To determine which set of changes in 2013 would keep the population growth of cheetahs consistent with the previous years (2011 and 2012), we need to follow a few steps.
### Step 1: Calculate Population Change for the Years 2011 and 2012
The population change for a given year can be determined using the formula:
[tex]\[ \text{Population Change} = \text{Births} + \text{Immigration} - \text{Deaths} - \text{Emigration} \][/tex]
#### Population Change for 2011
- Deaths: 2
- Births: 5
- Immigration: 6
- Emigration: 8
[tex]\[
\text{Population Change}_{2011} = 5 + 6 - 2 - 8 = 5 + 6 - 10 = 1
\][/tex]
#### Population Change for 2012
- Deaths: 1
- Births: 4
- Immigration: 1
- Emigration: 3
[tex]\[
\text{Population Change}_{2012} = 4 + 1 - 1 - 3 = 4 + 1 - 4 = 1
\][/tex]
### Step 2: Calculate the Average Population Change for 2011 and 2012
The average population change over these two years can be found by averaging the changes for each year:
[tex]\[
\text{Average Population Change} = \frac{\text{Population Change}_{2011} + \text{Population Change}_{2012}}{2} = \frac{1 + 1}{2} = \frac{2}{2} = 1
\][/tex]
### Step 3: Determine Which Change in 2013 Matches the Average
We need to check each set of values for 2013 to see which yields a population change closest to the average (1).
#### Option 1: 4 deaths, 7 births, 2 immigration, 6 emigration
[tex]\[
\text{Population Change}_{2013} = 7 + 2 - 4 - 6 = 9 - 10 = -1
\][/tex]
#### Option 2: 3 deaths, 6 births, 5 immigration, 7 emigration
[tex]\[
\text{Population Change}_{2013} = 6 + 5 - 3 - 7 = 11 - 10 = 1
\][/tex]
#### Option 3: 5 deaths, 2 births, 8 immigration, 3 emigration
[tex]\[
\text{Population Change}_{2013} = 2 + 8 - 5 - 3 = 10 - 8 = 2
\][/tex]
#### Option 4: 1 death, 5 births, 3 immigration, 7 emigration
[tex]\[
\text{Population Change}_{2013} = 5 + 3 - 1 - 7 = 8 - 8 = 0
\][/tex]
### Step 4: Compare Each Population Change to the Average
We compare each computed population change to the average population change (1):
- Option 1: -1 (difference of 2 away from average)
- Option 2: 1 (difference of 0 away from average)
- Option 3: 2 (difference of 1 away from average)
- Option 4: 0 (difference of 1 away from average)
The population change that matches the average most closely is option 2, with a population change of 1.
### Conclusion
The appropriate set of changes in 2013 that keeps the population growth consistent with previous years is:
3 deaths, 6 births, 5 immigration, 7 emigration.
