Create both a sine and cosine model for height of a passenger off of the ground for each of the following Ferris wheels.
1) customers must climb up 12 foot steps to get into the Ferris wheel (I.e) bottom of Ferris wheel is at the top of the steps
Diameter 67ft
Rotional speed 1 revolution every 80 seconds
2) customers gets into a Ferris wheel 17 feet below ground level (bottom of Ferris wheel is at that level)
Diameter 122ft
Rotional speed 1 revolution 150 seconds
My issue is that I don't know how to deal with the below and walking up steps and how to account for that when creating the functions. I just need the parameters for example
Amplitude =A
Vertical shift =K
Phase shift =h
Period =b
h(t)=acos(b(t-h))+k
h(t)=asin(b(t-h))+k
well, where is the center of the wheel above ground?
12 + 67/2 = 45.5 = center above ground
It does not say where the wheel starts so I will say it is at height = bottom
now in general
h = 45.5 + 33.5 sin( 2 pi t/T -p)
where T is period in seconds and p is phase in radians
Now I want it at the bottom or 12 ft when t = 0
that means sin (argument) = -1
or ( 2 pi t/T - p) = -pi/2
when t = 0
so
p = pi/2
so
h = 45.5 + 33.5 sin( 2 pi t/T - pi/2)
for the cos function, same deal but
h = 45.5 + 33.5 cos( 2 pi t/T -p)
now I want cos arg = -1 at t = 0
cos = -1 when p = pi or -pi radians
so
h = 45.5 + 33.5 cos( 2 pi t/T -pi)
To create the sine and cosine models for the height of a passenger off the ground on each Ferris wheel, we need to consider the starting position, amplitude, vertical shift, phase shift, and period.
First, let's define the parameters:
Amplitude (A): The maximum height above or below the starting position.
Vertical shift (K): The average or midline height.
Phase shift (h): The horizontal displacement or time delay.
Period (b): The time it takes for one complete cycle.
For the given Ferris wheels:
1) Ferris wheel with customers climbing 12-foot steps to get in:
- Starting position: The bottom of the Ferris wheel is at the top of the steps, so the initial height is 0 feet.
- Amplitude (A): The diameter of the Ferris wheel is 67 feet, so the radius is 33.5 feet. Since the Ferris wheel goes both above and below the starting position, the amplitude would be the radius, A = 33.5 feet.
- Vertical shift (K): Since the starting position is at the top of the steps, there is no vertical shift, K = 0.
- Phase shift (h): Since the Ferris wheel starts at the bottom of the steps, there is no phase shift, h = 0.
- Period (b): The rotational speed is 1 revolution every 80 seconds. Since one revolution covers 360 degrees or 2π radians, the period can be calculated as b = 80 seconds / (2π radians) ≈ 12.73 seconds.
Therefore, the sine model for the height of a passenger off the ground would be: h(t) = 33.5sin((2π/12.73)(t-0)) + 0.
The cosine model can be derived similarly: h(t) = 33.5cos((2π/12.73)(t-0)) + 0.
2) Ferris wheel with customers getting 17 feet below ground level:
- Starting position: The bottom of the Ferris wheel is 17 feet below ground level, so the initial height is -17 feet.
- Amplitude (A): The diameter of the Ferris wheel is 122 feet, so the radius is 61 feet. As the Ferris wheel goes both above and below the starting position, the amplitude would be the radius, A = 61 feet.
- Vertical shift (K): Since the starting position is 17 feet below ground level, the vertical shift would be the negative of the starting position, K = -17.
- Phase shift (h): Since the customers already get into the Ferris wheel at the starting position, there is no phase shift, h = 0.
- Period (b): The rotational speed is 1 revolution every 150 seconds. Calculating the period as b = 150 seconds / (2π radians) ≈ 23.84 seconds.
Therefore, the sine model for the height of a passenger off the ground would be: h(t) = 61sin((2π/23.84)(t-0)) - 17.
The cosine model would be: h(t) = 61cos((2π/23.84)(t-0)) - 17.
Keep in mind that these models assume a perfect circular motion and do not account for any external factors like air resistance or variations in speed during the ride.