The Colossus Ferris wheel debuted at the 1984 New Orleans World’s Fair. The ride is 180 ft tall, and passengers board the ride at an initial height of 15 ft above the ground. The height above ground, h, of a passenger on the ride is a periodic function of time, t. The graph displays the height above ground of the last passenger to board over the course of the 15 min ride.
1. What is the period of the sine function model? Interpret the period you found in the context of the operation of the Ferris wheel.
2. The duration of the ride is 15 min.
(a) How many times does the last passenger who boarded the ride make a complete loop on the Ferris wheel?
(b) What is the position of that passenger when the ride ends?
3. A power outage occurs 6 min after the ride started. Passengers must wait for their cage to be manually ranked into the lowest position in order to exit the ride. Sine function model: h= -82.5 cos 3pi(t)+97.5 where h is the height of the last passenger above the ground measured in feet and t is the time of operation of the ride in minutes.
(a) What is the height of the last passenger at the moment of the power outage? Verify your answer by evaluating the sine function model.
(b) Will the last passenger to board the ride need to wait in order to exit the ride? Explain

For the function [tex]f(x)=a*cos(bx+c)+d[/tex], the period is [tex]\frac{2\pi}{|b|}[/tex]. In the context of this problem, the period is [tex]\frac{2\pi}{|3\pi|}=\frac{2}{3}[/tex]. This means that after every [tex]\frac{2}{3}[/tex] minutes, the height of the passenger will be the same, which shows they have completed one loop of the Ferris wheel.

Problem 2A

Since the ride took 15 minutes and 1 loop takes [tex]\frac{2}{3}[/tex] minutes, then the number of loops made during the ride is [tex]15\div\frac{2}{3}=\frac{45}{2}=22\frac{1}{2}[/tex] loops.

Problem 2B

Plug [tex]t=15[/tex] into the function to determine the height of the passenger after 15 minutes:

[tex]h= -82.5*cos(3\pi t)+97.5[/tex]

[tex]h= -82.5*cos(3\pi(15))+97.5[/tex]

[tex]h= -82.5*cos(45\pi)+97.5[/tex]

[tex]h= -82.5*(-1)+97.5[/tex]

[tex]h=82.5+97.5[/tex]

[tex]h=180[/tex]

Therefore, the height of the passenger after 15 minutes is 180 feet above the ground.

Problem 3A

Plug [tex]t=6[/tex] into the function to determine the height of the last passenger after 6 minutes when the power outage occurs:

[tex]h= -82.5*cos(3\pi t)+97.5[/tex]

[tex]h= -82.5*cos(3\pi(6))+97.5[/tex]

[tex]h= -82.5*cos(18\pi)+97.5[/tex]

[tex]h= -82.5*(1)+97.5[/tex]

[tex]h=-82.5+97.5[/tex]

[tex]h=15[/tex]

Therefore, the last passenger is 15 feet above the ground after 6 minutes when the power outage occurs.

Problem 3B

The last passenger to board the ride won't need to wait in order to exit the ride because they are at the lowest point of the ride which is 15 feet above the ground. Therefore, they can get off immediately then.