# The population of a region is growing exponentially. There were 40 million people in 1980 (when t...

## Question:

The population of a region is growing exponentially. There were 40 million people in 1980 (when t = 0) and 50 million people in 1990. Find an exponential model for the population (in millions of people) at any time t, in years after 1980.

P(t) =

Predicted population in the year 2000 = _____ million people.

What is the doubling time?

Doubling time = _____ years.

## Exponential Population Growth:

A population is said to grow exponentially if the population at time {eq}\displaystyle{ \begin{align} t \end{align} } {/eq} is given by the function

{eq}\displaystyle{ \begin{align} P(t)=P_0 e^{kt} \end{align} } {/eq}

where {eq}\displaystyle{ \begin{align} P_0 \end{align} } {/eq} is the initial population when {eq}\displaystyle{ \begin{align} t=0 \end{align} } {/eq} and {eq}\displaystyle{ \begin{align} k>0 \end{align} } {/eq} is a constant.

When population growth is exponential, the doubling time for the population is the length of time it takes for the population to become {eq}\displaystyle{ \begin{align} 2P_0 \end{align} } {/eq}.

The initial population, in millions, is

{eq}\displaystyle{ \begin{align} P_0 = 40 \end{align} } {/eq}

After 10 years, the population is 50 million.

{eq}\displaystyle{ \begin{align} P(10)&=50\\ 40 e^{10k}&=50\\ e^{10k}&=\frac{5}{4}\\ 10k&=\ln \left( \frac{5}{4} \right)\\ k&=\frac{1}{10}\ln \left( \frac{5}{4} \right)\\ k&\approx 0.0223 \end{align} } {/eq}

So, the population {eq}\displaystyle{ \begin{align} t \end{align} } {/eq} years after 1980 is given by

{eq}\displaystyle{ \begin{align} P(t)=40 e^{0.0223t} \end{align} } {/eq}

To find the doubling time, solve for {eq}\displaystyle{ \begin{align} T \end{align} } {/eq}.

{eq}\displaystyle{ \begin{align} P(T)&=2P_0\\ 40e^{0.0223T}&=2(40)\\ e^{0.0223T}&=2\\ 0.0223T&=\ln (2)\\ T&=\frac{\ln (2)}{0.0223}\\ T&\approx 31 \text{ years} \end{align} } {/eq}

So, the doubling time is approximately 31 years. 