To determine the population of the town after a certain number of years given an initial population, an annual growth rate, and a specific period, you can use the formula for compound growth. Here’s a step-by-step detailed explanation:
1. Understand the Variables:
- [tex]\( X \)[/tex] represents the present population of the town.
- [tex]\( R \)[/tex] is the annual growth rate in percentage.
- [tex]\( Y \)[/tex] is the number of years over which the population grows.
2. Convert the Growth Rate:
- Since the growth rate is given in percentage, you need to convert it to a decimal form for calculations.
- The decimal form [tex]\( r \)[/tex] of the growth rate [tex]\( R \)[/tex] is calculated as:
[tex]\[ r = \frac{R}{100} \][/tex]
3. Population Growth Formula:
- The formula for calculating the future population after a certain number of years with compound growth is:
[tex]\[ P = X \times (1 + r)^Y \][/tex]
- Here, [tex]\( P \)[/tex] is the population after [tex]\( Y \)[/tex] years.
4. Substitute the Given Values:
- Let’s substitute the values:
- [tex]\( X = 1000 \)[/tex] (initial population)
- [tex]\( R = 5 \% \)[/tex] (annual growth rate)
- [tex]\( Y = 10 \)[/tex] years (time period)
5. Convert the Growth Rate:
- Convert [tex]\( R \% \)[/tex] to a decimal:
[tex]\[ r = \frac{5}{100} = 0.05 \][/tex]
6. Calculate the Future Population:
- Using the formula, the future population [tex]\( P \)[/tex] is calculated as:
[tex]\[ P = 1000 \times (1 + 0.05)^{10} \][/tex]
- This simplifies to:
[tex]\[ P = 1000 \times (1.05)^{10} \][/tex]
7. Exponent Calculation:
- Now, calculate [tex]\( (1.05)^{10} \)[/tex]:
[tex]\[ (1.05)^{10} \approx 1.628894626777442 \][/tex]
8. Multiply by the Initial Population:
- Finally, multiply this result by the initial population [tex]\( X = 1000 \)[/tex]:
[tex]\[ P \approx 1000 \times 1.628894626777442 \][/tex]
[tex]\[ P \approx 1628.894626777442 \][/tex]
Therefore, the population of the town after 10 years will be approximately 1628.894626777442 individuals.
