Question

a passenger is riding a ferris wheel. the graph shows the height, in feet, of the passenger as a function of time.
height above the ground
(feet)
move values to the boxes to create a function, h(t), that models the height, in feet, of the passenger after t minutes on the ferris wheel.

Explanation:

Step1: Determine the amplitude

The amplitude \(A\) of a cosine function \(y = A\cos(Bx - C)+D\) is half the distance between the maximum and minimum values. From the graph, the maximum height is \(3\) and the minimum is \(1\). So, \(A=\frac{3 - 1}{2}=1\). But since the function is of the form \(h(t)=-\frac{1}{5}\cos(Bt)+D\), we note the coefficient of the cosine term. The general form for a vertical - stretch/compression is considered. Here, the amplitude is related to the coefficient of the cosine function. The formula for the amplitude of \(y = a\cos(bt)+c\) is \(|a|\). We know that the mid - line \(D=\frac{3 + 1}{2}=2\).

Step2: Determine the period

The period \(T\) of a cosine function \(y=\cos(bt)\) is \(T=\frac{2\pi}{b}\). From the graph, the period \(T = 20\) (the time it takes for one full cycle). Using the formula \(T=\frac{2\pi}{b}\), if we assume the standard form related to the given structure \(h(t)=-\frac{1}{5}\cos(\frac{\pi}{10}t)+2\) (since for \(y = A\cos(Bt)+D\), \(B=\frac{2\pi}{T}\), and \(T = 20\), so \(B=\frac{2\pi}{20}=\frac{\pi}{10}\))

Answer:

\(h(t)=-\frac{1}{5}\cos(\frac{\pi}{10}t)+2\)