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Answer :
We begin by examining the sequence of cups of coffee recorded over several days:
[tex]$$
\text{Cups} = \{16,\;30,\;8,\;14,\;28,\;10,\;15,\;31,\;11,\;14,\;29,\;9\}
$$[/tex]
A closer look at the data suggests that the values can be split into groups of 4 numbers, which indicates that the data might repeat every 4 days. We thus consider a period of 4.
Next, for each 4-day segment we calculate the amplitude. The amplitude is defined as half the difference between the maximum and minimum values within each cycle.
1. First Segment (Days 1 to 4):
The cups consumed are
[tex]$$
\{16,\,30,\,8,\,14\}.
$$[/tex]
Here, the maximum is [tex]$30$[/tex] and the minimum is [tex]$8$[/tex]. The amplitude is:
[tex]$$
\frac{30 - 8}{2} = 11.
$$[/tex]
2. Second Segment (Days 5 to 8):
The cups consumed are
[tex]$$
\{28,\,10,\,15,\,31\}.
$$[/tex]
Here, the maximum is [tex]$31$[/tex] and the minimum is [tex]$10$[/tex]. The amplitude is:
[tex]$$
\frac{31 - 10}{2} = 10.5.
$$[/tex]
3. Third Segment (Days 9 to 12):
The cups consumed are
[tex]$$
\{11,\,14,\,29,\,9\}.
$$[/tex]
Here, the maximum is [tex]$29$[/tex] and the minimum is [tex]$9$[/tex]. The amplitude is:
[tex]$$
\frac{29 - 9}{2} = 10.
$$[/tex]
To obtain an overall measure of the amplitude, we can average the amplitudes of the segments:
[tex]$$
\text{Average Amplitude} = \frac{11 + 10.5 + 10}{3} \approx 10.5.
$$[/tex]
When rounded to the nearest whole number, the amplitude is about [tex]$10$[/tex].
Thus, the data set is approximately periodic with a period of [tex]$4$[/tex] days and an amplitude of about [tex]$10$[/tex].
[tex]$$
\text{Cups} = \{16,\;30,\;8,\;14,\;28,\;10,\;15,\;31,\;11,\;14,\;29,\;9\}
$$[/tex]
A closer look at the data suggests that the values can be split into groups of 4 numbers, which indicates that the data might repeat every 4 days. We thus consider a period of 4.
Next, for each 4-day segment we calculate the amplitude. The amplitude is defined as half the difference between the maximum and minimum values within each cycle.
1. First Segment (Days 1 to 4):
The cups consumed are
[tex]$$
\{16,\,30,\,8,\,14\}.
$$[/tex]
Here, the maximum is [tex]$30$[/tex] and the minimum is [tex]$8$[/tex]. The amplitude is:
[tex]$$
\frac{30 - 8}{2} = 11.
$$[/tex]
2. Second Segment (Days 5 to 8):
The cups consumed are
[tex]$$
\{28,\,10,\,15,\,31\}.
$$[/tex]
Here, the maximum is [tex]$31$[/tex] and the minimum is [tex]$10$[/tex]. The amplitude is:
[tex]$$
\frac{31 - 10}{2} = 10.5.
$$[/tex]
3. Third Segment (Days 9 to 12):
The cups consumed are
[tex]$$
\{11,\,14,\,29,\,9\}.
$$[/tex]
Here, the maximum is [tex]$29$[/tex] and the minimum is [tex]$9$[/tex]. The amplitude is:
[tex]$$
\frac{29 - 9}{2} = 10.
$$[/tex]
To obtain an overall measure of the amplitude, we can average the amplitudes of the segments:
[tex]$$
\text{Average Amplitude} = \frac{11 + 10.5 + 10}{3} \approx 10.5.
$$[/tex]
When rounded to the nearest whole number, the amplitude is about [tex]$10$[/tex].
Thus, the data set is approximately periodic with a period of [tex]$4$[/tex] days and an amplitude of about [tex]$10$[/tex].
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