Sure, let's break this down step-by-step and answer each part of the question with a detailed explanation.
Given:
- The town's population growth rate is [tex]\(10\%\)[/tex] per year.
- After 2 years, the total population is 36300, including 5800 people added by migration.
We use the following notations:
- [tex]\(P\)[/tex]: Initial population before any growth or migration.
- [tex]\(P_F\)[/tex]: Final population after [tex]\(T\)[/tex] years, including migration.
- [tex]\(R\)[/tex]: Growth rate (10\%).
- [tex]\(T\)[/tex]: Number of years (2 years).
- Migration population: 5800.
### (a) Write the formula to calculate [tex]\(P\)[/tex]
The formula to find the initial population [tex]\(P\)[/tex] after accounting for growth rate and time [tex]\(T\)[/tex] years is given by:
[tex]\[ P \cdot \left(1 + \frac{R}{100}\right)^T = P_F - \text{migration population} \][/tex]
Therefore, the formula for [tex]\(P\)[/tex] would be:
[tex]\[ P = \frac{P_F - \text{migration population}}{\left(1 + \frac{R}{100}\right)^T} \][/tex]
Where:
- [tex]\(P_F\)[/tex] = 36300
- Migration Population = 5800
- [tex]\(R\)[/tex] = 10%
- [tex]\(T\)[/tex] = 2 years
### (b) Write the population after 2 years
The total population after 2 years is given directly in the problem statement.
[tex]\[ P_F = 36300 \][/tex]
### (c) Find the population before 2 years
We will calculate the effective population growth without considering migration. Here’s the step-by-step computation:
1. Final population without migration:
[tex]\[ P_F - \text{migration population} = 36300 - 5800 \][/tex]
[tex]\[ P_{without_migration} = 30500 \][/tex]
2. Initial population [tex]\(P\)[/tex] before 2 years:
Using our formula to find the initial population:
[tex]\[ P = \frac{30500}{(1 + 0.10)^2} \][/tex]
The result is:
[tex]\[ P = 25206.61157024793 \][/tex]
So, the initial population before 2 years is approximately:
[tex]\[ P \approx 25206.61 \][/tex]
### (d) Determine the expected population after 4 years
To find the population after 4 years, we use the initial population found in part (c):
[tex]\[ P_{after\_4\_years} = P \cdot \left(1 + \frac{R}{100}\right)^4 \][/tex]
Substitute the initial population [tex]\(P\)[/tex] and growth rate:
[tex]\[ P_{after\_4\_years} = 25206.61157024793 \cdot (1 + 0.10)^4 \][/tex]
The result is:
[tex]\[ P_{after\_4\_years} = 36905.0 \][/tex]
### Summary:
(a) Formula to calculate [tex]\(P\)[/tex]:
[tex]\[ P = \frac{P_F - \text{migration population}}{\left(1 + \frac{R}{100}\right)^T} \][/tex]
(b) Population after 2 years:
[tex]\[ 36300 \][/tex]
(c) Initial population before 2 years:
[tex]\[ 25206.61157024793 \][/tex]
(d) Population after 4 years:
[tex]\[ 36905.0 \][/tex]
