1. **Problem:** Model the altitude of Nikki's seat on the Ferris wheel over time using a sine function. 2. **Given Data:** Time (s): 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60 Altitude (m): 2, 2.8, 5, 8, 11, 13.2, 14, 13.2, 11, 8, 5, 2.8, 2 3. **Step 1: Identify key parameters of the sinusoidal function** - The maximum altitude is 14 m. - The minimum altitude is 2 m. - The period is the time for one full cycle. From the data, the altitude returns to 2 m at 60 seconds, so the period $T = 60$ seconds. 4. **Step 2: Calculate amplitude and vertical shift** - Amplitude $A = \frac{\text{max} - \text{min}}{2} = \frac{14 - 2}{2} = 6$ - Vertical shift (midline) $D = \frac{\text{max} + \text{min}}{2} = \frac{14 + 2}{2} = 8$ 5. **Step 3: Write the general sine function form** $$ h(t) = A \sin(B(t - C)) + D $$ where - $A$ is amplitude, - $B = \frac{2\pi}{T}$ is the angular frequency, - $C$ is the phase shift, - $D$ is the vertical shift. 6. **Step 4: Calculate $B$** $$ B = \frac{2\pi}{60} = \frac{\pi}{30} $$ 7. **Step 5: Determine phase shift $C$** - At $t=0$, altitude is minimum (2 m), but sine function starts at zero. - Sine function reaches minimum at $\frac{3\pi}{2}$ radians. - So, set inside sine to $\frac{3\pi}{2}$ at $t=0$: $$ B(0 - C) = -B C = \frac{3\pi}{2} \implies C = -\frac{3\pi}{2B} = -\frac{3\pi/2}{\pi/30} = -45 $$ - Negative phase shift means shift right by 45 seconds. 8. **Step 6: Write the final equation** $$ h(t) = 6 \sin\left(\frac{\pi}{30}(t - 45)\right) + 8 $$ 9. **Interpretation:** This equation models Nikki's seat altitude over time, oscillating between 2 m and 14 m with a period of 60 seconds. **Final answer:** $$ h(t) = 6 \sin\left(\frac{\pi}{30}(t - 45)\right) + 8 $$