Question

Suppose a Ferris wheel, which completes one rotation in 20 minutes, has a diameter of 60 feet has a center 70 feet off the ground. Suppose the cart of the Ferris wheel begins at the bottom of Ferris wheel. Express the carts height as a function of time, in minutes.

Answer 1

In a diagram,

Considering the purple point in the diagram, we will model its height with respect to the ground as a function of time.

Let t_1 be between zero and 20 minutes; therefore, the position of the cart at that instant is

Therefore,

$y=2r\sin \theta$

where r is the radius of the circle (r=30m)

Then, we know that the Ferris wheel completes a rotation after 20 minutes, this is, 2*pi radians in 20 minutes; thus,

$\begin{gathered} \Rightarrow y=\frac{\sin (\frac{t}{20}\cdot\pi)}{r}=\frac{\sin (\frac{t}{20}\cdot\pi)}{30} \\ \Rightarrow y=\frac{\sin (\frac{t\pi}{20})}{30} \end{gathered}$

Notice that when t=0, y=0; this is the initial position of the cart, at which it is 40 ft above the ground. On the other hand, after 10 minutes, the sine function reaches its maximum value.

Then, the answer is

$undefined$