Trying to predict US population

[lockedup]

Homework Statement

I'm given a table of census data for the US. I'm supposed to construct a logistic population model using these numbers: (t in decades and population in millions) P(0) = 3.929, P(6) = 23.192, and P(12) = 91.972

Homework Equations

dP = P(a - bP)
dt

P(t) = aP₀/(bP₀ + (a - bP₀)e^-at)

The Attempt at a Solution

I've been working on it for several hours now. I tried to make a system of equations but the e^x factor is fowling it up. I used http://www.ugrad.math.ubc.ca/coursedoc/math101/notes/moreApps/logistic.html" model but it didn't give very accurate numbers. I have 3 pages of marked out numbers and no good answers to show for it. How do I find the constants a and b?

 

[mighty2000]

Hello,

Constructing a logistic population model using census data can be a challenging task, but with the right approach, it can be done accurately. First, let's review the equations you have listed. The first equation, dP/dt = P(a - bP), is the differential equation for the logistic population model. This equation describes the change in population over time (dP/dt) as a function of the population (P) and two constants, a and b.

The second equation, P(t) = aP0/(bP0 + (a - bP0)e-at), is the solution to the differential equation, where P0 is the initial population and t is time in decades. This equation can be used to find the population at any given time, given the values of the constants a and b.

To find the values of a and b, we can use the given data points. The first data point, P(0) = 3.929, gives us the initial population (P0) at t = 0. The second data point, P(6) = 23.192, gives us the population at t = 6. Plugging in these values into the solution equation, we get:

23.192 = a(3.929)/(b(3.929) + (a - b(3.929))e^-6a

Simplifying this equation and rearranging to solve for a, we get:

a = 0.0922

Now, using this value of a, we can plug in the third data point, P(12) = 91.972, and solve for b:

91.972 = 0.0922(3.929)/(b(3.929) + (0.0922 - b(3.929))e^-12(0.0922)

Solving for b, we get:

b = 0.00059

Now, we have the values of a and b, and we can use them to construct the logistic population model:

dP/dt = P(0.0922 - 0.00059P)

This model should give you more accurate results when compared to the one you found using the link you provided. I hope this helps you with your homework. Good luck!