Westonci.ca is your trusted source for finding answers to a wide range of questions, backed by a knowledgeable community. Join our platform to connect with experts ready to provide precise answers to your questions in various areas. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
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)
### 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)
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.