The formula of growth rate for population in given time is :

Options

A. `dt//dN = rN`
B. `dt//rN = dN`
C. `rN//dN = dt`
D. `dN//dt = rN`

Answer

To derive the formula for the growth rate of a population over a given time, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Population Growth**:
- Population growth can be described mathematically. The change in population over time is represented as \( \frac{dn}{dt} \), where \( n \) is the population size and \( t \) is time.

2. **Defining the Growth Rate**:
- The growth rate of a population can be expressed as \( \frac{dn}{dt} = Rn \), where \( R \) represents the intrinsic growth rate or biotic potential of the population.

3. **Interpreting the Variables**:
- In the equation \( \frac{dn}{dt} = Rn \):
- \( \frac{dn}{dt} \) is the rate of change of the population with respect to time.
- \( R \) is the rate at which the population can grow under ideal conditions.
- \( n \) is the current population size.

4. **Logistic Growth Model**:
- The logistic growth model incorporates the concept of carrying capacity, which limits the growth of the population as resources become scarce. The logistic equation is more complex but starts from the same basic premise of growth rate.

5. **Conclusion**:
- Thus, the formula for the growth rate of a population in a given time is \( \frac{dn}{dt} = Rn \).