Question

WILL MAKE BRAINLIEST!!! A Ferris wheel has a diameter of 220 feet and the center of the wheel is 125 feet above the ground. The wheel makes a complete turn every 2 minutes. Write a cosine function to express the height h (in feet) of a passenger on the wheel as a function of time t (in minutes) ), given that the passenger is at the lowest point on the wheel when t=0

Answer 1

Since the radius of a circle is half its diameter, the radius of our Ferris wheel is $r= \frac{220}{2} =110$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:
$\frac{1*360}{120}=3$ 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:
$t= \frac{180}{3}$
$t=60$
Notice that the height at the top is the diameter of the wheel plus the height above the ground, so $h=220+125=345$.

To model the situation we are going to use the cosine function, but notice that $cos (\alpha)$ is 1 when $\alpha =0$ and -1 wen $\alpha =180$. Since we want the opposite, we are going to use negative cosine.
Notice that we want $\alpha =180$ when $t=60$, so we are going to use $-cos(3t)$. Next, we are going to multiply our cosine by the radius of our wheel: $-110cos(3t)$, and last but not least we are going to add the sum of the radius of the wheel plus the height above the ground:
$h=110+125-110cos(3t)$
$h=225-110cos(3t)$

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:
$h=225-110cos(3t)$
$h=225-110cos(3*0)$
$h=125$ 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:
$h=225-110cos(3t)$
$h=225-110cos(3*60)$
$h=225-110cos(180)$
$h=345$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: $h=225-110cos(3t)$