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.

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.