Sure, let's derive the relationship between the initial population [tex]\( P \)[/tex], the population after [tex]\( T \)[/tex] years [tex]\( P_T \)[/tex], the number of years [tex]\( T \)[/tex], and the growth rate [tex]\( R \% \)[/tex].
To solve this, we can refer to the formula for compound interest, which is generally used to model population growth. The compound interest formula can be adapted to calculate the population growth over time.
1. Identifying the Variables:
- [tex]\( P \)[/tex]: Initial population
- [tex]\( P_T \)[/tex]: Population after [tex]\( T \)[/tex] years
- [tex]\( T \)[/tex]: Number of years
- [tex]\( R \)[/tex]: Growth rate (expressed as a percentage)
2. Formula Setup:
The compound interest formula that fits this scenario is:
[tex]\[ P_T = P \times \left(1 + \frac{R}{100}\right)^T \][/tex]
Here, [tex]\( \left(1 + \frac{R}{100}\right)^T \)[/tex] represents the growth factor after [tex]\( T \)[/tex] years, where the rate [tex]\( R \% \)[/tex] is compounded annually.
3. Step-by-Step Derivation:
- In year 1, the population becomes:
[tex]\[ P_1 = P \times \left(1 + \frac{R}{100}\right) \][/tex]
- In year 2, the population becomes:
[tex]\[ P_2 = P_1 \times \left(1 + \frac{R}{100}\right) = P \times \left(1 + \frac{R}{100}\right) \times \left(1 + \frac{R}{100}\right) = P \times \left(1 + \frac{R}{100}\right)^2 \][/tex]
- This pattern continues, and after [tex]\( T \)[/tex] years, the population is:
[tex]\[ P_T = P \times \left(1 + \frac{R}{100}\right)^T \][/tex]
4. Conclusion:
The relation between the initial population [tex]\( P \)[/tex], the population after [tex]\( T \)[/tex] years [tex]\( P_T \)[/tex], and the growth rate [tex]\( R \% \)[/tex] over [tex]\( T \)[/tex] years is:
[tex]\[ P_T = P \times \left(1 + \frac{R}{100}\right)^T \][/tex]
Therefore, the equation that represents the relationship is:
[tex]\[ \boxed{P_T = P \left(\frac{R}{100} + 1\right)^T} \][/tex]
