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After experiencing a period of rapid increase beginning in 2005, the populations of swallows in two different national parks level off. The approximate populations are modeled here, where [tex]\(x\)[/tex] is time, in years, since 2005.

Park A:
\begin{tabular}{|c|c|}
\hline Years since 2005 & Swallow Population \\
\hline 0 & 1,370 \\
\hline 1 & 1,870 \\
\hline 2 & 2,077 \\
\hline 3 & 2,236 \\
\hline 4 & 2,370 \\
\hline 5 & 2,488 \\
\hline 6 & 2,595 \\
\hline 7 & 2,693 \\
\hline 9 & 2,784 \\
\hline 10 & 2,870 \\
\hline
\end{tabular}

Park B:
\begin{tabular}{|c|c|}
\hline Years since 2005 & Swallow Population \\
\hline 0 & 1,450 \\
\hline 1 & 1,930 \\
\hline 2 & 2,140 \\
\hline 3 & 2,310 \\
\hline 4 & 2,460 \\
\hline 5 & 2,590 \\
\hline 6 & 2,710 \\
\hline 7 & 2,820 \\
\hline 9 & 2,920 \\
\hline 10 & 3,010 \\
\hline
\end{tabular}



Answer :

GinnyAnswer
To solve this problem and determine when the population of swallows in Park A levels off, we need to look at the differences in population between consecutive years. When these differences become consistently small, we can conclude that the population is leveling off.

### Step-by-Step Solution:

1. Examine the given population data for Park A:

[tex]\[ \begin{array}{|c|c|} \hline \text{Years since 2005} & \text{Swallow Population} \\ \hline 0 & 1370 \\ 1 & 1870 \\ 2 & 2077 \\ 3 & 2236 \\ 4 & 2370 \\ 5 & 2488 \\ 6 & 2595 \\ 7 & 2693 \\ 9 & 2784 \\ 10 & 2870 \\ \hline \end{array} \][/tex]

2. Calculate the change in population between each consecutive pair of years:

[tex]\[ \Delta \text{Population} = \text{Population}[x+1] - \text{Population}[x] \][/tex]

3. Determine the year-to-year differences:

[tex]\[ \begin{align*} \text{1st year difference} & = 1870 - 1370 = 500 \\ \text{2nd year difference} & = 2077 - 1870 = 207 \\ \text{3rd year difference} & = 2236 - 2077 = 159 \\ \text{4th year difference} & = 2370 - 2236 = 134 \\ \text{5th year difference} & = 2488 - 2370 = 118 \\ \text{6th year difference} & = 2595 - 2488 = 107 \\ \text{7th year difference} & = 2693 - 2595 = 98 \\ \text{9th year difference} & = 2784 - 2693 = 91 \\ \text{10th year difference} & = 2870 - 2784 = 86 \\ \end{align*} \][/tex]

The differences form the following series:

[tex]\[ [500, 207, 159, 134, 118, 107, 98, 91, 86] \][/tex]

4. Identify the point where the population differences become consistently small:

From the data:

- Initially, the differences are relatively large (500 to 134).
- After the 4th year (starting from the 5th year), the differences become smaller and more consistent (118, 107, 98, 91, 86).

5. Conclusion:

The population of swallows in Park A begins to level off from the 5th year onward, as indicated by the consistently smaller and more stable differences in population (below 120).

Given these steps, we have successfully determined that the population of swallows in Park A starts leveling off approximately from the 5th year since 2005.

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