Logarithm applications

Question

a logistic model for the word population is P(t)=73.2/(6.1+5.9e^-0.02t) where t=0 indicates the year 2000 and the population is measured in billions.

a) what world population does this model predict for the year 2200?

b)in what year is the world population 10 billion?

C)according to this model, what size does the world population approches as tim goes on?

Answer 1

P(t) = 73.2/(6.1 + 5.9e^-0.02t)

 

a). population in 2200 = P(200) = 73.2/(6.1+5.9e^-4)

 

                                                 = 11.8 billion

 

b). The population reaches 10 billion when

 

       10 = 73.2/(6.1+5.9e^-0.02t)

 

       61+59e^-0.02t = 73.2

 

                e^-0.02t = 0.2067797

 

                  -0.02t = ln(0.2067797)

 

                         t = 78.8 years after 2000 (in the year 2078)

 

c).  As t gets larger and larger, e^-0.02t gets closer and closer to 0.

 

      So, P(t) approaches 73.2/(6.1 + 5.9(0)) = 12 billion