The logistic growth model shown below represents the number of families that own a home in a certain small (but growing) city t years after 1990. a) What is th

Question

The logistic growth model shown below represents the number of families that own a home in a certain small (but growing) city t years after 1990.
a) What is the maximum number of families that will own a home in this city?
b) How many families owned a home in 1990?
c) In what year did 5475 families own a home?

 

H(t)=6000/1+4e^-0.67t

Answer 1

All previous steps as given by Mark

 

4(e^(-0.67t))=0.09589041

 

e^(-0.67t)=0.02397

 

-0.67t=ln(0.02397)

 

 -0.67t=-3.73095

 

t=5.5685

Answer 2

As t → ∝, 1 + 4e^-0.67t → 1, therefore the maximum is 6000.

 

In 1990 t = 0 and the denominator = 5

H(1990) = 1200

 

5475 = 6000 / (1 + 4e^-0.67t )

5475 (1 + 4e^-0.67t) = 6000

1 + 4e^-0.67t = 6000 / 5475

1 + 4e^-0.67t ≈ 1.09589041

4e^-067t ≈ 0.09589041

e^-067t ≈ 0.239726025

ln e^-0.67t ≈ ln 0.239726025

-0.67t ≈ -1.42825857

t ≈ 2.13172921