College algebra (logarithms)

Question

the growth model H(t)= 8000/1+4e^-0.67t 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 2626 families own a home?

Answer 1

Please include parentheses in the problem statement.  I assume you meant

 

H(t) = 800 / (1 + 4e^-0.67t)

 

(a)

 

The maximum occurs when the denominator is minimum.  Note that at t=0 the denominator is maximum (i.e. 1+4e⁰ = 5), and as t→∞ the denominator approaches a minimum value of 1 (i.e. 1+4e^-∞ = 1+4(0) = 1).  Hence, the maximum is 800/1 = 800.

 

(b)

 

In 1990, t=0, so substitute t=0 and simplify.

 

H(0) = 800 / (1 + 4e⁰) = 800/5 = 160

 

(c)

 

Is 2626 the correct number here?  I suspect it is a typo, because you start at H(0) = 160, and as t→∞, H(t)→800.  There is no way to get to 2626.

 

 

(c) Set 2626 = H(t) and solve for t. 2626 = 8000 / (1 + 4e-0.67t) 1 + 4e-0.67t = 3.046458 e-0.67t = 0.511615 t = -ln(0.511615) / 0.67 = 1.000273 That is years after 1990, so that puts you just a little past 1991, into very early 1992.