To determine whether the given data represent exponential or logistic growth, we need to evaluate the growth patterns in the population over the years.
Let's start by looking at the given data:
[tex]\[
\begin{array}{|c|c|}
\hline \text{Year} & \text{Population Size} \\
\hline 2012 & 5 \\
\hline 2013 & 25 \\
\hline 2014 & 125 \\
\hline 2015 & 185 \\
\hline 2016 & 205 \\
\hline
\end{array}
\][/tex]
### Step-by-Step Solution:
1. Calculate the growth factors for each subsequent year:
- Growth factor from 2012 to 2013: [tex]\(\frac{25}{5} = 5\)[/tex]
- Growth factor from 2013 to 2014: [tex]\(\frac{125}{25} = 5\)[/tex]
- Growth factor from 2014 to 2015: [tex]\(\frac{185}{125} \approx 1.48\)[/tex]
- Growth factor from 2015 to 2016: [tex]\(\frac{205}{185} \approx 1.108\)[/tex]
So, the growth factors are:
[tex]\([5, 5, 1.48, 1.108]\)[/tex]
2. Check if the growth factors indicate exponential or logistic growth:
- Exponential growth is characterized by consistent or increasing growth factors over time.
- Logistic growth is characterized by decreasing growth factors over time as the population approaches a carrying capacity.
3. Analyze the pattern of growth factors:
- From 2012 to 2013: Growth factor = 5
- From 2013 to 2014: Growth factor = 5 (same as previous year)
- From 2014 to 2015: Growth factor ≈ 1.48 (significantly less than previous years)
- From 2015 to 2016: Growth factor ≈ 1.108 (slightly less than the previous year)
4. Determine the type of growth:
Based on the calculated growth factors:
[tex]\([5, 5, 1.48, 1.108]\)[/tex]
- The growth factors are consistently greater than 1, which is a characteristic of exponential growth.
- However, the growth factors are not constant, and they show a decreasing pattern, which suggests a logistic growth trend.
In conclusion, the data appears to exhibit both properties that could hint at exponential growth (since all growth factors are greater than 1) and logistic growth (since the growth factors are decreasing). However, because the growth factors are not consistently the same and they are decreasing, the dominant observation here is that the population growth exhibits the characteristics of exponential growth.
Therefore, the data represent:
Exponential growth
