1. **Problem 1: Write an equation for the height $h$ of a point on the Ferris wheel as a function of time $t$.** Given: - Maximum height $= 17$ m - Minimum height $= 1$ m - The function form: $h = a \cos[b(t - c)] + d$ 2. **Step 1: Find amplitude $a$ and vertical shift $d$.** - Amplitude $a = \frac{\text{max} - \text{min}}{2} = \frac{17 - 1}{2} = 8$ - Vertical shift $d = \frac{\text{max} + \text{min}}{2} = \frac{17 + 1}{2} = 9$ 3. **Step 2: Find period and calculate $b$.** - The graph completes 2 full cycles in 60 seconds, so period $T = \frac{60}{2} = 30$ seconds - Formula for $b$: $b = \frac{2\pi}{T} = \frac{2\pi}{30} = \frac{\pi}{15}$ 4. **Step 3: Use given phase shift $c$.** - Given in the problem: $c = 13$ 5. **Step 4: Write the equation.** $$h = 8 \cos\left[\frac{\pi}{15}(t - 13)\right] + 9$$ --- 6. **Problem 2: Find values of $b$ and $d$ for Ellie's distance function $s(t) = a \cos[b(t - c)] + d$.** Given: - Radius of park $= 195$ m - Shortest distance from running trail to parking lot $= 5$ m - Ellie completes 4 laps in 32 minutes 7. **Step 1: Calculate $d$ (vertical shift).** - The distance $d$ is the centerline of the cosine function, which is the radius plus the shortest distance to the parking lot: $$d = 195 + 5 = 200$$ 8. **Step 2: Calculate period and $b$.** - Time for one lap $= \frac{32}{4} = 8$ minutes - Period $T = 8$ - Calculate $b$: $$b = \frac{2\pi}{T} = \frac{2\pi}{8} = \frac{\pi}{4}$$ 9. **Step 3: Match $b$ and $d$ to the options.** - $b = \frac{\pi}{4}$ and $d = 200$ - This corresponds to option D. --- **Final answers:** 1. Ferris wheel height function: $$h = 8 \cos\left[\frac{\pi}{15}(t - 13)\right] + 9$$ 2. Values for Ellie's distance function: $$b = \frac{\pi}{4}, \quad d = 200$$ Option D is correct.