The logistic growth model H(t)=2000/1+4e^-0.73t represents the number of families that own a home in a certain small(but growing) city t years after 2000.

Question

A) What is the maximum number of families that will own a home in this city?

B) How many families owned a home in 2000?

C) In what year did 1645 families own a home?

Answer 1

A logistic curve function, used to model growth with a fixed maximum, has an equation given by

P(t) = A/[1 + Be^-kt] which has 3 separate parameters: A represents the maximum y-value, because as t→∞, the denominator approaches 1, and the function's graph has a horizontal asymptote at y = A.

The initial value, P₀ = A(1+B), since e⁰=1. k controls the rate at which the function approaches its max.

A) Max # of families = 2,000

B) H(0) = 400, the # of families that owned homes in the year 2000

C) Set H(t) = 1,645 and solve for t by using natural logs:

2000 / [1 + 4e^-.73t] = 1645

[1 + 4e^-.73t] = 1.2158

4e^-.73t = .2158

e^-.73t = .05395

-.73t = ln(.05395) = -2.9197

t ~ 4 so in 2004 about 1,645 families will own homes.