Question

The center of a Ferris wheel is 55 feet above the ground. The Ferris wheel completes one revolution every 40 seconds. The distance between the lowest and highest points of the Ferris wheel is 100 feet.

A trigonometric function models the height of the seat that starts closest to the ground when the wheel begins moving. (a) What is the period of the function?

(b) What is the amplitude of the function?

(c) What is the midline of the function?

(d) Sketch the first two periods of the graph of the function. Plot and label the key points of the graph. Label each axis with an appropriate scale

Answer 1

The period of the function is 40 seconds, the amplitude is 50 feet, the midline is 55 feet, and the graph will be a sinusoidal curve repeating every 40 seconds.

Step-by-step explanation:

(a) Period of the function:

The period of a trigonometric function represents the time it takes for one complete cycle. In this case, the Ferris wheel completes one revolution every 40 seconds. Since one revolution equals a full cycle, the period of the function is 40 seconds.

(b) Amplitude of the function:

The amplitude of a trigonometric function represents the maximum displacement from the midline. In this case, the distance between the lowest and highest points of the Ferris wheel is 100 feet. Since the center of the Ferris wheel is 55 feet above the ground, the amplitude is half of the distance between the lowest and highest points, which is 50 feet.

(c) Midline of the function:

The midline of a trigonometric function represents the average value of the function. In this case, the center of the Ferris wheel is 55 feet above the ground, so the midline of the function is 55 feet.

(d) Sketch of the graph:

The graph of the function will be a sinusoidal curve. The lowest point will be 55 - 50 = 5 feet above the midline, and the highest point will be 55 + 50 = 105 feet above the midline. The graph will repeat every 40 seconds, with the first two periods shown on the graph with appropriately labeled key points.