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:

Step1: Find the amplitude

The amplitude \(A\) of a sinusoidal function for a Ferris - wheel problem is half of the diameter. Given the diameter \(d = 50\) meters, then \(A=\frac{d}{2}\).
\(A=\frac{50}{2}=25\)

Step2: Find the mid - line

The mid - line \(y = k\) is the height of the center of the Ferris wheel. The Ferris wheel is boarded from a platform that is \(4\) meters above the ground, and the radius (which is related to the center height) is \(25\) meters. So \(k=4 + 25\).
\(k = 29\)

Step3: Find the period

The period \(T\) is the time it takes for the Ferris wheel to complete one full revolution. Given that the wheel completes \(1\) full revolution in \(4\) minutes, so \(T = 4\)

Step4: Find the height at \(t = 2\) minutes

We can use the cosine function \(h(t)=A\cos(\omega t)+k\), where \(\omega=\frac{2\pi}{T}\). Since \(T = 4\), \(\omega=\frac{\pi}{2}\). The function is \(h(t)=- 25\cos(\frac{\pi}{2}t)+29\) (negative because at \(t = 0\), \(h(0)=4\) which is the minimum of the cosine - type function).
When \(t = 2\), we substitute \(t\) into the function:

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Answer:

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