1. **State the problem:** We need to write an equation for the height $y$ of Aaron's Ferris wheel car as a function of time $t$ in minutes. 2. **Analyze the graph:** The height oscillates between a minimum of about 15 feet and a maximum of about 395 feet. 3. **Find the amplitude:** Amplitude $A$ is half the distance between max and min heights: $$A = \frac{395 - 15}{2} = \frac{380}{2} = 190$$ 4. **Find the midline (vertical shift):** Midline $D$ is the average of max and min heights: $$D = \frac{395 + 15}{2} = \frac{410}{2} = 205$$ 5. **Determine the period:** The graph completes two full cycles in 20 minutes, so one period $T$ is: $$T = \frac{20}{2} = 10$$ 6. **Find the angular frequency:** $$\omega = \frac{2\pi}{T} = \frac{2\pi}{10} = \frac{\pi}{5}$$ 7. **Determine phase shift:** The maximum height occurs at $t=9$ minutes. The cosine function has its maximum at $t=0$, so phase shift $\phi$ satisfies: $$\omega t - \phi = 0 \Rightarrow \phi = \omega \times 9 = \frac{\pi}{5} \times 9 = \frac{9\pi}{5}$$ 8. **Write the equation:** Using cosine (since max at $t=9$), the height is: $$y = A \cos\left(\omega t - \phi\right) + D = 190 \cos\left(\frac{\pi}{5} t - \frac{9\pi}{5}\right) + 205$$ **Final answer:** $$y = 190 \cos\left(\frac{\pi}{5} t - \frac{9\pi}{5}\right) + 205$$