a) The minimum height, this is, at the bottom of the Ferris wheel is 13 meters above the ground.
b) The maximum height, this is, at the top of the Ferris wheel is 135 meters above the ground.
c) A seat takes [tex]\frac{1}{3}[/tex] hours to complete one full rotation.
d) The height of a seat in a Ferris wheel is modeled after [tex]y(t) = 61\cdot \sin \left(6\pi\cdot t - 0.5\pi\right)+74[/tex], whose graphic representation in time is included below as attachment.
Procedure - Modeling of the motion of a Ferris wheel by trigonometric functions
a) Minimum height
The minimum height, this is, at the bottom of the Ferris wheel is 13 meters above the ground. [tex]\blacksquare[/tex]
b) Maximum height
The maximum height, this is, at the top of the Ferris wheel is 135 meters above the ground. [tex]\blacksquare[/tex]
c) Period of rotation
According to the statement, we know the frequency of rotation, that is, the number of rotations per unit time, this is, one hour. The period ([tex]T[/tex]), in hours per rotation, is inverse of the frequency ([tex]f[/tex]), in rotations per hour:
[tex]T = \frac{1}{f}[/tex] (1)
If we know that [tex]f = 3\,\frac{rotations}{hour}[/tex], then the period of the Ferris wheel is:
[tex]T = \frac{1}{3}\,h[/tex]
A seat takes [tex]\frac{1}{3}[/tex] hours to complete one full rotation. [tex]\blacksquare[/tex]
d) Plotting the function
Mathematically speaking, a sinusoidal function representing the clockwise rotation of the Ferris wheel is described below:
[tex]y(t) = \frac{D}{2}\cdot \sin \left(-\frac{2\pi\cdot t}{T} + \phi \right) + \left(y_{min} + \frac{D}{2}\right)[/tex] (2)
Where:
- [tex]y(t)[/tex] - Current height of the seat above the ground, in meters.
- [tex]D[/tex] - Diameter of the Ferris wheel, in meters.
- [tex]y_{min}[/tex] - Height of the bottom of the Ferris wheel, in meters.
- [tex]\phi[/tex] - Phase angle, in radians.
Note - Please notice that the negative sign aside the angular frequency ([tex]\frac{2\pi\cdot t}{T}[/tex]) represents the counterclockwise rotation of the Ferris wheel.
If we know that [tex]D = 122\,m[/tex], [tex]T = \frac{1}{3}\,h[/tex], [tex]\phi = -\frac{\pi}{2}\,rad[/tex] and [tex]y_{min} = 13\,m[/tex], then the function of the height of the Ferris wheel is:
[tex]y(t) = 61\cdot \sin \left(6\pi\cdot t - 0.5\pi\right)+74[/tex] (3)
Now we proceed to graph for [tex]0 \le h \le \frac{50}{60}\,h[/tex] (1 hour = 60 minutes), of which we present an image attached below. [tex]\blacksquare[/tex]
To learn more on Ferris wheels, we kindly invite to check this verified question: https://brainly.com/question/16571298
