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Answer :
To solve the question about the sinusoidal relationship between time (in seconds) and velocity (in centimeters per second), we need to determine three properties of the cosine function that models the data.
### 1. Period of the Cosine Function
The period of a function is the length of one complete cycle of the wave. In a sinusoidal function, the period is the time it takes for the function to repeat itself.
Looking at the [tex]$x$[/tex] values provided, we observe that the function appears to complete one full cycle from 0 seconds to 120 seconds. This means the period of the cosine function is 120 seconds.
### 2. Equation of the Midline
The midline of a cosine function is the horizontal line that passes through the middle of the wave, equally bisecting the maximum and minimum points of the function.
To find the midline, we take the average of the maximum and minimum values of the given velocities [tex]$v(x)$[/tex]. From the data:
- The maximum velocity is 72 cm/s.
- The minimum velocity is 8 cm/s.
The midline is calculated as follows:
[tex]\[
\text{Midline} = \frac{\text{Max} + \text{Min}}{2} = \frac{72 + 8}{2} = 40
\][/tex]
Thus, the equation of the midline is [tex]\(y = 40\)[/tex].
### 3. Amplitude of the Cosine Function
The amplitude of a sinusoidal function is the distance from the midline to a peak (either maximum or minimum value). It represents half of the total height of the wave from a peak to a trough.
Using the maximum and minimum values:
- Maximum velocity [tex]\(v_{\max} = 72\)[/tex] cm/s
- Minimum velocity [tex]\(v_{\min} = 8\)[/tex] cm/s
The amplitude is calculated as:
[tex]\[
\text{Amplitude} = \frac{\text{Max} - \text{Min}}{2} = \frac{72 - 8}{2} = 32
\][/tex]
So, the amplitude of the cosine function is 32.
In summary:
- The period of the cosine function is 120.
- The equation of the midline is [tex]\(y = 40\)[/tex].
- The amplitude of the cosine function is 32.
These values complete the description of the sinusoidal function based on the given tabular data.
### 1. Period of the Cosine Function
The period of a function is the length of one complete cycle of the wave. In a sinusoidal function, the period is the time it takes for the function to repeat itself.
Looking at the [tex]$x$[/tex] values provided, we observe that the function appears to complete one full cycle from 0 seconds to 120 seconds. This means the period of the cosine function is 120 seconds.
### 2. Equation of the Midline
The midline of a cosine function is the horizontal line that passes through the middle of the wave, equally bisecting the maximum and minimum points of the function.
To find the midline, we take the average of the maximum and minimum values of the given velocities [tex]$v(x)$[/tex]. From the data:
- The maximum velocity is 72 cm/s.
- The minimum velocity is 8 cm/s.
The midline is calculated as follows:
[tex]\[
\text{Midline} = \frac{\text{Max} + \text{Min}}{2} = \frac{72 + 8}{2} = 40
\][/tex]
Thus, the equation of the midline is [tex]\(y = 40\)[/tex].
### 3. Amplitude of the Cosine Function
The amplitude of a sinusoidal function is the distance from the midline to a peak (either maximum or minimum value). It represents half of the total height of the wave from a peak to a trough.
Using the maximum and minimum values:
- Maximum velocity [tex]\(v_{\max} = 72\)[/tex] cm/s
- Minimum velocity [tex]\(v_{\min} = 8\)[/tex] cm/s
The amplitude is calculated as:
[tex]\[
\text{Amplitude} = \frac{\text{Max} - \text{Min}}{2} = \frac{72 - 8}{2} = 32
\][/tex]
So, the amplitude of the cosine function is 32.
In summary:
- The period of the cosine function is 120.
- The equation of the midline is [tex]\(y = 40\)[/tex].
- The amplitude of the cosine function is 32.
These values complete the description of the sinusoidal function based on the given tabular data.
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