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).