a ferris wheel is 40 meters in diameter and boarded from a platform that is 3 meters above the ground. the…

Explanation:

Step1: Find the amplitude

The amplitude of a sinusoidal - like function for a Ferris - wheel problem is half of the diameter. Given the diameter $d = 40$ meters, so $A=\frac{d}{2}$. $A=\frac{40}{2}=20$

Step2: Find the midline

The midline is the average of the maximum and minimum heights. The minimum height is the height of the platform, which is 3 meters above the ground, and the maximum height is $3 + 40=43$ meters above the ground. The midline $y$ is $y=\frac{3 + 43}{2}=23$.

Step3: Find the period

The period is the time it takes for the Ferris - wheel to complete one full revolution. Given that the wheel completes 1 full revolution in 6 minutes, so the period $T = 6$ minutes.

Step4: Find the height at $t = 3$ minutes

We can model the height function as a sinusoidal function of the form $h(t)=A\sin(\omega t+\varphi)+k$, where $A = 20$, $k = 23$, $\omega=\frac{2\pi}{T}=\frac{2\pi}{6}=\frac{\pi}{3}$, and since the six - o'clock position is level with the loading platform, $\varphi =-\frac{\pi}{2}$. So $h(t)=20\sin(\frac{\pi}{3}t-\frac{\pi}{2})+23$. When $t = 3$, we have $\frac{\pi}{3}\times3-\frac{\pi}{2}=\pi-\frac{\pi}{2}=\frac{\pi}{2}$. Then $h(3)=20\sin(\frac{\pi}{2})+23=20\times1 + 23=43$.

Answer:

Amplitude: 20 meters Midline: $y = 23$ meters Period: 6 minutes Height at $t = 3$ minutes: 43 meters