kumalsamir89
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The population of a town increases every year by [tex]$10\%$[/tex]. At the end of two years, the total population of the town was given. It includes 5800 people who were added by migration.

(a) If the initial population is [tex]$P$[/tex] and the population after [tex]$T$[/tex] years is [tex]$P_T$[/tex], write the formula to calculate [tex]$P$[/tex].

[tex]\[ P_T = P \left(1 + \frac{R}{100}\right)^T \][/tex]

(b) Write the population after 2 years.

(c) Find the population before 2 years.

(d) Determine the simple population growth in four years.

Ans:
(a) [tex]\[ P_T = P \left(1 + \frac{R}{100}\right)^T \][/tex]
(b) 36300
(c)
(d)

Sagot :

GinnyAnswer
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]
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