Math Problem Statement

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

Solution

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

dPdt=0.03142P0.0001489P2\frac{dP}{dt} = 0.03142P - 0.0001489P^2 with the initial condition P(0)=3.9P(0) = 3.9.

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

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

where rr is the growth rate and KK is the carrying capacity. In this case:

  • r=0.03142r = 0.03142
  • The term 0.0001489P2-0.0001489P^2 gives us the value of KK, since K=r0.0001489K = \frac{r}{0.0001489}.

Step 1: Finding the Carrying Capacity KK

We can find KK using the relation K=rbK = \frac{r}{b}, where b=0.0001489b = 0.0001489:

K=0.031420.0001489K = \frac{0.03142}{0.0001489}

Step 2: General Solution to the Logistic Equation

The general solution to a logistic differential equation is:

P(t)=K1+(KP01)ertP(t) = \frac{K}{1 + \left( \frac{K}{P_0} - 1 \right)e^{-rt}}

where:

  • KK is the carrying capacity,
  • P0P_0 is the initial population P(0)P(0),
  • rr is the growth rate, and
  • tt is the time in years.

We can use this formula to find the predicted population in 1930 (t=19301790=140t = 1930 - 1790 = 140).

Let's calculate this.The carrying capacity KK for the population is approximately 211.0141211.0141 million. The predicted population in 1930, according to the logistic equation, is approximately 127.67165298577991127.67165298577991 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 KK 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 tt when the population reaches half the carrying capacity?
  5. How does the value of rr influence the speed at which the population approaches KK?

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

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Math Problem Analysis

Mathematical Concepts

Differential Equations
Logistic Growth
Population Dynamics

Formulas

Logistic differential equation: dP/dt = rP(1 - P/K)
General solution to logistic growth: P(t) = K / (1 + [(K/P0) - 1]e^(-rt))
Carrying capacity: K = r / b

Theorems

Logistic Growth Theorem

Suitable Grade Level

College Level (Differential Equations)