A function of the form Pt abt represents the population in

A function of the form P(t) = ab^t represents the population (in million) of the given country t years after January 1, 2000. a. write an equivalent function using base e; that is, write a function of the form P(t) = P_0e^kt. Also, determine the population of each country for the year 2000. b. The population of the two given countries is very close for the year 2000, but their growth rates are different. Use the model to approximate the year during which the population of each country would reach 10 5 million. c. Haiti had fewer people in the year 2000 than Sweden. Why would Haiti reach a population of 10.5 million sooner?

Solution

Haiti = 8.5(1.0158)^t

Sweden = 9.00(1.0048)^t

a) k = 1-1.0158 = 0.0158

Haiti

P(t)= 8.5 e^(0.0158t)

Sewedn : k = 1.0048 -1.0

= 0.0048

P(t) = 9e^(0.0048t)

b) P(t) = 10.5 million for each

P(t)= 8.5 e^(0.0158t)

10.5 = 8.5e^(0.0158t)

1.235 = 0.0158t

take natural log on both sides:

t = 13.35 years

In the year 2014 population reaches 10.5 million

P(t) = 9e^(0.0048t)

10.5 = 9e^(0.0048t)

1.167 = e^(0.0048t)

0.154 = 0.0048t

t = 32.11 years

In the year 2033 population reaches 10.5 million

c) Haiti reaches 10.5 milion population earlier than Sweden because its growth rate is more than Sweden