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