1. **Problem Statement:** We need to model the height $h(t)$ of a passenger on a Ferris wheel over time $t$ seconds. Given: - Maximum height $= 42$ m - Minimum height $= 2$ m - One full rotation every $60$ seconds - Passenger starts at the lowest point 2. **Formula and Important Rules:** The height of a point on a Ferris wheel can be modeled by a sinusoidal function: $$h(t) = A \sin(B(t - C)) + D$$ where: - $A$ is the amplitude (half the distance between max and min heights) - $B$ relates to the period by $B = \frac{2\pi}{\text{period}}$ - $C$ is the phase shift (horizontal shift) - $D$ is the midline (average of max and min heights) 3. **Calculate Amplitude $A$:** $$A = \frac{\text{max height} - \text{min height}}{2} = \frac{42 - 2}{2} = 20$$ 4. **Calculate Midline $D$:** $$D = \frac{\text{max height} + \text{min height}}{2} = \frac{42 + 2}{2} = 22$$ 5. **Calculate Period and $B$:** Period $T = 60$ seconds $$B = \frac{2\pi}{T} = \frac{2\pi}{60} = \frac{\pi}{30}$$ 6. **Determine Phase Shift $C$:** Passenger starts at the lowest point, which corresponds to the minimum of the sine function. Since $\sin(x)$ has minimum at $x = \frac{3\pi}{2}$, we set: $$B(t - C) = \frac{3\pi}{2} \text{ at } t=0$$ $$\Rightarrow \frac{\pi}{30}(0 - C) = \frac{3\pi}{2}$$ $$\Rightarrow -\frac{\pi}{30}C = \frac{3\pi}{2}$$ $$\Rightarrow C = -\frac{3\pi}{2} \times \frac{30}{\pi} = -45$$ So phase shift $C = -45$ seconds. 7. **Final Equation:** $$h(t) = 20 \sin\left(\frac{\pi}{30}(t + 45)\right) + 22$$ This equation models the height of the passenger over time.