
</span>
Notice that the height at the top is the diameter of the wheel plus the height above the ground, so
.
To model the situation we are going to use the cosine function, but notice that
is 1 when
and -1 wen
. Since we want the opposite, we are going to use negative cosine.
Notice that we want
when
, so we are going to use
. Next, we are going to multiply our cosine by the radius of our wheel:
, and last but not least we are going to add the sum of the radius of the wheel plus the height above the ground:
Now that we have our height function lets check if everything is working:
<span>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:
</span>
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:
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:
