A ferris wheel is 10 meters in diameter and boarded from a platform that is 5 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. Write an equation for \( h=f(t) \). \( f(t)= \)

To model the height \( h \) of a person on the Ferris wheel as a function of time \( t \), we can use a sinusoidal function. ### Step 1: Determine the parameters of the function 1. **Diameter and Radius**: The Ferris wheel has a diameter of 10 meters, so the radius \( r \) is: \[ r = \frac{10}{2} = 5 \text{ meters} \] 2. **Center Height**: The center of the Ferris wheel is located 5 meters (the height of the platform) plus the radius of the wheel: \[ \text{Center height} = 5 + 5 = 10 \text{ meters} \] 3. **Period**: The Ferris wheel completes one full revolution in 4 minutes, so the period \( T \) is: \[ T = 4 \text{ minutes} \] 4. **Angular Frequency**: The angular frequency \( \omega \) is given by: \[ \omega = \frac{2\pi}{T} = \frac{2\pi}{4} = \frac{\pi}{2} \text{ radians per minute} \] ### Step 2: Write the height function The general form of the sinusoidal function for height \( h \) can be expressed as: \[ h(t) = A \sin(\omega t + \phi) + D \] where: - \( A \) is the amplitude, - \( \omega \) is the angular frequency, - \( \phi \) is the phase shift, - \( D \) is the vertical shift (the center height). ### Step 3: Assign values 1. **Amplitude \( A \)**: The amplitude is equal to the radius of the Ferris wheel: \[ A = 5 \] 2. **Vertical Shift \( D \)**: The vertical shift is the height of the center of the Ferris wheel: \[ D = 10 \] 3. **Phase Shift \( \phi \)**: Since the loading platform is at the 6 o'clock position (the lowest point of the Ferris wheel), we can use a cosine function (which starts at its maximum) and shift it down. To start at the lowest point, we can use: \[ \phi = -\frac{\pi}{2} \quad \text{(to shift the sine function down)} \] ### Final Function Putting it all together, we can write the height function as: \[ h(t) = 5 \cos\left(\frac{\pi}{2} t - \frac{\pi}{2}\right) + 10 \] Using the identity \( \cos(x - \frac{\pi}{2}) = \sin(x) \), we can simplify this to: \[ h(t) = 5 \sin\left(\frac{\pi}{2} t\right) + 10 \] Thus, the final equation for the height \( h \) as a function of time \( t \) is: \[ \boxed{h(t) = 5 \sin\left(\frac{\pi}{2} t\right) + 10} \]