Answer :
Alright, let's tackle each part of the question step-by-step.
### Given Data
- Population in 2012 ([tex]\( P_{2012} \)[/tex]): 10,840,334
- Population in 2024 ([tex]\( P_{2024} \)[/tex]): 14,414,910
- Time period: 2024 - 2012 = 12 years
### a) Geometric Annual Growth Rate
The geometric annual growth rate can be found using the formula:
[tex]\[ r = \left( \frac{P_{2024}}{P_{2012}} \right)^{\frac{1}{n}} - 1 \][/tex]
where [tex]\( n \)[/tex] is the number of years.
Plugging the values in, we get:
[tex]\[ r = \left( \frac{14,414,910}{10,840,334} \right)^{\frac{1}{12}} - 1 \][/tex]
[tex]\[ r \approx 0.024033 = 2.4033\% \][/tex]
### b) Linear Annual Growth Rate
The linear annual growth rate is determined by measuring the total population change and dividing it by the number of years:
[tex]\[ \text{Linear Growth Rate} = \frac{P_{2024} - P_{2012}}{n} \][/tex]
So:
[tex]\[ \text{Linear Growth Rate} = \frac{14,414,910 - 10,840,334}{12} \][/tex]
[tex]\[ \text{Linear Growth Rate} \approx 297,881.33 \][/tex]
### c) Exponential Annual Growth Rate
The exponential annual growth rate can be calculated using the continuous growth rate formula:
[tex]\[ r_e = \frac{\ln \left( \frac{P_{2024}}{P_{2012}} \right)}{n} \][/tex]
So:
[tex]\[ r_e = \frac{\ln \left( \frac{14,414,910}{10,840,334} \right)}{12} \][/tex]
[tex]\[ r_e \approx 0.023749 = 2.3749\% \][/tex]
### d) Project the Population in the Year 2050
First, we determine the number of years from 2024 to 2050:
[tex]\[ \text{Years to 2050} = 2050 - 2024 = 26 \][/tex]
Using the geometric growth rate found in part (a), we project the population:
[tex]\[ P_{2050} = P_{2024} \times (1 + r)^{\text{Years to 2050}} \][/tex]
Substituting the values:
[tex]\[ P_{2050} = 14,414,910 \times (1 + 0.024033)^{26} \][/tex]
[tex]\[ P_{2050} \approx 26,728,752.57 \][/tex]
### Summary of Answers
a) Geometric annual growth rate: [tex]\(\approx 2.4033\% \)[/tex]
b) Linear annual growth rate: [tex]\(\approx 297,881.33 \text{ people per year} \)[/tex]
c) Exponential annual growth rate: [tex]\(\approx 2.3749\% \)[/tex]
d) Projected population in 2050: [tex]\(\approx 26,728,752.57 \text{ people} \)[/tex]
This concludes the step-by-step solution for the given question.
### Given Data
- Population in 2012 ([tex]\( P_{2012} \)[/tex]): 10,840,334
- Population in 2024 ([tex]\( P_{2024} \)[/tex]): 14,414,910
- Time period: 2024 - 2012 = 12 years
### a) Geometric Annual Growth Rate
The geometric annual growth rate can be found using the formula:
[tex]\[ r = \left( \frac{P_{2024}}{P_{2012}} \right)^{\frac{1}{n}} - 1 \][/tex]
where [tex]\( n \)[/tex] is the number of years.
Plugging the values in, we get:
[tex]\[ r = \left( \frac{14,414,910}{10,840,334} \right)^{\frac{1}{12}} - 1 \][/tex]
[tex]\[ r \approx 0.024033 = 2.4033\% \][/tex]
### b) Linear Annual Growth Rate
The linear annual growth rate is determined by measuring the total population change and dividing it by the number of years:
[tex]\[ \text{Linear Growth Rate} = \frac{P_{2024} - P_{2012}}{n} \][/tex]
So:
[tex]\[ \text{Linear Growth Rate} = \frac{14,414,910 - 10,840,334}{12} \][/tex]
[tex]\[ \text{Linear Growth Rate} \approx 297,881.33 \][/tex]
### c) Exponential Annual Growth Rate
The exponential annual growth rate can be calculated using the continuous growth rate formula:
[tex]\[ r_e = \frac{\ln \left( \frac{P_{2024}}{P_{2012}} \right)}{n} \][/tex]
So:
[tex]\[ r_e = \frac{\ln \left( \frac{14,414,910}{10,840,334} \right)}{12} \][/tex]
[tex]\[ r_e \approx 0.023749 = 2.3749\% \][/tex]
### d) Project the Population in the Year 2050
First, we determine the number of years from 2024 to 2050:
[tex]\[ \text{Years to 2050} = 2050 - 2024 = 26 \][/tex]
Using the geometric growth rate found in part (a), we project the population:
[tex]\[ P_{2050} = P_{2024} \times (1 + r)^{\text{Years to 2050}} \][/tex]
Substituting the values:
[tex]\[ P_{2050} = 14,414,910 \times (1 + 0.024033)^{26} \][/tex]
[tex]\[ P_{2050} \approx 26,728,752.57 \][/tex]
### Summary of Answers
a) Geometric annual growth rate: [tex]\(\approx 2.4033\% \)[/tex]
b) Linear annual growth rate: [tex]\(\approx 297,881.33 \text{ people per year} \)[/tex]
c) Exponential annual growth rate: [tex]\(\approx 2.3749\% \)[/tex]
d) Projected population in 2050: [tex]\(\approx 26,728,752.57 \text{ people} \)[/tex]
This concludes the step-by-step solution for the given question.
