Question

a ferris wheel is 50 meters in diameter and boarded from a platform that is 4 meters above the ground. the six oclock position on the ferris wheel is level with the loading platform. the wheel completes 1 full revolution in 4 minutes. the function h = f(t) gives your height in meters above the ground t minutes after the wheel begins to turn. what is the amplitude? meters what is the midline? y = meters what is the period? minutes how high are you off of the ground after 2 minutes? meters question help: video

Explanation:

Amplitude

The amplitude \(A\) of a sinusoidal function for a Ferris - wheel height model is half of the diameter of the Ferris wheel.
Given the diameter \(d = 50\) meters.

$$A=\frac{d}{2}$$

$$A=\frac{50}{2}=25$$

Mid - line

The mid - line \(y = k\) is the height of the center of the Ferris wheel above the ground.
The radius \(r=25\) meters and the platform height is \(4\) meters.

$$k = 4 + 25$$

$$k=29$$

Period

The period \(P\) is the time taken for one full revolution.
Given the wheel completes 1 full revolution in \(4\) minutes. So \(P = 4\) minutes.

Height at \(t = 2\) minutes

We can use the cosine function \(h(t)=A\cos(\omega t)+k\), where \(\omega=\frac{2\pi}{P}\) and \(P = 4\), so \(\omega=\frac{\pi}{2}\)

$$h(t)=- 25\cos(\frac{\pi}{2}t)+29$$

When \(t = 2\)

$$h(2)=-25\cos(\frac{\pi}{2}\times2)+29$$

Since \(\cos(\pi)=- 1\)

$$h(2)=-25\times(-1)+29$$

$$h(2)=25 + 29$$

$$h(2)=54$$

Answer:

Amplitude: \(25\) meters
Midline: \(y = 29\) meters
Period: \(4\) minutes
Height at \(t = 2\) minutes: \(54\) meters