Ferris Wheel Height

by | Aug 4, 2023 | mathematics

 

For this Critical Thinking assignment. you will be exploring a real-world example that can be modeled using a periodic function.
Begin by reading the following prompt:
A Ferris wheel is 25 meters in diameter and boarded from a platform that is 1 meter 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 10 minutes. The function h(t) gives a person’s height in meters above the ground t minutes after the wheel begins to turn.
Part I: Complete the following steps:
1. Find the amplitude, midline, and period of h(t).
2. Find the domain and range of the function h(t) and
3. Find a formula for the height function h(t).
4. State the phase shift and vertical translation, if applicable.
5. How high off the ground is a person after 5 minutes?
6. Use the GeoGebra Graphing Calculator tool to model this situation. (Refer to this tutorial as needed: https://www.geogebra.org/m/XUv5mXTm n.) Save your GeoGebra work as a .pdf file for submission.
Part II: Based on your work in Part I. discuss the following:
1. In your own words, discuss why this situation can be modelled with a periodic function and how the information provided relates to the amplitude, midline. and period of the function h(t).
2. Discuss why the domain and range you found in Part I makes sense in the context of this problem.
3. Discuss how you found the height off the ground of the person after 5 minutes.
4. Discuss how your answers in Part I would be affected if:
a. The diameter of the Ferris wheel increased.
b. The time it takes for the Ferris wheel to complete 1 full revolution decreases.
5. Provide at least two other real-world situations that can be modeled using a periodic function and respond to the following:
a. What common characteristics do the real-world scenarios you chose share?
b. What did you look for in the way that the real-world scenario can be modeled?
c. How can you verify that the real-world scenarios you chose can be modeled by a periodic function?

 

 

Sample solution

Dante Alighieri played a critical role in the literature world through his poem Divine Comedy that was written in the 14th century. The poem contains Inferno, Purgatorio, and Paradiso. The Inferno is a description of the nine circles of torment that are found on the earth. It depicts the realms of the people that have gone against the spiritual values and who, instead, have chosen bestial appetite, violence, or fraud and malice. The nine circles of hell are limbo, lust, gluttony, greed and wrath. Others are heresy, violence, fraud, and treachery. The purpose of this paper is to examine the Dante’s Inferno in the perspective of its portrayal of God’s image and the justification of hell. 

In this epic poem, God is portrayed as a super being guilty of multiple weaknesses including being egotistic, unjust, and hypocritical. Dante, in this poem, depicts God as being more human than divine by challenging God’s omnipotence. Additionally, the manner in which Dante describes Hell is in full contradiction to the morals of God as written in the Bible. When god arranges Hell to flatter Himself, He commits egotism, a sin that is common among human beings (Cheney, 2016). The weakness is depicted in Limbo and on the Gate of Hell where, for instance, God sends those who do not worship Him to Hell. This implies that failure to worship Him is a sin.


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God is also depicted as lacking justice in His actions thus removing the godly image. The injustice is portrayed by the manner in which the sodomites and opportunists are treated. The opportunists are subjected to banner chasing in their lives after death followed by being stung by insects and maggots. They are known to having done neither good nor bad during their lifetimes and, therefore, justice could have demanded that they be granted a neutral punishment having lived a neutral life. The sodomites are also punished unfairly by God when Brunetto Lattini is condemned to hell despite being a good leader (Babor, T. F., McGovern, T., & Robaina, K. (2017). While he commited sodomy, God chooses to ignore all the other good deeds that Brunetto did.

Finally, God is also portrayed as being hypocritical in His actions, a sin that further diminishes His godliness and makes Him more human. A case in point is when God condemns the sin of egotism and goes ahead to commit it repeatedly. Proverbs 29:23 states that “arrogance will bring your downfall, but if you are humble, you will be respected.” When Slattery condemns Dante’s human state as being weak, doubtful, and limited, he is proving God’s hypocrisy because He is also human (Verdicchio, 2015). The actions of God in Hell as portrayed by Dante are inconsistent with the Biblical literature. Both Dante and God are prone to making mistakes, something common among human beings thus making God more human.

To wrap it up, Dante portrays God is more human since He commits the same sins that humans commit: egotism, hypocrisy, and injustice. Hell is justified as being a destination for victims of the mistakes committed by God. The Hell is presented as being a totally different place as compared to what is written about it in the Bible. As a result, reading through the text gives an image of God who is prone to the very mistakes common to humans thus ripping Him off His lofty status of divine and, instead, making Him a mere human. Whether or not Dante did it intentionally is subject to debate but one thing is clear in the poem: the misconstrued notion of God is revealed to future generations.

 

References

Babor, T. F., McGovern, T., & Robaina, K. (2017). Dante’s inferno: Seven deadly sins in scientific publishing and how to avoid them. Addiction Science: A Guide for the Perplexed, 267.

Cheney, L. D. G. (2016). Illustrations for Dante’s Inferno: A Comparative Study of Sandro Botticelli, Giovanni Stradano, and Federico Zuccaro. Cultural and Religious Studies, 4(8), 487.

Verdicchio, M. (2015). Irony and Desire in Dante’s” Inferno” 27. Italica, 285-297.

Sample Answer

Sample Answer

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.

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