In a hypothetical population of 100 individual having 'r' = 0.5/ female/ year, what will be the population size in 6 years (with e = 2.72) showing exponential rate of growth?

Options

A. 1218
B. 739
C. 2012
D. 448

Answer

To solve the problem of determining the population size in 6 years for a hypothetical population of 100 individuals with an intrinsic growth rate (r) of 0.5 per female per year, we will use the exponential growth formula:

### Step-by-Step Solution:

1. **Identify the Variables**:
- Initial population size (N0) = 100
- Intrinsic growth rate (r) = 0.5
- Time (T) = 6 years
- Base of the natural logarithm (e) = 2.72

2. **Use the Exponential Growth Formula**:
The formula for exponential growth is given by:
\[ N_t = N_0 \cdot e^{(r \cdot T)} \]
where:
- \(N_t\) = population size at time T
- \(N_0\) = initial population size
- \(e\) = base of the natural logarithm
- \(r\) = intrinsic growth rate
- \(T\) = time in years

3. **Plug in the Values**:
Substitute the known values into the formula:
\[ N_t = 100 \cdot 2.72^{(0.5 \cdot 6)} \]

4. **Calculate the Exponent**:
First, calculate \(r \cdot T\):
\[ r \cdot T = 0.5 \cdot 6 = 3 \]

5. **Calculate the Exponential Term**:
Now calculate \(2.72^3\):
\[ N_t = 100 \cdot 2.72^3 \]

6. **Calculate \(2.72^3\)**:
Using a calculator or performing the multiplication:
\[ 2.72^3 \approx 20.0855 \]

7. **Final Calculation**:
Now multiply by the initial population size:
\[ N_t = 100 \cdot 20.0855 \approx 2008.55 \]
Rounding this to the nearest whole number gives:
\[ N_t \approx 2009 \]

8. **Conclusion**:
The population size after 6 years will be approximately **2009** individuals.