Question

If t is in years since 1990, one model for the population of the world, P, in billions, is P=40/1+11e^-0.08t

a) What does this model predict for the maximum sustainable population of the world? Enter the exact answer.

This model predicts that t is very large, the population is ________ billion.

b) According to this model, when will the earth's population reach 10 billion? 39.9 billion?

Round your answer to the nearest year.
The population of the world should be 10 billion in ______.
The population of the world should be 39.9 billion in ______.

Answer 1

a ) If t is very large:
40 / ( 1 + 11 e^(-0.08 t ) )  ≈ 40 / ( 1 + 0 ) = 40 / 1 = 40
This model predicts that t is very large, the population is 40 billion.
b )
10 = 40 / ( 1 + 11 e^(-0.08 t ) )
1 + 11 e^(-0.08 t ) = 4
11 e^(-0.08 t ) = 3
e^(-0.08 t ) = 0.27272
ln 0.27272 = - 0.08 t
- 1.299 = - 0.008 t
t = 1.298 / 0.08 = 16.24 ≈ 16 years
The population of the world should be 10 billion in 16 years.
39.9 = 40 / ( 1 + 11 e^(-0.08 t) )
1 + 11 e^(-0.08 t) = 1.0025062
11 e^(-0.08 t) = 0.0025062
e^(-0.08 t ) = 0.0002278
ln 0.0002278 = - 0.008 t
-8.387 = - 0.008 t
t = 104.838 ≈ 105
The population of the world should be 39.9 billion in 105 years.