Logistic Math Probelm

Question

The gross domestic product (GDP) of the United States has shown logistic growth from 1970 through 1992. The gross domestic product G (in billions of dollars) can be modeled by the equation

G = 9200/1+8.03e^{- 0.121t}

where t is the number of years since 1970.

1. When did the GDP reach its point of Maximum growth?

Answer 1

The brute force way is to take the derivative twice and set the second derivative to zero. This will find the time of maximum growth which happens at 9200/2 (Half of the carrying capacity.)

Another way is to write the logistic differential equation for your situation:

dG/dt = .121(G)(9200-G)

Take the second derivative

d2G/dt2 = d/dt(dG/dt) = (-.121G + (9200-G)(.121)) dG/dt by product rule and chain rule.

This is zero when G = (9200 -G) or G = 9200/2

Answer 2

Please use parenthesis to clearly mark which parts of the function are in the numerator and denominator respectively.

The function shown can be rewritten:

9200 exp(0.121t) / [ exp(0.121t) + 8.03]

which is unbounded....

the max will be at 1992.