The problem involves solving the differential equation given for a population model. The differential equation is:

$\frac{dP}{dt} = 0.5 \cdot \frac{P}{500 - P}$

Where $P$ is the population at time $t$, and $t$ is measured in years. The question asks for the range of values of $P$ for which the population is increasing, which corresponds to when $\frac{dP}{dt} > 0$.

Step 1: Understand the model

The differential equation describes the rate of change of the population, and it depends on the current population $P$. The population increases when the rate of change, $\frac{dP}{dt}$, is positive, and decreases when it is negative.

Step 2: Find conditions for population increase

For population growth to occur, we need the rate of change to be positive:

$\frac{dP}{dt} > 0$

This gives the inequality:

$0.5 \cdot \frac{P}{500 - P} > 0$

Since 0.5 is a positive constant, we can safely ignore it when determining the inequality. Therefore, we have:

$\frac{P}{500 - P} > 0$

Step 3: Analyze the inequality

The expression $\frac{P}{500 - P}$ is positive when both the numerator and the denominator are either both positive or both negative.

• The numerator $P$ is positive when $P > 0$.
• The denominator $500 - P$ is positive when $P < 500$.

Thus, $\frac{P}{500 - P} > 0$ when $0 < P < 500$.

Step 4: Conclusion

The population is increasing when $P$ is between 0 and 500. Therefore, the correct answer is:

(A) $0 < P < 500$.

Would you like further details on the solution or have any other questions about this topic?

Here are some related questions for further exploration:

1. What would happen if the population exceeds 500?
2. How does the differential equation change if the rate of growth is modified?
3. What is the long-term behavior of the population based on this model?
4. How can you solve this type of differential equation using separation of variables?
5. What does the constant 0.5 represent in the population growth model?

Tip: Understanding the direction of the differential equation's sign can help in determining the nature of the population's change—whether it's increasing or decreasing.