Question

According to this model the world population will be 9 billion in? Round to the nearest whole number as needed

Answer 1

A logistic growth model for the world population is:

$f(x)=\frac{12.57}{1+4.11\cdot e^{-0.026x}}$

Where x is the number of years since 1958 and f(x) is expressed in billions

It's required to find the year for which the world population will be f(x) = 9 billion.

Then we set up the equation:

$9=\frac{12.57}{1+4.11\cdot e^{-0.026x}}$

Cross-multiplying:

$9(1+4.11\cdot e^{-0.026x})=12.57$

Dividing by 9:

$1+4.11\cdot e^{-0.026x}=1.3967$

Subtracting 1 and dividing by 4.11:

$\begin{gathered} 4.11\cdot e^{-0.026x}=1.3967-1=0.3967 \\ e^{-0.026x}=0.0965126 \end{gathered}$

Taking logarithms on both sides:

$\begin{gathered} -0.026x=\log0.0965126 \\ Dividing\text{ by -0.026:} \\ x=\frac{\log0.0965126}{-0.026} \\ x=90 \end{gathered}$

According to this model, the world population will be 9 billion in 90 years from 1958, that is, in the year 2048