The logistic growth model Ht 2000 14e061t represents the num

The logistic growth model H(t)= 2000/ 1+4e^0.61t represents the number of families that own a home in a certain small (but growing) city t years after 1970. a) What is the maximum number of families that will own a home in this city? b.) How many families owned a home in 1970? c) In what year did 1968 families own a home?

Solution

h(t) = 2000 / ( 1+ 4e^-0.61t)

a) the maximum number of families that will own a home in this city are

2000 since as t tends to infinity the bottom expression becomes 1

b) families that own home in 1970 ( plug t = 0 )

h(t) = 2000 / ( 1+ 4e^-0.61t)

h(t) = 2000 / ( 1 + 4e^0) = 2000/ 5

h(t) = 400

400 families owned home in 1970

c) plugging h(t) = 1968 and finding t

1968 = 2000 / ( 1+ 4e^-0.61t)

( 1+ 4e^-0.61t) = 2000 / 1968

( 1+ 4e^-0.61t) = 1.01626

subtracting 1 from both sides

4e^-0.61t = 0.01626

dividing both sides by 4

e^-0.61t = 0.004065

-0.61t = ln 0.004065

t = 9.025

hence, in the year 1970 + 9 = 1979 , 1968 families own a home