Question

The world population in 1999 was approximately P₀ = 6.067 billion people, and in 2000, it was approximately P₁ = 6.145 billion people.

Logistic Growth Model. Scientists often use a different function to model population growth when there are limited resources, as is the case with our planet. To do this, we will model the world population (in billions of people) in year t by the function L(t) = P0K / P₀ + (K − P₀)e^-rbt , where P₀ and t are the same as above, r_b is a number called the base growth rate, and K is a number called the carrying capacity of the environment. Scientists estimate that the carrying capacity of the Earth is K = 15. Based on that, and the population data given above, you can calculate that r_b is about 0.022.

(a) Write down the function L(t) explicitly.

(b) Graph L(t) on the same axes as your function P(t) from Problem 1.

(c) According to this model, what would the population be in 2019? In 2049? In 2099?

(d) According to this model, how fast (in billions of people per year) is the world population growing in 2019? In 2049?

Answer 1

--------(1)

(a)

P₀=6067 billion people

P₁=6145 billion people

t=0 corresponding to 1999

t=1 corresponding to 2000

at t=1;P(1)=6145

taking log of both sides

-----(2)

(c) 2019 (t=20)

billions people

2049 (t=50)

billions people

2099 (t=100)

billions people

(d)

Differentiating with respect to t

[Because ]

Population growing in 2019 means t=20

billions people per year

Population growing in 2049 means t=50

billions people per year