An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (1)
The period is found by identifying a complete cycle in the data: 24 − 8 = 16 .
The midline is calculated as the average of the maximum and minimum values: 2 34.5 + 4 = 19.25 .
The amplitude is calculated as half the difference between the maximum and minimum values: 2 34.5 − 4 = 15.25 .
Therefore, the period is 16 , the midline is y = 19.25 , and the amplitude is 15.25 .
Explanation
Understanding the Problem We are given a table of data that shows a sinusoidal relationship between time x and velocity v ( x ) . Our goal is to determine the period, midline, and amplitude of the cosine function that models this data.
Determining the Period First, let's find the period. Looking at the table, we can see that the velocity values start repeating after x = 8 . Specifically, v ( 8 ) = 20 and v ( 24 ) = 20 . Thus, one complete cycle occurs from x = 8 to x = 24 . Therefore, the period is 24 − 8 = 16 .
Calculating the Midline Next, we need to find the midline. The midline is the horizontal line that runs midway between the maximum and minimum values of the function. From the table, the maximum velocity is 34.5 and the minimum velocity is 4. The midline is calculated as the average of the maximum and minimum values: Midline = 2 Maximum + Minimum = 2 34.5 + 4 = 2 38.5 = 19.25 . Therefore, the equation of the midline is y = 19.25 .
Calculating the Amplitude Now, let's find the amplitude. The amplitude is the distance from the midline to the maximum (or minimum) value. It is calculated as half the difference between the maximum and minimum values: Amplitude = 2 Maximum − Minimum = 2 34.5 − 4 = 2 30.5 = 15.25 . Therefore, the amplitude of the cosine function is 15.25.
Final Answer In conclusion, the period of the cosine function is 16, the equation of the midline is y = 19.25 , and the amplitude of the cosine function is 15.25.
Examples
Understanding sinusoidal relationships is crucial in many real-world applications. For example, in electrical engineering, alternating current (AC) voltage and current vary sinusoidally with time. The period determines the frequency of the AC signal, the midline represents the average voltage level, and the amplitude indicates the peak voltage. Similarly, in acoustics, sound waves can be modeled as sinusoidal functions, where the period corresponds to the pitch, the midline represents the ambient air pressure, and the amplitude corresponds to the loudness of the sound. By analyzing these parameters, engineers and scientists can design and optimize systems involving AC circuits, audio equipment, and other wave-based phenomena.
