Please Help Emergency
A Ferris wheel with a radius of 20 meters completes one rotation every 30 seconds. The
bottom of the Ferris wheel is 2 meters above the ground. Write a sinusoidal function that
models the height of a passenger above the ground as a function of time , where is
measured in seconds. Determine the height of the passenger after 7 seconds.

We can find the sinusoidal function of the Ferris wheel with a radius of 20 m and completes 1 rotation every 30 seconds by using this formula:

[tex]\boxed{y=Asin(\omega t+\phi_0)+y_0}[/tex]

where:

[tex]y=\texttt{height}[/tex]
[tex]A=\texttt{amplitude (max.height)}[/tex]
[tex]\omega=\texttt{angular velocity}[/tex]
[tex]t=\texttt{time}[/tex]
[tex]\phi_0=\texttt{initial phase}[/tex]
[tex]y_0=\texttt{initial height of equilibrium position}[/tex]

Refer to the picture, we can find that the equilibrium position (y₀) is the total of Ferris wheel distance from ground and the radius, therefore:

[tex]y_0=2+20=22\ m[/tex]

From the picture, we can also find the amplitude (A) and the initial phase (φ₀):

Amplitude is is the maximum height from the equilibrium position, which is 20 m.
Initial phase is the phase of the wave when t = 0, which is -90° or -π/2. (with assumption the Ferris wheel start at the lowest point)

"The Ferris wheel completes one rotation every 30 seconds" means the period (T) of the Ferris wheel equals to 30 s. We can find the angular velocity (ω) by using this formula:

[tex]\boxed{\omega=\frac{2\pi}{T} }[/tex]

[tex]\begin{aligned}\\\omega&=\frac{2\pi}{30} \\\\&=\frac{\pi}{15} \ rad/s\end{aligned}[/tex]

By putting in all the information, we can come up with the sinusoidal function:

[tex]\begin{aligned}y&=Asin(\omega t+\phi_0)+y_0\\\\y&=20\cdot sin\left(\frac{\pi}{15} t+\left(-\frac{\pi}{2} \right)\right)+22\\\\\bf y&\bf=20\cdot sin\left(\frac{\pi}{15} t-\frac{\pi}{2} \right)+22\end{aligned}[/tex]

To determine the height of the passenger after 7 seconds, we substitute t with 7:

[tex]\begin{aligned}y&=20\cdot sin\left(\frac{\pi}{15} t-\frac{\pi}{2} \right)+22\\\\&=20\cdot sin\left(\frac{\pi}{15} (7)-\frac{\pi}{2} \right)+22\\\\&=20\cdot sin\left(-\frac{\pi}{30} \right)+22\\\\&\approx\bf 21.94\ m\end{aligned}[/tex]