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Answer :
We first examine the data set of cups of coffee consumed over 12 days:
[tex]$$
\begin{array}{|l|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\text{Day} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ \hline
\text{Cups of coffee} & 16 & 30 & 8 & 14 & 28 & 10 & 15 & 31 & 11 & 14 & 29 & 9 \\
\hline
\end{array}
$$[/tex]
Step 1. Identify the Period
Notice that the consumption pattern from days 1 to 6 is very similar to the pattern from days 7 to 12:
- Day 1 (16 cups) is close to Day 7 (15 cups).
- Day 2 (30 cups) is close to Day 8 (31 cups).
- Day 3 (8 cups) is close to Day 9 (11 cups).
- Day 4 (14 cups) equals Day 10 (14 cups).
- Day 5 (28 cups) is close to Day 11 (29 cups).
- Day 6 (10 cups) is close to Day 12 (9 cups).
This resemblance indicates that the data repeats approximately every [tex]$6$[/tex] days. Thus, the period of the data is:
[tex]$$
\text{Period} = 6 \text{ days}
$$[/tex]
Step 2. Determine the Amplitude
For periodic data, the amplitude is defined as half the difference between the maximum and minimum values.
- First, identify the maximum and minimum numbers in the data:
- Maximum value: [tex]$31$[/tex] cups
- Minimum value: [tex]$8$[/tex] cups
- Next, compute the amplitude as follows:
[tex]$$
\text{Amplitude} = \frac{\text{Maximum} - \text{Minimum}}{2} = \frac{31 - 8}{2} = \frac{23}{2} = 11.5 \text{ cups}
$$[/tex]
Final Answer:
The data set is approximately periodic with a period of [tex]$6$[/tex] days and an amplitude of [tex]$11.5$[/tex] cups.
[tex]$$
\begin{array}{|l|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\text{Day} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ \hline
\text{Cups of coffee} & 16 & 30 & 8 & 14 & 28 & 10 & 15 & 31 & 11 & 14 & 29 & 9 \\
\hline
\end{array}
$$[/tex]
Step 1. Identify the Period
Notice that the consumption pattern from days 1 to 6 is very similar to the pattern from days 7 to 12:
- Day 1 (16 cups) is close to Day 7 (15 cups).
- Day 2 (30 cups) is close to Day 8 (31 cups).
- Day 3 (8 cups) is close to Day 9 (11 cups).
- Day 4 (14 cups) equals Day 10 (14 cups).
- Day 5 (28 cups) is close to Day 11 (29 cups).
- Day 6 (10 cups) is close to Day 12 (9 cups).
This resemblance indicates that the data repeats approximately every [tex]$6$[/tex] days. Thus, the period of the data is:
[tex]$$
\text{Period} = 6 \text{ days}
$$[/tex]
Step 2. Determine the Amplitude
For periodic data, the amplitude is defined as half the difference between the maximum and minimum values.
- First, identify the maximum and minimum numbers in the data:
- Maximum value: [tex]$31$[/tex] cups
- Minimum value: [tex]$8$[/tex] cups
- Next, compute the amplitude as follows:
[tex]$$
\text{Amplitude} = \frac{\text{Maximum} - \text{Minimum}}{2} = \frac{31 - 8}{2} = \frac{23}{2} = 11.5 \text{ cups}
$$[/tex]
Final Answer:
The data set is approximately periodic with a period of [tex]$6$[/tex] days and an amplitude of [tex]$11.5$[/tex] cups.
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