During the period from 1790 to 1930​, a​ country's population​ P(t) (t in​ years) grew from 3.9 million to 121.1 million. Throughout this​ period, P(t) remained close to the solution of the initial value problem (dP/dt) = 0.03142P - 0.0001489P^2. P(0) = 3.9.

What 1930 population does this logistic equation​ predict?

calculate without approximating anything, also the final answer should not be rounded, leave as long as it is

We are tasked with predicting the population in 1930 using the logistic differential equation:

\[
\frac{dP}{dt} = 0.03142P - 0.0001489P^2
\]
with the initial condition \( P(0) = 3.9 \).

This is a logistic equation, which typically has the form:

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

where \( r \) is the growth rate and \( K \) is the carrying capacity. In this case:

- \( r = 0.03142 \)
- The term \( -0.0001489P^2 \) gives us the value of \( K \), since \( K = \frac{r}{0.0001489} \).

### Step 1: Finding the Carrying Capacity \( K \)
We can find \( K \) using the relation \( K = \frac{r}{b} \), where \( b = 0.0001489 \):

\[
K = \frac{0.03142}{0.0001489}
\]

### Step 2: General Solution to the Logistic Equation
The general solution to a logistic differential equation is:

\[
P(t) = \frac{K}{1 + \left( \frac{K}{P_0} - 1 \right)e^{-rt}}
\]

where:
- \( K \) is the carrying capacity,
- \( P_0 \) is the initial population \( P(0) \),
- \( r \) is the growth rate, and
- \( t \) is the time in years.

We can use this formula to find the predicted population in 1930 (\( t = 1930 - 1790 = 140 \)).

Let's calculate this.The carrying capacity \( K \) for the population is approximately \( 211.0141 \) million. The predicted population in 1930, according to the logistic equation, is approximately \( 127.67165298577991 \) million. This result is not rounded, as requested.

If you need further clarifications or details, feel free to ask!

Here are 5 related questions to deepen your understanding:
1. How does the logistic growth model differ from exponential growth?
2. What is the significance of the carrying capacity \( K \) in population growth models?
3. How would the population change if the initial population were higher or lower than 3.9 million?
4. Can you solve for the time \( t \) when the population reaches half the carrying capacity?
5. How does the value of \( r \) influence the speed at which the population approaches \( K \)?

**Tip:** The logistic growth model is often used in ecology to model populations with limited resources, as it naturally accounts for a saturation point.

During the period from 1790 to 1930​, a​ country's population​ P(t) (t in​ years) grew from 3.9 million to 121.1 million. Throughout this​ period, P(t) remained close to the solution of the initial value problem (dP/dt) = 0.03142P - 0.0001489P^2. P(0) = 3.9. What 1930 population does this logistic equation​ predict? Calculate without approximating anything, also the final answer should not be rounded, leave as long as it is.

Learn how to predict a country's population in 1930 using the logistic differential equation (dP/dt = 0.03142P - 0.0001489P^2) with an initial population of 3.9 million in 1790. This problem illustrates logistic growth, carrying capacity, and the dynamics of population models. Suitable for college-level students studying differential equations.

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