During the period from 1790 to 1930​, a​ country's population​ P(t) (t in​ years) grew from 4.1 million to 121.2 million. Throughout this​ period, P(t) remained close to the solution of the initial value problem (dP/dt) = 0.03127P - 0.0001485P^2. P(0) = 4.1.

What 1930 population does this logistic equation​ predict?

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

\[
\frac{dP}{dt} = 0.03127P - 0.0001485P^2
\]

with the initial condition \( P(0) = 4.1 \) million, where \( t = 0 \) corresponds to the year 1790.

The logistic equation 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 of the population. From the given equation, we can identify:

\[
r = 0.03127, \quad \text{and} \quad \frac{r}{K} = 0.0001485
\]

To find the carrying capacity \( K \), we solve for \( K \):

\[
K = \frac{r}{0.0001485} = \frac{0.03127}{0.0001485}
\]

I'll calculate this first, then we can use the solution for the logistic growth equation to find the population in 1930 (corresponding to \( t = 1930 - 1790 = 140 \) years).The carrying capacity \( K \) is approximately \( 210.57 \) million.

Next, we solve the logistic growth equation. The general solution for a logistic equation of the form:

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

is given by:

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

where \( P_0 = 4.1 \) million is the initial population at \( t = 0 \), and \( K \) is the carrying capacity. Using this, we will compute \( P(140) \), which corresponds to the population in 1930.The logistic equation predicts a population of approximately \( 129.01 \) million in 1930.

Would you like more details or have any further questions?

Here are some related questions:
1. How does the logistic growth model account for population limits?
2. What is the significance of the carrying capacity \( K \) in population growth?
3. How can you derive the general solution of the logistic growth equation?
4. What factors might cause deviations from the logistic model in real-world population growth?
5. How would the population growth change if the growth rate \( r \) were different?

**Tip:** The logistic growth model is useful for scenarios where growth is initially exponential but slows as the population approaches a maximum limit (carrying capacity).

During the period from 1790 to 1930​, a​ country's population​ P(t) (t in​ years) grew from 4.1 million to 121.2 million. Throughout this​ period, P(t) remained close to the solution of the initial value problem (dP/dt) = 0.03127P - 0.0001485P^2. P(0) = 4.1. What 1930 population does this logistic equation​ predict?

Explore how to predict population in 1930 using the logistic growth equation, based on the given initial value problem (dP/dt = 0.03127P - 0.0001485P^2) with P(0) = 4.1. Learn about the logistic growth model, carrying capacity, and the solution for population dynamics. Suitable for university-level students.

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