Question

You are riding a Ferris wheel that turns for 180 seconds. Your height \(h\) (in feet) above the ground at any time \(t\) (in seconds) can be modeled by the equation $$ h=85 \sin \frac{\pi}{20}(t-10)+90 $$ a. Graph the function. b. How many cycles does the Ferris wheel make in 180 seconds? c. What are your maximum and minimum heights?

Short answer

The Ferris wheel makes 4.5 cycles in 180 seconds. The maximum height is 175 feet and the minimum height is 5 feet.

Step-by-step solution

Identify Key Sine Function Attributes

The given function is in the form \(h = a \sin(b(t-h))+k\) where a is the amplitude, \( \frac{2 \pi}{b} \) is the period, h is the horizontal shift, and k is the vertical shift. From the given function \(h=85 \sin(\frac{\pi}{20}\cdot(t-10))+90\), we can identify that the amplitude is 85 and that the Ferris wheel is vertically shifted by 90 feet. The function is stretched horizontally by a factor of \(\frac{1}{\frac{\pi}{20}}\), yielding a period of 40 seconds. The graph is also shifted to the right by 10 units representing a delay of 10 seconds.

Graph the function

By taking into consideration the parameters identified, plot the graph. The entire function repeats every 40 seconds and reaches its maximum and minimum every 20 seconds. It starts 10 seconds into the ride and this will affect the shape of the initial part of the graph.

Calculate the number of cycles in 180 seconds

Since the period of the Ferris wheel's revolution - that is, one complete cycle - is 40 seconds. This means that in 180 seconds, the Ferris wheel completes \( \frac{180}{40} = 4.5 \) cycles

Identify the maximum and minimum heights

The maximum height is reached when the value of the sine function is 1 and the minimum when it is -1. Therefore, by substituting these values into the height function, we get the maximum height to be \(85*1+90=175\) feet and the minimum height to be \(85*(-1)+90=5\) feet

Key concepts

Sine Function

A sine function is a type of trigonometric function that describes oscillating movements, like waves. It is represented mathematically as \( y = a \sin(bx - c) + d \). Here,

• \( a \) is the amplitude, which determines the peak height of the wave.
• \( b \) affects the period of the wave, deciding how long one complete cycle takes.
• \( c \) shifts the graph horizontally.
• \( d \) moves the graph vertically.

The sine function repeats its shape periodically as it goes from -1 to 1 and back. In real-life scenarios, it can be used to model phenomena such as sound waves, light waves, and in this case, the motion of a Ferris wheel.

Amplitude

Amplitude refers to the height of the wave in the sine function. It is the maximum distance from the middle of the wave to its peak or valley.
In the equation \( y = a \sin(bx - c) + d \), \( a \) represents the amplitude.
When someone says a function has a larger amplitude, it means the wave stretches higher and lower from its central axis. For our Ferris wheel example, the amplitude is 85, meaning the height oscillates 85 feet above and below the central position. Determining amplitude helps us understand how extreme the variations in the wave are, which in practical terms read as how high or low you are on the Ferris wheel during each cycle.

Period of a Function

The period of a function indicates how long it takes for the wave pattern to complete one full cycle and start repeating itself.
In the sine equation \( y = a \sin(bx - c) + d \), the period is calculated as \( \frac{2\pi}{b} \).For our example, where \( b = \frac{\pi}{20} \), the period of the Ferris wheel journey is 40 seconds. This means every 40 seconds, the wheel completes a full rotation, and your height above the ground goes through the same set of values. Determining the period is crucial when graphing the sine function, as it tells you how often the cycle repeats within a given timeframe. Hence, it helps in predicting and understanding the motion dynamics of the Ferris wheel ride.

Graphing Trigonometric Functions

Graphing trigonometric functions, like the sine function, helps to visually represent the oscillating patterns they model. To graph a sine function, follow these steps:

• Identify the amplitude to determine vertical stretching.
• Calculate the period to know the length of one complete cycle.
• Notice any horizontal or vertical shifts to place the graph accurately.

For the Ferris wheel function, the graph starts 10 seconds into the ride due to the horizontal shift. It repeats every 40 seconds, hitting maximum and minimum heights in between. By gradually plotting these points and connecting them in a smooth, wave-like motion, you can sketch the sine graph efficiently. This representation makes it easier to see how the height changes over time and visualize the overall pattern of the trigonometric function.