The integral form of the exponential growth equation as `N_t=N_0e^(rt)`
A. population density after time t
B. population density at time zero
C. intrinsic rate of natural increase
D. the base of natural logarithms (2.71828)
Identify A,B,C and D fromt eh given equation.

Options

A. A-r,B-e,C-No,D-`N_t`
B. `A-N_t,B-N_0,C-r,D-e`
C. `A-N_0,B-N_t,C-r,D-e`
D. `A-N_0,B-N_t,C-e,D-r`

Answer

To identify A, B, C, and D from the given exponential growth equation \( N_t = N_0 e^{rt} \), we can break down the equation and analyze each component step by step.

### Step-by-Step Solution:

1. **Understanding the Equation**:
The equation \( N_t = N_0 e^{rt} \) represents the exponential growth of a population over time. Here, \( N_t \) is the population size at time \( t \), \( N_0 \) is the initial population size, \( r \) is the intrinsic rate of natural increase, and \( e \) is the base of natural logarithms.

2. **Identifying A**:
- \( N_t \) represents the population density after time \( t \).
- Therefore, **A = \( N_t \)**.

3. **Identifying B**:
- \( N_0 \) represents the population density at time zero (the initial population size).
- Therefore, **B = \( N_0 \)**.

4. **Identifying C**:
- The intrinsic rate of natural increase is represented by \( r \) in the equation.
- Therefore, **C = \( r \)**.

5. **Identifying D**:
- The base of natural logarithms is denoted by \( e \), which is approximately 2.71828.
- Therefore, **D = \( e \)**.

### Summary of Identifications:
- **A = \( N_t \)** (population density after time \( t \))
- **B = \( N_0 \)** (population density at time zero)
- **C = \( r \)** (intrinsic rate of natural increase)
- **D = \( e \)** (the base of natural logarithms)

### Conclusion:
Based on the identifications, the correct option is the second one, which states:
- A is \( N_t \)
- B is \( N_0 \)
- C is \( r \)
- D is \( e \)