Question

A buoy bobs up and down as waves go past. The vertical displacement \(y\) (in feet) of the buoy with respect to sea level can be modeled by \(y=1.75 \cos \frac{\pi}{3} t\), where \(t\) is the time (in seconds). Find and interpret the period and amplitude in the context of the problem. Then graph the function.

Short answer

The amplitude of the function is 1.75 feet, which indicates the buoy moves 1.75 feet above and below sea level. The period is 6 seconds, meaning the buoy completes one cycle of motion every 6 seconds.

Step-by-step solution

Identify the Amplitude

The amplitude of the cosine function is the absolute value of the coefficient of the cosine term. In this case, \[A = |1.75|\] so the amplitude of the function is 1.75. This means the buoy moves 1.75 feet above and below sea level.

Identify the Period

The period of a cosine function is given by \(\frac{2\pi}{B}\) where B is the coefficient of t. In this function, \[B = \frac{\pi}{3}\]. Substituting B into the formula, we get that the period of the function is \[\frac{2\pi}{\frac{\pi}{3}} = 2*3 = 6\] seconds. This means it takes 6 seconds for the buoy to complete one cycle of bobbing up and down.

Graph the Function

Now we will depict the behavior of the buoy over time graphically. On the x-axis, place 't' (time) and on the y-axis 'y' (displacement). The function y=1.75 cos(\(\frac{\pi}{3}\)t) has a maximum point at (0,1.75), a minimum point at (2,-1.75), and zero-crossings (i.e., the buoy is at sea level) at (1,0) and (3,0). Repeat this pattern for each 6-second cycle.

Key concepts

Amplitude

When working with trigonometric functions like the cosine function, the term "amplitude" plays a critical role in understanding the motion or movement being described. In the equation \( y = 1.75 \cos \frac{\pi}{3} t \), the amplitude refers to how far from its central position the buoy moves. This central position is termed as the "mean position" or "equilibrium position."

The amplitude is determined by the coefficient of the cosine or sine function, which in this case is 1.75. Since we take the absolute value when considering amplitude, the amplitude of 1.75 tells us that the buoy moves 1.75 feet above and below sea level in its bobbing motion. This vertical span is essential as it gives us an idea of the extent of the buoy's motion caused by the passing waves.

To summarize, amplitude indicates the maximum distance that the buoy reaches from sea level, highlighting the peak influence of wave energy on the buoy's position.

Period of a Function

The "period" of a trigonometric function indicates how long it takes for the function to complete one full cycle. In terms of real-world phenomena, for our buoy, the period tells us how long it takes to complete one full up-and-down bob.

For the formula \( y = 1.75 \cos \frac{\pi}{3} t \), the period is derived from the coefficient of \( t \), which in this scenario, is \( \frac{\pi}{3} \). The formula to calculate the period \( T \) for a cosine function is given by \( T = \frac{2\pi}{B} \), where \( B \) is the coefficient of \( t \). Inserting the given values:
- \( T = \frac{2\pi}{\frac{\pi}{3}} = 2 \times 3 = 6 \) seconds
This means it takes 6 seconds for the buoy to go through one complete cycle of moving from sea level to its maximum height, back through sea level to its minimum height, and returning to sea level again. Such information is invaluable for predicting the buoy's motion and understanding the wave patterns affecting it.

Graphing Trigonometric Functions

Graphing the function \( y = 1.75 \cos \frac{\pi}{3} t \) provides a visual representation of the buoy's motion over time. To create the graph, the time \( t \) is plotted on the horizontal axis (x-axis), and the vertical displacement \( y \) is plotted on the vertical axis (y-axis).

In this function:

• The maximum displacement occurs at the start of the cycle when \( t = 0 \), and the value of \( y \) is 1.75.
• At \( t = 1 \), the buoy returns to sea level (\( y = 0 \)), passing through its equilibrium.
• At \( t = 2 \), the buoy reaches its lowest point at -1.75 feet.
• By \( t = 3 \), it is back at sea level.
• This pattern repeats every 6 seconds, as calculated with the period.

Plotting these points creates a smooth, wave-like curve known as a cosine curve. Understanding this graph helps visualize the regular, rhythmic motion characteristic of cosine functions and, in this context, the predictable movement of the buoy caused by waves.