$\"\"$

\

Relative growth rate is $\"\"$ , $\"\"$ is a constant.

\

Initial relative growth rate is $\"\"$ , where $\"\"$ is initial value of $\"\"$ , $\"\"$ is change in $\"\"$ and $\"\"$ is change in $\"\"$ .

\

Initial growth rate is $\"\"$ .

\

Initial growth rate is $\"\"$ millions.

\

$\"\"$ .

\

$\"\"$ .

\

The initial population is $\"\"$ .

\

Substitute $\"\"$ , $\"\"$ and $\"\"$ in $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

Relative growth rate is $\"\"$ .

\

The logistic differential equation is $\"\"$ , where $\"\"$ is a carrying capacity.

\

Substitute $\"\"$ and $\"\"$ .

\

$\"\"$ , $\"\"$ is in billions.

\

The logistic differential equation is $\"\"$ , $\"\"$ is in billions.

\

$\"\"$

\

(b) Use the logistic model to estimate the world population in the year 2000 and compare with the actual population of 6.1

\

billion.

\

The logistic model equation is $\"\"$ , where $\"\"$ .

\

Substitute $\"\"$ and $\"\"$ in $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

Substitute $\"\"$ , $\"\"$ , $\"\"$ and $\"\"$ in $\"\"$ .

\

The logistic model equation is $\"\"$ .

\

The population in year $\"\"$ , $\"\"$ .

\

Substitute $\"\"$ in $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

The population in year $\"\"$ is $\"\"$ .

\

$\"\"$

\

(c) Use the logistic model to predict the world population in the years 2100 and 2500.

\

The logistic model equation is $\"\"$ .

\

The population in year $\"\"$ , $\"\"$ .

\

Substitute $\"\"$ in $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

The population in year $\"\"$ is $\"\"$ .

\

The population in year $\"\"$ , $\"\"$ .

\

Substitute $\"\"$ in $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

The population in year $\"\"$ is $\"\"$ .

\

$\"\"$

\

(d) What are your predictions if the carrying capacity is 50 billion.

\

The carrying capacity is $\"\"$ .

\

The logistic model equation is $\"\"$ , where $\"\"$ .

\

Substitute $\"\"$ and $\"\"$ in $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

Substitute $\"\"$ , $\"\"$ , $\"\"$ and $\"\"$ in $\"\"$ .

\

The logistic model equation is $\"\"$ .

\

The population in year $\"\"$ , $\"\"$ .

\

Substitute $\"\"$ in $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

The population in year $\"\"$ is $\"\"$ .

\

$\"\"$

\

The population in year $\"\"$ , $\"\"$ .

\

Substitute $\"\"$ in $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

The population in year $\"\"$ is $\"\"$ .

\

$\"\"$

\

The population in year $\"\"$ , $\"\"$ .

\

Substitute $\"\"$ in $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

The population in year $\"\"$ is $\"\"$ .

\

$\"\"$

\

(a) The logistic differential equation is $\"\"$ , $\"\"$ is in billions.

\

(b) The population in year $\"\"$ is $\"\"$ .

\

(c)

\

The population in year $\"\"$ is $\"\"$ .

\

The population in year $\"\"$ is $\"\"$ .

\

(d)

\

The population in year $\"\"$ is $\"\"$ .

\

The population in year $\"\"$ is $\"\"$ .

\

The population in year $\"\"$ is $\"\"$ .