Question 1.
The Rwanda population was estimated at 10840334 at the 2012 cen
14414910 at the 2024 census.
a) Calculate the geometric annual growth rate.
b) Compute the linear annual growth rate
c) Calculate the exponential annual growth rate.
d) Project the population of Rwanda in the year 2050.



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.