In a population of 2,500 whose intrinsic rate of natural increase (r) is 0.2 per capita per year is exhibiting exponential growth. If the number of individuals at a given moment is 2,000 what will the population size after one year?

Options

A. 2500
B. 2300
C. 2600
D. 2400

Answer

To solve the problem of predicting the population size after one year given the intrinsic rate of natural increase and the current population size, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Current population size (N₀) = 2000
- Intrinsic rate of natural increase (r) = 0.2 per capita per year
- Time period (t) = 1 year

2. **Use the Exponential Growth Formula**:
The formula for exponential growth is given by:
\[ N(t) = N_0 \times e^{(r \times t)} \]
where:
- \(N(t)\) = population size at time t
- \(N_0\) = initial population size
- \(r\) = intrinsic rate of increase
- \(t\) = time in years
- \(e\) = base of the natural logarithm (approximately equal to 2.71828)

3. **Substitute the Values into the Formula**:
Plugging in the values we have:
\[ N(1) = 2000 \times e^{(0.2 \times 1)} \]

4. **Calculate the Exponential Factor**:
First, calculate \(0.2 \times 1 = 0.2\). Now calculate \(e^{0.2}\):
\[ e^{0.2} \approx 1.2214 \]

5. **Calculate the Future Population Size**:
Now substitute \(e^{0.2}\) back into the equation:
\[ N(1) = 2000 \times 1.2214 \approx 2442.8 \]
Since population size must be a whole number, we round it to 2443.

6. **Conclusion**:
The population size after one year will be approximately 2443. However, since the options provided in the question do not include this number, we can see that the closest option is 2400, which is likely a rounding or estimation based on the context of the question.

### Final Answer:
The population size after one year is approximately 2400.