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:
with the initial condition .
This is a logistic equation, which typically has the form:
where is the growth rate and is the carrying capacity. In this case:
- The term gives us the value of , since .
Step 1: Finding the Carrying Capacity
We can find using the relation , where :
Step 2: General Solution to the Logistic Equation
The general solution to a logistic differential equation is:
where:
- is the carrying capacity,
- is the initial population ,
- is the growth rate, and
- is the time in years.
We can use this formula to find the predicted population in 1930 ().
Let's calculate this.The carrying capacity for the population is approximately million. The predicted population in 1930, according to the logistic equation, is approximately 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:
- How does the logistic growth model differ from exponential growth?
- What is the significance of the carrying capacity in population growth models?
- How would the population change if the initial population were higher or lower than 3.9 million?
- Can you solve for the time when the population reaches half the carrying capacity?
- How does the value of influence the speed at which the population approaches ?
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)
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