Discover a wealth of knowledge at Westonci.ca, where experts provide answers to your most pressing questions. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
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]
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]
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.
