A ferris wheel, with a diameter of 40 m , makes a full rotation in 3 minutes. Passengers board at the bottom of the ferris wheel, which is 2 m above ground. a) Find the equation that best models the height of a passenger. b) Find the height of a passenger who has been riding the ferris wheel for 150 seconds.

To model the height of a passenger on the Ferris wheel over time, we'll analyze the given information step by step. ### Given: - **Diameter of Ferris Wheel:** 40 meters - **Radius (A):** \( \frac{40}{2} = 20 \) meters - **Time for one full rotation (Period, \( T \)):** 3 minutes = 180 seconds - **Height of the bottom of the Ferris wheel above ground:** 2 meters ### Part a) Finding the Height Equation **1. Determine the Center Height (D):** Since the bottom of the Ferris wheel is 2 meters above the ground and the radius is 20 meters, the center of the Ferris wheel is: \[ D = 2\, \text{meters} + 20\, \text{meters} = 22\, \text{meters} \] **2. Amplitude (A):** The amplitude represents the maximum deviation from the center height, which is equal to the radius: \[ A = 20\, \text{meters} \] **3. Angular Frequency (B):** The angular frequency relates to how quickly the Ferris wheel makes a full rotation: \[ B = \frac{2\pi}{T} = \frac{2\pi}{180} = \frac{\pi}{90}\, \text{radians per second} \] **4. Phase Shift (C):** Since passengers board at the **lowest point** at \( t = 0 \), the cosine function (which typically starts at a maximum) needs to be shifted to start at a minimum. This can be achieved by introducing a phase shift of \( \pi \): \[ C = \pi \] **5. Constructing the Equation:** Combining these components, the height \( h(t) \) as a function of time \( t \) is: \[ h(t) = D - A \cdot \cos(Bt) \] \[ h(t) = 22\, \text{m} - 20\, \text{m} \cdot \cos\left(\frac{\pi t}{90}\right) \] **Final Equation:** \[ h(t) = 22 - 20\,\cos\left(\frac{\pi t}{90}\right) \] *where \( h(t) \) is the height in meters and \( t \) is the time in seconds.* ### Answer to Part a): The height \( h(t) \) (in meters) of a passenger at time \( t \) (in seconds) is \[ h(t) = 22 - 20\,\cos\left(\frac{\pi t}{90}\right) \].