Logistic Growth Model, in what year did 875 families own a home?

Question

H(t)= 1,000/1+3e^-0.76t

The logistic growth model represents the number of families that own a home in a city after 2000, what year did 875 own a home?

Answer 1

In this question, we're given a number of families owning homes, 875 and are asked to find the expected number of years that 875 families would own a home. In this case our H(t) = 875.

So we have it that:

1000

875 = -----------------------

1 + 3e^-0.76t

We can solve for t algebraically.

3e^-0.76t = 1000/875 - 1

Simplifying, we get that:

3e^-0.76t = 125/875 = 1/7

e^-0.76t = 1/21

Taking natural logarithms on both sides, we have that:

-0.76t = ln(1/21) = -ln21 (Since ln a/b = ln a - ln b and any base logarithm of 1 equals zero)

t = ln21/0.76 = 4.00595

Therefore, in 2004, 875 families would own a home.