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Sagot :
To determine which scenario in 2013 would maintain the same rate of population growth as in 2011 and 2012, let's break down the process step-by-step.
### Step 1: Calculate the population change for 2011 and 2012.
Given data for 2011:
- Deaths: 2
- Births: 5
- Immigration: 6
- Emigration: 8
Population change in 2011:
[tex]\[ \text{Change}_{2011} = (\text{Births}_{2011} + \text{Immigration}_{2011}) - (\text{Deaths}_{2011} + \text{Emigration}_{2011}) \][/tex]
[tex]\[ \text{Change}_{2011} = (5 + 6) - (2 + 8) \][/tex]
[tex]\[ \text{Change}_{2011} = 11 - 10 \][/tex]
[tex]\[ \text{Change}_{2011} = 1 \][/tex]
Given data for 2012:
- Deaths: 1
- Births: 4
- Immigration: 1
- Emigration: 3
Population change in 2012:
[tex]\[ \text{Change}_{2012} = (\text{Births}_{2012} + \text{Immigration}_{2012}) - (\text{Deaths}_{2012} + \text{Emigration}_{2012}) \][/tex]
[tex]\[ \text{Change}_{2012} = (4 + 1) - (1 + 3) \][/tex]
[tex]\[ \text{Change}_{2012} = 5 - 4 \][/tex]
[tex]\[ \text{Change}_{2012} = 1 \][/tex]
Since the population growth change in both 2011 and 2012 is 1, we need the same population growth change of 1 for 2013.
### Step 2: Evaluate each scenario for 2013.
#### Scenario 1: 4 deaths, 7 births, 2 immigration, 6 emigration
[tex]\[ \text{Change}_{2013} = (7 + 2) - (4 + 6) \][/tex]
[tex]\[ \text{Change}_{2013} = 9 - 10 \][/tex]
[tex]\[ \text{Change}_{2013} = -1 \][/tex]
#### Scenario 2: 3 deaths, 6 births, 5 immigration, 7 emigration
[tex]\[ \text{Change}_{2013} = (6 + 5) - (3 + 7) \][/tex]
[tex]\[ \text{Change}_{2013} = 11 - 10 \][/tex]
[tex]\[ \text{Change}_{2013} = 1 \][/tex]
#### Scenario 3: 5 deaths, 2 births, 8 immigration, 3 emigration
[tex]\[ \text{Change}_{2013} = (2 + 8) - (5 + 3) \][/tex]
[tex]\[ \text{Change}_{2013} = 10 - 8 \][/tex]
[tex]\[ \text{Change}_{2013} = 2 \][/tex]
#### Scenario 4: 1 death, 5 births, 3 immigration, 7 emigration
[tex]\[ \text{Change}_{2013} = (5 + 3) - (1 + 7) \][/tex]
[tex]\[ \text{Change}_{2013} = 8 - 8 \][/tex]
[tex]\[ \text{Change}_{2013} = 0 \][/tex]
From the calculations above, Scenario 2 results in a population change of 1, which is consistent with the population changes in 2011 and 2012.
### Therefore, the correct scenario to keep the population growth of cheetahs the same in 2013 as in 2011 and 2012 is:
3 deaths, 6 births, 5 immigration, 7 emigration
### Step 1: Calculate the population change for 2011 and 2012.
Given data for 2011:
- Deaths: 2
- Births: 5
- Immigration: 6
- Emigration: 8
Population change in 2011:
[tex]\[ \text{Change}_{2011} = (\text{Births}_{2011} + \text{Immigration}_{2011}) - (\text{Deaths}_{2011} + \text{Emigration}_{2011}) \][/tex]
[tex]\[ \text{Change}_{2011} = (5 + 6) - (2 + 8) \][/tex]
[tex]\[ \text{Change}_{2011} = 11 - 10 \][/tex]
[tex]\[ \text{Change}_{2011} = 1 \][/tex]
Given data for 2012:
- Deaths: 1
- Births: 4
- Immigration: 1
- Emigration: 3
Population change in 2012:
[tex]\[ \text{Change}_{2012} = (\text{Births}_{2012} + \text{Immigration}_{2012}) - (\text{Deaths}_{2012} + \text{Emigration}_{2012}) \][/tex]
[tex]\[ \text{Change}_{2012} = (4 + 1) - (1 + 3) \][/tex]
[tex]\[ \text{Change}_{2012} = 5 - 4 \][/tex]
[tex]\[ \text{Change}_{2012} = 1 \][/tex]
Since the population growth change in both 2011 and 2012 is 1, we need the same population growth change of 1 for 2013.
### Step 2: Evaluate each scenario for 2013.
#### Scenario 1: 4 deaths, 7 births, 2 immigration, 6 emigration
[tex]\[ \text{Change}_{2013} = (7 + 2) - (4 + 6) \][/tex]
[tex]\[ \text{Change}_{2013} = 9 - 10 \][/tex]
[tex]\[ \text{Change}_{2013} = -1 \][/tex]
#### Scenario 2: 3 deaths, 6 births, 5 immigration, 7 emigration
[tex]\[ \text{Change}_{2013} = (6 + 5) - (3 + 7) \][/tex]
[tex]\[ \text{Change}_{2013} = 11 - 10 \][/tex]
[tex]\[ \text{Change}_{2013} = 1 \][/tex]
#### Scenario 3: 5 deaths, 2 births, 8 immigration, 3 emigration
[tex]\[ \text{Change}_{2013} = (2 + 8) - (5 + 3) \][/tex]
[tex]\[ \text{Change}_{2013} = 10 - 8 \][/tex]
[tex]\[ \text{Change}_{2013} = 2 \][/tex]
#### Scenario 4: 1 death, 5 births, 3 immigration, 7 emigration
[tex]\[ \text{Change}_{2013} = (5 + 3) - (1 + 7) \][/tex]
[tex]\[ \text{Change}_{2013} = 8 - 8 \][/tex]
[tex]\[ \text{Change}_{2013} = 0 \][/tex]
From the calculations above, Scenario 2 results in a population change of 1, which is consistent with the population changes in 2011 and 2012.
### Therefore, the correct scenario to keep the population growth of cheetahs the same in 2013 as in 2011 and 2012 is:
3 deaths, 6 births, 5 immigration, 7 emigration
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