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The problem provides a differential equation model for the population \( P(t) \) of a suburb of a large city:

\[
\frac{dP}{dt} = P(10^{-1} - 10^{-7}P)
\]
with the initial condition \( P(0) = 4000 \), where \( t \) is measured in months.

### Step-by-step explanation:

1. **Find the population \( P(t) \) at any time \( t \):**

   The solution to the given initial-value problem is a separable differential equation. The general form of the solution given is:
   
   \[
   P(t) = \frac{400}{0.0004 + 0.0996e^{-0.1t}}
   \]

   This expression is the population as a function of time \( t \), which models how the population evolves over time based on the parameters in the equation.

2. **What is the limiting value of the population?**

   The limiting value of the population refers to what happens as \( t \to \infty \), which is when the population stabilizes.

   As \( t \to \infty \), the exponential term \( e^{-0.1t} \) tends to 0. Therefore, the equation for \( P(t) \) becomes:

   \[
   P(t) \to \frac{400}{0.0004} = 1,000,000
   \]

   So, the limiting population value is \( 1,000,000 \).

3. **At what time will the population be equal to one-half of this limiting value?**

   To find when the population reaches half of the limiting value (which is \( 500,000 \)), we set \( P(t) = 500,000 \) and solve for \( t \) in the equation:

   \[
   500,000 = \frac{400}{0.0004 + 0.0996e^{-0.1t}}
   \]

   Simplifying the equation step-by-step, solving for \( t \), leads to the answer:

   \[
   t \approx 55.2 \text{ months}
   \]

Thus, the population will reach half of the limiting value in approximately 55.2 months.

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### Would you like more details on any of the steps? Here are some questions to deepen understanding:

1. What is the process for solving separable differential equations?
2. How does the exponential decay function \( e^{-0.1t} \) affect the population model?
3. Why does the population stabilize at \( 1,000,000 \) as \( t \to \infty \)?
4. Can the model be used to predict when the population will be close to the limiting value (e.g., 90%)?
5. What would happen to the limiting population if the parameters in the equation changed?

**Tip**: When working with exponential models, remember that \( e^{-kt} \) quickly approaches zero as \( t \) increases, making it useful for modeling growth and decay in populations.

Explore the solution to a population model using a differential equation, including finding the limiting value and the time required to reach half the population limit. This detailed explanation covers separable differential equations, exponential decay, and limiting values.

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