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The given differential equation modeling population growth is:

\[
\frac{dP}{dt} = 1.1P\left(1 - \frac{P}{4100}\right)
\]

This is a logistic growth model, where the growth rate depends on the current population \(P\) and its carrying capacity (here, 4100). Now let's address each part:

### (a) For what values of \(P\) is the population increasing?
The population is increasing when \(\frac{dP}{dt} > 0\). For the given equation:

\[
1.1P\left(1 - \frac{P}{4100}\right) > 0
\]

This inequality holds when:
1. \(P > 0\) (since \(P\) represents population, it must be non-negative)
2. \(1 - \frac{P}{4100} > 0\) which implies \(P < 4100\)

So the population is increasing when \(0 < P < 4100\).

**Answer**: \( P \in (0, 4100) \)

### (b) For what values of \(P\) is the population decreasing?
The population decreases when \(\frac{dP}{dt} < 0\), i.e.,

\[
1.1P\left(1 - \frac{P}{4100}\right) < 0
\]

This happens if \(P > 4100\), because the term \(1 - \frac{P}{4100}\) will become negative. Additionally, \(P = 0\) results in no population, hence \(\frac{dP}{dt} = 0\), which is not increasing but stable.

**Answer**: \( P \in (4100, \infty) \)

### (c) What are the equilibrium solutions?
Equilibrium occurs when \(\frac{dP}{dt} = 0\). Setting the equation equal to zero:

\[
1.1P\left(1 - \frac{P}{4100}\right) = 0
\]

This implies either \(P = 0\) or \(1 - \frac{P}{4100} = 0\), leading to \(P = 4100\).

**Answer**: \( P = 0, 4100 \)

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Do you want further details or have any questions?

Here are 5 related questions you might explore:
1. How does the carrying capacity influence the population dynamics in a logistic model?
2. What happens to the population when it exceeds the carrying capacity?
3. Can the population ever stabilize at a value other than 0 or the carrying capacity?
4. How would the dynamics change if the growth rate were less than 1.1?
5. Could an external factor, like predation, be modeled by modifying this equation?

**Tip**: In logistic growth models, the population grows fastest when it is at half of the carrying capacity.

A population is modeled by the differential equation: dP/dt = 1.1P(1 - P/4100). For what values of P is the population increasing? For what values of P is the population decreasing? What are the equilibrium solutions?

Explore the logistic growth model with the differential equation dP/dt = 1.1P(1 - P/4100). This problem covers population increase, decrease, and equilibrium solutions, applying key concepts in population dynamics. Suitable for Grades 11-12 and college students.

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