Which one of the following populations will show the greatest population increase in the next year, assuming exponential growth?

Options

A. Population A, with 20,000 individuals when r= 0.020
B. Population B, with 500,000 individuals when r=0.040
C. Population C, with 2 million individuals when r=0.002
D. Population D, with 30 million individuals when r=0.001

Answer

To determine which population will show the greatest increase in the next year under the assumption of exponential growth, we need to consider the following steps:

### Step-by-Step Solution:

1. **Understand Exponential Growth**:
Exponential growth occurs when the growth rate of a population is proportional to its current size. This means that as the population increases, the growth rate also increases.

2. **Identify Key Factors**:
The key factors that influence exponential growth are:
- The initial population size (N)
- The growth rate (r)

3. **Exponential Growth Formula**:
The formula for exponential growth can be expressed as:
\[ N(t) = N_0 \times e^{rt} \]
Where:
- \(N(t)\) is the population at time t
- \(N_0\) is the initial population size
- \(r\) is the intrinsic growth rate
- \(e\) is the base of the natural logarithm
- \(t\) is time

4. **Compare the Given Options**:
To find out which population will show the greatest increase, we need to look at the initial population sizes and growth rates of each option provided (Option 1, Option 2, Option 3, and Option 4).

5. **Calculate the Expected Increase**:
For each option, calculate the expected population increase using the exponential growth formula. This will involve plugging in the values of \(N_0\) and \(r\) for each option.

6. **Identify the Greatest Increase**:
After calculating the expected population for each option, compare the results to identify which one shows the greatest increase.

### Conclusion:
The option with the highest calculated population increase will be the one that shows the greatest population increase in the next year assuming exponential growth.

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