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. 448
B. 1212
C. 739
D. 2012

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 females per year, we will use the formula for exponential growth:

\[ N_t = N_0 \cdot e^{rt} \]

Where:
- \( N_t \) = population size after time \( t \)
- \( N_0 \) = initial population size
- \( r \) = intrinsic growth rate
- \( t \) = time in years
- \( e \) = base of the natural logarithm (approximately 2.72)

### Step-by-Step Solution:

1. **Identify the Variables:**
- Initial population size (\( N_0 \)) = 100 individuals
- Intrinsic growth rate (\( r \)) = 0.5 females/year
- Time (\( t \)) = 6 years
- Base of the natural logarithm (\( e \)) = 2.72

2. **Plug the Values into the Formula:**
\[ N_t = 100 \cdot e^{(0.5 \cdot 6)} \]

3. **Calculate the Exponent:**
- First, calculate \( 0.5 \cdot 6 = 3 \).
- So, the equation now becomes:
\[ N_t = 100 \cdot e^{3} \]

4. **Calculate \( e^3 \):**
- Using the approximate value of \( e \):
\[ e^3 \approx 2.72^3 \approx 20.0855 \]

5. **Calculate the Population Size:**
\[ N_t = 100 \cdot 20.0855 \approx 2008.55 \]
- Rounding this to the nearest whole number gives us approximately 2009.

6. **Final Population Size:**
- Therefore, the population size after 6 years is approximately **2009 individuals**.

### Summary of Calculation:
- \( N_t = 100 \cdot e^{(0.5 \cdot 6)} \)
- \( N_t = 100 \cdot e^3 \)
- \( N_t \approx 100 \cdot 20.0855 \)
- \( N_t \approx 2009 \)