Question

a ferris wheel is 10 meters in diameter and boarded from a platform that is 3 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 6 minutes. the function h = f(t) gives your height in meters above the ground t minutes after the wheel begins to turn. write an equation for h = f(t). f(t) =

Explanation:

Step1: Determine the amplitude, period, and midline

• Amplitude (\(A\)): The diameter of the Ferris wheel is \(10\) meters, so the radius \(r = 5\) meters. The amplitude \(A=5\).
• Midline (\(D\)): The Ferris wheel is boarded from a platform \(3\) meters above the ground. The mid - line \(D = 3 + 5=8\) (since the center of the Ferris wheel is at \(r\) meters above the boarding platform).
• Period (\(T\)): The Ferris wheel completes \(1\) full revolution in \(T = 6\) minutes. Using the formula \(B=\frac{2\pi}{T}\), we get \(B=\frac{2\pi}{6}=\frac{\pi}{3}\).

Step2: Choose the trigonometric function form

Since the Ferris wheel starts at the lowest point (six o'clock position), we use the cosine function. The general form of a cosine function is \(h = A\cos(Bt)+D\). But for a function that starts at the minimum value, we can write it as \(h=-A\cos(Bt)+D\)

Step3: Substitute the values of \(A\), \(B\), and \(D\) into the function

Substitute \(A = 5\), \(B=\frac{\pi}{3}\), and \(D = 8\) into the function \(h=-A\cos(Bt)+D\).

We get \(h=- 5\cos(\frac{\pi}{3}t)+8\)

Answer:

\(f(t)=-5\cos(\frac{\pi}{3}t)+8\)