What does 'K' represent in the equation given below?
`(dN)/(dt)=rN((K-N)/(K))`

Options

A. 1)Population density at time 't'
B. 2)Intrinsic rate of natural increase
C. 3)Carrying capacity
D. 4)The base of natural logarithm

Answer

To solve the question, we need to analyze the equation provided and identify what 'K' represents in the context of the logistic population growth model.

### Step-by-Step Solution:

1. **Understand the Equation**: The equation given is `(dN)/(dt) = rN((K-N)/(K))`. This is a differential equation that describes how a population changes over time.

2. **Identify the Variables**:
- **N**: This represents the population density at time 't'.
- **r**: This is the intrinsic rate of natural increase of the population.
- **K**: This is the variable we need to identify.

3. **Analyze the Role of K**: In the context of the logistic growth model, 'K' is known as the carrying capacity. The carrying capacity is the maximum population size that an environment can sustain indefinitely without being degraded.

4. **Eliminate Incorrect Options**:
- **Option 1**: Population density at time t is represented by 'N', not 'K'.
- **Option 2**: The intrinsic rate of natural increase is represented by 'r', not 'K'.
- **Option 4**: The base of the natural logarithm is 'e', not 'K'.

5. **Conclusion**: Since 'K' represents the carrying capacity in the logistic growth model, the correct answer is that 'K' represents the carrying capacity.

### Final Answer:
**K represents the carrying capacity.**

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