Answer to Question #286921 in Calculus for Jvb

Question

5 A Ferris Wheel in Las Vegas, Nevada, opened in March 2014. The 550

ft tall wheel has a diameter of 5290 ft A ride on its one of its 28 passenger cars last 30 minutes the time it takes the wheel to complete one full rotation Riders board the passenger cars at the bottom of the wheel Assume that once the wheel is in motion it maintains a constant speed for the 30-minutes ride and is rotating in a counter clockwise direction. If you were on this ride how high would you be above the ground after 20 minutes?

Answer 1

Maximum height of the wheel from the ground =550 ft

Diameter of the wheel =520 ft

Minimum height of the wheel from the ground = Maximum height - Diameter = 550-520= 30 ft

So, the centre Point, $d=\frac{\text { Max }+\text { Min }}{2}=\frac{550+30}{2}=290$

With these 3 points we can draw the graph of the height of a passenger car on the Wheel

Let H be the height and T be the time

So When T=0, H=30 ft

 $\begin{aligned} &{T}=15, {H}=550 \\ &{~T}=7.5, {H}=290 \\ &{~T}=22.5, {H}=290 \end{aligned}$

To find a sinusoidal function we need to find the distance travelled in a minute Let B be the distance

$\frac{2 \pi}{B}=30 \Rightarrow 30 B=2 \pi \Rightarrow B=\frac{2 \pi}{30}=\frac{\pi}{15}$

So the height H at any time can be formulated as

$H=-260 \operatorname{Cos}\left(\frac{\pi}{15} t\right)+290$ Negative sign is used because the wheel is traveling in anti-clockwise direction.

t is the time in minutes. 

Height after 20 minutes

Here t = 20

Substitute in the above formulated equation and solve

$\begin{aligned} &H=-260 \operatorname{Cos}\left(\frac{\pi}{15} * 20\right)+290 \\ &H=-260 \operatorname{Cos}\left(\frac{4 \pi}{3}\right)+290 \\ &H=-260 * \frac{-1}{2}+290=130+290=420\ \mathrm{ft} \end{aligned}$

So height after 20 minute = 420 ft