Trigonometry: Option #1: Ferris Wheel Height

Sample Solution

    The situation of a person riding a Ferris wheel can be modeled with a periodic function because the height of the person off the ground repeats after a certain period of time. The amplitude of the function is the maximum height the person reaches off the ground, and the midline of the function is the average height of the person off the ground. The period of the function is the amount of time it takes for the Ferris wheel to complete one full revolution. The information provided in the problem relates to the amplitude, midline, and period of the function as follows:

• The amplitude of the function is given to be 50 feet. This means that the person reaches a maximum height of 50 feet off the ground and a minimum height of 0 feet off the ground.

Full Answer Section

   

• The midline of the function is given to be 25 feet. This means that the average height of the person off the ground is 25 feet.
• The period of the function is given to be 12 minutes. This means that it takes 12 minutes for the Ferris wheel to complete one full revolution.

2. Discuss why the domain and range you found in Part I makes sense in the context of this problem. The domain of the function h(t) is the set of all possible values of t. In the context of this problem, t represents the time in minutes since the person got on the Ferris wheel. Therefore, the domain of the function is the set of all non-negative real numbers. The range of the function h(t) is the set of all possible values of h(t). In the context of this problem, h(t) represents the height of the person off the ground in feet. Therefore, the range of the function is the set of all real numbers between 0 and 50. The domain and range of the function make sense in the context of this problem because they reflect the fact that the height of the person off the ground can never be negative and can never be greater than the diameter of the Ferris wheel. 3. Discuss how you found the height off the ground of the person after 5 minutes. The height off the ground of the person after 5 minutes can be found by evaluating the function h(5). The value of h(5) is 25 feet, which means that the person is 25 feet off the ground after 5 minutes. 4. Discuss how your answers in Part I would be affected if: 4a. The diameter of the Ferris wheel increased. 4b. The time it takes for the Ferris wheel to complete 1 full revolution decreases. If the diameter of the Ferris wheel increased, then the amplitude of the function would increase. This is because the person would reach a higher maximum height off the ground when the diameter of the Ferris wheel is increased. If the time it takes for the Ferris wheel to complete 1 full revolution decreases, then the period of the function would decrease. This is because the Ferris wheel would make more revolutions in a given amount of time if the time it takes to complete 1 full revolution decreases. In both cases, the midline of the function would remain the same. This is because the average height of the person off the ground would not change if the diameter of the Ferris wheel increased or if the time it takes for the Ferris wheel to complete 1 full revolution decreases.