A ferris wheel is 45 meters in diameter and boarded from a platform that is 2 meters above the ground. The six o'clock position on the ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 4 minutes. The function \( h=f(t) \) gives your height in meters above the ground \( t \) minutes after the wheel begins to turn. What is the Amplitude? What is the Midline? \( y=\square \) meters What is the Period? How High are you off of the ground after 2 minutes?

To analyze the Ferris wheel's height function \( h = f(t) \), we can break down the information given: 1. **Diameter of the Ferris wheel**: 45 meters - Therefore, the **radius** \( r \) is half of the diameter: \[ r = \frac{45}{2} = 22.5 \text{ meters} \] 2. **Height of the loading platform**: 2 meters above the ground. 3. **Midline**: The midline of the Ferris wheel's height function is the average height of the highest and lowest points of the wheel. The highest point is at the top of the wheel, which is: \[ \text{Highest point} = \text{Height of platform} + \text{Radius} = 2 + 22.5 = 24.5 \text{ meters} \] The lowest point is at the bottom of the wheel, which is: \[ \text{Lowest point} = \text{Height of platform} - \text{Radius} = 2 - 22.5 = -20.5 \text{ meters} \text{ (not possible, so it is actually 0)} \] Since the wheel cannot go below the ground, the lowest point is actually at 0 meters. Thus, the midline is: \[ \text{Midline} = \frac{24.5 + 0}{2} = 12.25 \text{ meters} \] So, the midline is: \[ y = 12.25 \text{ meters} \] 4. **Amplitude**: The amplitude is half the distance between the highest and lowest points: \[ \text{Amplitude} = \frac{\text{Highest point} - \text{Lowest point}}{2} = \frac{24.5 - 0}{2} = 12.25 \text{ meters} \] 5. **Period**: The period is the time it takes for the Ferris wheel to complete one full revolution, which is given as 4 minutes. 6. **Height after 2 minutes**: The height function can be modeled as: \[ h(t) = A \sin(B(t - C)) + D \] where: - \( A \) is the amplitude, - \( B \) is related to the period, - \( C \) is the phase shift (not needed here), - \( D \) is the midline. Given: - Amplitude \( A = 22.5 \) - Midline \( D = 12.25 \) - Period \( T = 4 \) minutes, so \( B = \frac{2\pi}{T} = \frac{2\pi}{4} = \frac{\pi}{2} \) The function can be expressed as: \[ h(t) = 22.5 \sin\left(\frac{\pi}{2} t - \frac{\pi}{2}\right) + 12.25 \] Since the sine function starts at 0, we can adjust it to start at the midline: \[ h(t) = 22.5 \cos\left(\frac{\pi}{2} t\right) + 12.25 \] Now, we can find the height after 2 minutes: \[ h(2) = 22.5 \cos\left(\frac{\pi}{2} \cdot 2\right) + 12.25 \] \[ h(2) = 22.5 \cos(\pi) + 12.25 \] \[ h(2) = 22.5 \cdot (-1) + 12.25 \] \[ h(2) = -22.5 + 12.25 = -10.25 \text{ meters (not possible, so it is actually 0)} \] Therefore, the height after 2 minutes is: \[ h(2) = 12.25 \text{ meters} \] ### Summary: - **Amplitude**: 22.5 meters - **Midline**: \( y = 12.25 \) meters - **Period**: 4 minutes - **Height after 2 minutes**: 12.25 meters