Problem 6:
*I understand how to complete the problem but I am stuck on trying to find the common difference for population 1. It's the only thing I need help with for this problem.*

The table shows two sloth populations growing over time.

time (years since 1990)
0
1
2
3
4
5
6
7
8
population 1 (thousands)
0, 90.0
1, 76.5
2, 65.0
3, 55.3
4, 47.0
population 2 (thousands)
0, 39
1, 37
2, 35
3, 33
4, 31
1. Describe a pattern in how each population changed from one year to the next.
2. These patterns continued for many years. Based on this information, fill in the extra rows in the table.
3. On the same axes, draw graphs of the two populations over time.
4. Does Population 2 ever equal Population 1? If so, when? Explain or show your reasoning.

Graphs and functions can be used to illustrate real life events.

The table is given as:

[tex]\left[\begin{array}{cccccccccc}Time&0&1&2&3&4&5&6&7&8\\P_1&90.0&76.5&65.0&55.3&47.0\\P_2&39&37&35&33&31\end{array}\right][/tex]

(a) The pattern in each population

Population 1

This population follows a geometric pattern.

The common ratio (r) is calculated by dividing the subsequent values of population 1.

We have:

[tex]r = \frac{P_1(1)}{P_1(0)}[/tex]

This gives

[tex]r = \frac{76.5}{90.0}[/tex]

[tex]r = 0.85[/tex]

Express as percentage

[tex]r = 85\%[/tex]

Next, we calculate the scale factor (k)

[tex]k = 1 - r[/tex]

[tex]k = 1 - 85\%[/tex]

[tex]k = 15\%[/tex]

Hence, the pattern of population 1 is: the population decreases by 15% each year

Population 2

This population follows an arithmetic pattern.

The common difference (d) is calculated by subtracting the subsequent values of population 1.

We have:

[tex]d = P_2(1) - P_2(0)[/tex]

[tex]d = 37 - 39[/tex]

[tex]d = -2[/tex]

-2 indicates that the population reduces, each year by 2000

(b) Complete the table

For population 1, we keep multiplying current population by 85%.

So, we have:

[tex]P_1(5) = 47.0 \times 85\% = 40.0[/tex]

[tex]P_1(6) = 40.0 \times 85\% = 34.0[/tex]

[tex]P_1(7) = 34.0 \times 85\% = 28.9[/tex]

[tex]P_1(8) = 28.9 \times 85\% = 24.6[/tex]

For population 2, we keep adding -2 to the current population

So, we have:

[tex]P_2(5) = 31 -2 = 29[/tex]

[tex]P_2(6) = 29 -2 = 27[/tex]

[tex]P_2(7) = 27 -2 = 25[/tex]

[tex]P_2(8) = 25 -2 = 23[/tex]

So, the complete table is:

[tex]\left[\begin{array}{cccccccccc}Time&0&1&2&3&4&5&6&7&8\\P_1&90.0&76.5&65.0&55.3&47.0&40.0&34.0&28.9&24.6\\P_2&39&37&35&33&31&29&27&25&23\end{array}\right][/tex]

(c) The graph of the two populations

First, we determine the equations of both populations

Population 1

We have:

[tex]P_1(0) = 90.0[/tex]

[tex]r = 0.85[/tex]

The nth term of a geometric sequence is:

[tex]P_1(n) = P_1(0) \times r^n[/tex]

This gives

[tex]P_1(n) = 90.0 \times 0.85^n[/tex]

Population 1

We have:

[tex]P_2(0) = 39[/tex]

[tex]d = -2[/tex]

The nth term of an arithmetic sequence is:

[tex]P_2(n) = P_1(0) + nd[/tex]

This gives

[tex]P_2(n) = 39 -2 \times n[/tex]

[tex]P_2(n) = 39 -2n[/tex]

See attachment for the graphs of:

[tex]P_1(n) = 90.0 \times 0.85^n[/tex] and [tex]P_2(n) = 39 -2n[/tex]

(d) Determine if both populations ever have the same value

From the attached image, the graph of both populations intersect at:

[tex](8.891,21.218)[/tex] and [tex](16.337,6.327)[/tex]

This means that both populations have equal values, twice

Read more about graphs and functions at:

https://brainly.com/question/18806107

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