To create the function $h = f(t)$ for the height of a rider above the ground, let us analyze the problem step by step:

Key Information:

1. Diameter: 35 meters → Radius $r = \frac{35}{2} = 17.5$ meters.
2. Platform Height: The loading platform is 2 meters above the ground.
3. Period: The wheel completes one full revolution in 8 minutes, so the angular velocity is $\omega = \frac{2\pi}{8} = \frac{\pi}{4}$ radians per minute.
4. Position: At $t = 0$, the rider is at the lowest point of the ferris wheel.

Function Setup:

The ferris wheel motion is sinusoidal. The rider's height $h(t)$ as a function of time $t$ can be modeled as: $h(t) = A \sin(\omega t + \phi) + D$ where:

• $A$: Amplitude (equal to the radius of the ferris wheel, $17.5$).
• $\omega$: Angular velocity ($\frac{\pi}{4}$).
• $\phi$: Phase shift (determined based on starting position).
• $D$: Vertical shift (platform height + radius, $2 + 17.5 = 19.5$).

Finding Phase Shift $\phi$:

• At $t = 0$, the rider is at the lowest point ($h(0) = 2$).
• Substituting $t = 0$ into the equation: $2 = 17.5 \sin(\phi) + 19.5$ Simplify: $\sin(\phi) = \frac{2 - 19.5}{17.5} = -1$ Hence, $\phi = -\frac{\pi}{2}$.

Final Equation:

$h(t) = 17.5 \sin\left(\frac{\pi}{4}t - \frac{\pi}{2}\right) + 19.5$ Simplify: $h(t) = 17.5 \cos\left(\frac{\pi}{4}t\right) + 19.5$

Summary:

The height $h(t)$ in meters above the ground is given by: $h(t) = 17.5 \cos\left(\frac{\pi}{4}t\right) + 19.5$

Would you like a breakdown of the steps or help with related questions?

Expansion Questions:

1. What would the equation be if the period of the wheel changed to 6 minutes?
2. How does the phase shift ($\phi$) affect the position of the rider at $t = 0$?
3. Can we rewrite the equation using sine instead of cosine? How?
4. What is the maximum and minimum height a rider reaches during the motion?
5. How would the equation change if the platform height were raised to 5 meters?

Tip:

When modeling sinusoidal motion, always identify amplitude, vertical shift, angular velocity, and initial conditions to construct the equation systematically!