Let's break down the components of the sinusoidal function [tex]\( h(t) \)[/tex] modeling Amare's height above the ground.
1. Amplitude: The Ferris wheel's diameter is 50 meters, so the radius is half of that, which is 25 meters. Hence, the amplitude of the sinusoidal function is 25 meters.
2. Angular frequency: The Ferris wheel takes six minutes to complete three revolutions. Therefore, one full revolution takes [tex]\( \frac{6}{3} \)[/tex] minutes, which is 2 minutes. The angular frequency [tex]\( \omega \)[/tex] in terms of [tex]\( \pi \)[/tex] is calculated using:
[tex]\[
\omega = \frac{2\pi}{\text{period}} = \frac{2\pi}{2} = \pi
\][/tex]
3. Phase shift: Since Amare starts at the lowest point of the Ferris wheel, the phase shift corresponds to a downward shift, which is [tex]\( -\frac{\pi}{2} \)[/tex].
4. Vertical shift: The vertical shift accounts for the initial height above the ground plus the radius of the Ferris wheel. Thus, the vertical shift is:
[tex]\[
\text{Vertical shift} = \text{initial height} + \text{radius} = 4 + 25 = 29
\][/tex]
Putting all these components together, the function can be written as follows:
[tex]\[
h(t) = 25 \cdot \sin(\pi t - \frac{\pi}{2}) + 29
\][/tex]
In the format provided by the question:
[tex]\(h(t) =\)[/tex]
- [tex]\(25\)[/tex] [tex]\(\cdot \sin (\)[/tex]
- [tex]\(\pi t +\)[/tex]
- [tex]\(-\frac{\pi}{2}) +\)[/tex]
- [tex]\(29\)[/tex]
So the completed function is:
[tex]\(h(t) = 25 \cdot \sin(\pi t - \frac{\pi}{2}) + 29\)[/tex].
