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A population grows according to an exponential growth model. The initial population is [tex]P_0 = 3[/tex], and the growth rate is [tex]r = 0.45[/tex].

Then:
[tex]\[
\begin{array}{l}
P_1 = \square \\
P_2 = \square
\end{array}
\][/tex]

Find an explicit formula for [tex]P_n[/tex]. Your formula should involve [tex]n[/tex].
[tex]\[
P_n = \square
\][/tex]

Use your formula to find [tex]P_{11}[/tex].
[tex]\[
P_{11} = \square
\][/tex]

Give all answers accurate to at least one decimal place.


Sagot :

To solve the problem of finding the population at various points in time given an exponential growth model, let's go step-by-step.

### Step-by-Step Solution:

1. Initial Population and Growth Rate
- Initial population, [tex]\( P_0 = 3 \)[/tex]
- Growth rate, [tex]\( r = 0.45 \)[/tex]

2. Exponential Growth Model Formula
In an exponential growth model, the population at any time [tex]\( n \)[/tex] can be determined using the formula:

[tex]\[ P_n = P_0 \times (1 + r)^n \][/tex]

Therefore, the explicit formula for [tex]\( P_n \)[/tex] is:

[tex]\[ P_n = 3 \times (1 + 0.45)^n \][/tex]

3. Calculating [tex]\( P_1 \)[/tex]

To find the population at [tex]\( n = 1 \)[/tex]:

[tex]\[ P_1 = 3 \times (1 + 0.45)^1 \][/tex]

Simplifying the expression inside the parentheses:

[tex]\[ P_1 = 3 \times 1.45 \][/tex]

Multiplying the numbers together:

[tex]\[ P_1 = 4.35 \][/tex]

4. Calculating [tex]\( P_2 \)[/tex]

To find the population at [tex]\( n = 2 \)[/tex]:

[tex]\[ P_2 = 3 \times (1 + 0.45)^2 \][/tex]

Simplifying the expression inside the parentheses:

[tex]\[ P_2 = 3 \times 1.45^2 \][/tex]

Calculating [tex]\( 1.45^2 \)[/tex]:

[tex]\[ P_2 = 3 \times 2.1025 \][/tex]

Multiplying the numbers together:

[tex]\[ P_2 = 6.3075 \][/tex]

5. Calculating [tex]\( P_{11} \)[/tex]

To find the population at [tex]\( n = 11 \)[/tex]:

[tex]\[ P_{11} = 3 \times (1 + 0.45)^{11} \][/tex]

Simplifying the expression inside the parentheses:

[tex]\[ P_{11} = 3 \times 1.45^{11} \][/tex]

Using the exponentiation:

[tex]\[ P_{11} = 3 \times 59.57280159036047 \][/tex]

Multiplying the numbers together:

[tex]\[ P_{11} = 178.7184047710814 \][/tex]

6. Final Results

[tex]\[ \begin{array}{l} P_1 = 4.35 \\ P_2 = 6.3075 \end{array} \][/tex]

The explicit formula for [tex]\( P_n \)[/tex]:

[tex]\[ P_n = 3 \times (1.45)^n \][/tex]

Finding [tex]\( P_{11} \)[/tex]:

[tex]\[ P_{11} = 178.7 \][/tex]

Thus, summarizing the results:

- [tex]\( P_1 = 4.35 \)[/tex]
- [tex]\( P_2 = 6.3075 \)[/tex]
- The explicit formula for [tex]\( P_n \)[/tex]: [tex]\( P_n = 3 \times (1.45)^n \)[/tex]
- [tex]\( P_{11} = 178.7 \)[/tex] (rounded to one decimal place)