Certainly! Let's solve this step-by-step using the exponential growth model for population growth.
### Given Data
- Initial population, [tex]\( P_0 = 8 \)[/tex]
- Growth rate, [tex]\( r = 0.25 \)[/tex]
### Step-by-Step Solution
#### 1. Calculating [tex]\( P_1 \)[/tex]:
The population after one time period ([tex]\( n = 1 \)[/tex]) can be calculated using the formula:
[tex]\[ P_1 = P_0 \times (1 + r) \][/tex]
Substituting the given values:
[tex]\[ P_1 = 8 \times (1 + 0.25) \][/tex]
[tex]\[ P_1 = 8 \times 1.25 \][/tex]
[tex]\[ P_1 = 10.0 \][/tex]
So, [tex]\[ P_1 = 10.0 \][/tex]
#### 2. Calculating [tex]\( P_2 \)[/tex]:
The population after two time periods ([tex]\( n = 2 \)[/tex]) can be calculated by using the population at [tex]\( P_1 \)[/tex] and applying the growth rate again:
[tex]\[ P_2 = P_1 \times (1 + r) \][/tex]
Substituting the previously calculated [tex]\( P_1 \)[/tex]:
[tex]\[ P_2 = 10.0 \times 1.25 \][/tex]
[tex]\[ P_2 = 12.5 \][/tex]
So, [tex]\[ P_2 = 12.5 \][/tex]
#### 3. Finding the Explicit Formula for [tex]\( P_n \)[/tex]:
In general, the population after [tex]\( n \)[/tex] time periods can be expressed as:
[tex]\[ P_n = P_0 \times (1 + r)^n \][/tex]
Substituting the given values [tex]\( P_0 = 8 \)[/tex] and [tex]\( r = 0.25 \)[/tex]:
[tex]\[ P_n = 8 \times (1 + 0.25)^n \][/tex]
[tex]\[ P_n = 8 \times 1.25^n \][/tex]
So, the explicit formula is:
[tex]\[ P_n = 8 \times 1.25^n \][/tex]
#### 4. Calculating [tex]\( P_{11} \)[/tex]:
Now, let's use the formula to calculate the population after 11 time periods ([tex]\( n = 11 \)[/tex]):
[tex]\[ P_{11} = 8 \times 1.25^{11} \][/tex]
Calculating the value:
[tex]\[ P_{11} = 8 \times 93.13225746154785 \][/tex]
So, [tex]\[ P_{11} = 93.13225746154785 \][/tex]
### Summary
- [tex]\( P_1 = 10.0 \)[/tex]
- [tex]\( P_2 = 12.5 \)[/tex]
- Explicit formula: [tex]\( P_n = 8 \times 1.25^n \)[/tex]
- [tex]\( P_{11} = 93.13225746154785 \)[/tex]
