Modeling a Ferris Wheel Ride with a Periodic Function

Part I

1. Amplitude, Midline, and Period of h(t):
   • The amplitude represents half the difference between the maximum and minimum values of the function. Since the Ferris wheel has a diameter of 25 meters, the radius is half of that, which is 12.5 meters. Therefore, the amplitude is 12.5 meters.
   • The midline is the horizontal line that represents the average or center value of the function. In this case, since the loading platform is 1 meter above the ground, the midline is located at a height of 1 meter.
   • The period represents the time it takes for one complete cycle or revolution of the function. Given that the Ferris wheel completes one full revolution in 10 minutes, the period of h(t) is 10 minutes.
2. Domain and Range of h(t):
   • The domain refers to all possible input values for the function. In this case, since we are measuring time in minutes after the wheel begins to turn, the domain of h(t) is all real numbers greater than or equal to 0.
   • The range represents all possible output values for the function. Since the lowest point on the Ferris wheel is 1 meter above the ground and the highest point is 25 meters above the ground, the range of h(t) is [1, 25].
3. Formula for the Height Function h(t):
   • The general formula for a periodic function with amplitude A, midline M, period P, phase shift C, and vertical translation D is given by:

     h(t) = A * sin(2π/P * (t - C)) + M + D
     

     Applying this formula to our situation, we have:

     h(t) = 12.5 * sin(2π/10 * (t - 0)) + 1
     

     Simplifying further:

     h(t) = 12.5 * sin(π/5 * t) + 1
     

4. Phase Shift and Vertical Translation:
   • In this case, there is no phase shift or vertical translation. The function h(t) starts at the loading platform, which is the reference point for time t = 0, and there is no vertical translation added to the function.
5. Height after 5 minutes:
   • To find the height off the ground after 5 minutes, we substitute t = 5 into the function h(t):

     h(5) = 12.5 * sin(π/5 * 5) + 1
          = 12.5 * sin(π) + 1
          = 12.5 * 0 + 1
          = 1
     

     Therefore, a person is 1 meter off the ground after 5 minutes.
6. GeoGebra Graphing Calculator Tool:
   • Please refer to the attached GeoGebra PDF file for the graphical representation of the situation using the GeoGebra Graphing Calculator tool.

Part II

1. Modeling with a Periodic Function:
   • This situation can be modeled with a periodic function because the height of a person on the Ferris wheel repeats in a predictable pattern as the wheel completes each revolution. The information provided regarding the diameter, height of the loading platform, and time it takes for one revolution allows us to determine the amplitude, midline, and period of the periodic function that represents the height.
2. Domain and Range Relevance:
   • The domain of h(t) being all real numbers greater than or equal to 0 makes sense in the context of this problem because we are measuring time in minutes after the wheel begins to turn. It allows us to track the height of the person as time progresses.
   • The range of h(t) being [1, 25] makes sense because the lowest point on the Ferris wheel is 1 meter above the ground (the height of the loading platform), and the highest point is 25 meters above the ground (the diameter of the Ferris wheel).
3. Height after 5 Minutes:
   • To find the height off the ground of the person after 5 minutes, we substituted t = 5 into the height function h(t). The resulting value of 1 meter indicates that after 5 minutes, the person is at the same height as the loading platform.
4. Effect of Changes in the Ferris Wheel: a. If the diameter of the Ferris wheel increased, the amplitude of the height function h(t) would also increase. The larger diameter would result in a greater difference between the highest and lowest points on the wheel. b. If the time it takes for the Ferris wheel to complete one full revolution decreases, the period of h(t) would decrease accordingly. The shorter time would result in a faster repetition of the height pattern, compressing the graph horizontally.
5. Other Real-World Scenarios Modeled with Periodic Functions: a. Tides: The rise and fall of tides throughout the day can be modeled using a periodic function. The amplitude represents the difference between high and low tide levels, while the period represents the time it takes for one complete cycle of high and low tides. b. Seasons: The changing seasons throughout the year can be modeled with a periodic function. The amplitude represents the difference in temperature or daylight hours between summer and winter, while the period represents the time it takes for one complete cycle of seasons.
   • Common Characteristics:
     • Both scenarios exhibit repetition or cycles in their patterns.
     • The amplitudes represent the magnitude or difference in the variable being measured.
     • The periods represent the time it takes for one complete cycle of the phenomenon.
   • Verification of Modeling:
     • For tides, historical data on high and low tide levels can be analyzed to identify patterns and confirm the periodic nature of the phenomenon.
     • For seasons, long-term climate data, including temperature and daylight hours, can be analyzed to observe the cyclical patterns and verify the use of a periodic function.

Sample Answer