Answer:
The height of the rider as a function of time is [tex]h(t) = 15 + 82.5\cdot (1-\cos 0.168t) \,[ft][/tex], where time is measured in seconds.
Step-by-step explanation:
Given that Ferris wheel rotates at constant rate and rider begins at the bottom of the wheel, the height of the rider as a function of time is modelled after this expression:
[tex]h(t) = h_{bottom} + (1-\cos \omega t)\cdot r_{w}[/tex]
Where:
[tex]h_{bottom}[/tex] - Height of the bottom with respect to ground, measured in feet.
[tex]\omega[/tex] - Angular speed of the ferris wheel, measured in radians per second.
[tex]t[/tex] - Time, measured in seconds.
[tex]r_{w}[/tex] - Radius of the Ferris wheel, measured in feet.
The angular speed of the ferris wheel, measured in radians per second, is obtained from the following expression:
[tex]\omega = \frac{\pi}{30}\cdot \dot n[/tex]
Where:
[tex]\dot n[/tex] - Angular speed of the ferris wheel, measured in revolutions per minute.
If [tex]\dot n = 1.6\,rpm[/tex], then:
[tex]\omega = \frac{\pi}{30}\cdot (1.6\,rpm)[/tex]
[tex]\omega \approx 0.168\,\frac{rad}{s}[/tex]
Now, given that [tex]h_{bottom} = 15\,ft[/tex], [tex]r_{w} = 82.5\,ft[/tex] and [tex]\omega \approx 0.168\,\frac{rad}{s}[/tex], the height of the rider as a function of time is:
[tex]h(t) = 15 + 82.5\cdot (1-\cos 0.168t) \,[ft][/tex]
