High School

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Is the data set approximately periodic? If so, what are its period and amplitude?

[tex]
\[
\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Day & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\
\hline
\begin{tabular}{l}
Cups of \\
coffee
\end{tabular} & 16 & 30 & 8 & 14 & 28 & 10 & 15 & 31 & 11 & 14 & 29 & 9 \\
\hline
\end{tabular}
\]
[/tex]

A. Not periodic

B. Periodic with a period of 4 and an amplitude of about 7.5

C. Periodic with a period of 4 and an amplitude of about 10

D. Periodic with a period of 3 and an amplitude of about 10

Answer :

We start by examining whether the coffee consumption data repeats in a regular pattern. One way to check for periodicity is to see if the pattern repeats every few days. Here, we explore a period of 4 days.

Given the data for 12 days:

[tex]$$
\begin{array}{c|cccccccccccc}
\text{Day:} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\[6pt]
\text{Cups of coffee:} & 16 & 30 & 8 & 14 & 28 & 10 & 15 & 31 & 11 & 14 & 29 & 9 \\
\end{array}
$$[/tex]

We group the days based on a period of 4. That is, we separate the days into four groups, where each group contains days that are 4 days apart:

1. Group 0 (Phase 0): Days 1, 5, and 9
Values: [tex]$16$[/tex], [tex]$28$[/tex], [tex]$11$[/tex]
- Maximum value: [tex]$28$[/tex]
- Minimum value: [tex]$11$[/tex]
- Amplitude for Group 0 is calculated as half the difference of these extremes:

[tex]$$
\text{Amplitude}_0 = \frac{28 - 11}{2} = \frac{17}{2} = 8.5
$$[/tex]

2. Group 1 (Phase 1): Days 2, 6, and 10
Values: [tex]$30$[/tex], [tex]$10$[/tex], [tex]$14$[/tex]
- Maximum: [tex]$30$[/tex]
- Minimum: [tex]$10$[/tex]
- Amplitude for Group 1:

[tex]$$
\text{Amplitude}_1 = \frac{30 - 10}{2} = \frac{20}{2} = 10.0
$$[/tex]

3. Group 2 (Phase 2): Days 3, 7, and 11
Values: [tex]$8$[/tex], [tex]$15$[/tex], [tex]$29$[/tex]
- Maximum: [tex]$29$[/tex]
- Minimum: [tex]$8$[/tex]
- Amplitude for Group 2:

[tex]$$
\text{Amplitude}_2 = \frac{29 - 8}{2} = \frac{21}{2} = 10.5
$$[/tex]

4. Group 3 (Phase 3): Days 4, 8, and 12
Values: [tex]$14$[/tex], [tex]$31$[/tex], [tex]$9$[/tex]
- Maximum: [tex]$31$[/tex]
- Minimum: [tex]$9$[/tex]
- Amplitude for Group 3:

[tex]$$
\text{Amplitude}_3 = \frac{31 - 9}{2} = \frac{22}{2} = 11.0
$$[/tex]

To get an overall measure of the amplitude of this approximately periodic behavior, we compute the average amplitude across these four groups:

[tex]$$
\text{Average Amplitude} = \frac{8.5 + 10.0 + 10.5 + 11.0}{4} = \frac{40.0}{4} = 10.0
$$[/tex]

Thus, we conclude that the data is approximately periodic with a period of 4 days and an amplitude of about 10 cups of coffee.

Final Answer: Periodic with a period of 4 and an amplitude of about 10.

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