1. **State the problem:** We need to write an equation for the height $h = f(t)$ of a person on a ferris wheel as a function of time $t$ in minutes. 2. **Given information:** - Diameter of ferris wheel = 15 meters, so radius $r = \frac{15}{2} = 7.5$ meters. - Platform height = 8 meters above ground. - One full revolution takes 30 minutes. - The height function $h = f(t)$ models height above ground. 3. **Formula for sinusoidal height on a ferris wheel:** $$h(t) = A \sin(B(t - C)) + D$$ where: - $A$ is amplitude (radius of wheel), - $B = \frac{2\pi}{T}$ with $T$ the period (time for one revolution), - $C$ is horizontal shift (phase shift), - $D$ is vertical shift (height of center above ground). 4. **Calculate parameters:** - Amplitude $A = 7.5$ meters. - Period $T = 30$ minutes, so $$B = \frac{2\pi}{30} = \frac{\pi}{15}$$ 5. **Vertical shift $D$:** The center of the wheel is the radius above the platform height: $$D = 8 + 7.5 = 15.5$$ 6. **Horizontal shift $C$:** The wheel is boarded at the three o'clock position, which corresponds to the lowest point on the wheel if we use sine starting at zero. The sine function starts at zero height at $t=0$, but we want the function to start at the lowest point. The sine function reaches its minimum at $\frac{3\pi}{2}$ radians, so we set: $$B(t - C) = \frac{3\pi}{2} \text{ at } t=0 \implies -BC = \frac{3\pi}{2} \implies C = -\frac{3\pi}{2B} = -\frac{3\pi}{2 \cdot \frac{\pi}{15}} = -\frac{3\pi}{2} \cdot \frac{15}{\pi} = -\frac{45}{2} = -22.5$$ 7. **Final equation:** $$h(t) = 7.5 \sin\left(\frac{\pi}{15}(t + 22.5)\right) + 15.5$$ This models the height above ground $t$ minutes after the wheel begins to turn, starting at the lowest point (boarding platform).