Answer :
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)
### 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)
