A country's population in 1994 was 195 million. In 2002 it was 199 million. How do you estimate the population in 2016 using the exponential growth formula P=Ae^(kt)?

May 28, 2017

$207$ million

Explanation:

For this problem, we have the final population $P$, the starting population $A$, and the time passed $t$.

We still need to determine the value of the growth factor $k$.

This can be done using the information for the years $1994$ and $2002$.

In this case, $A$ will be $195$ million (the population in $1994$), $P$ will be $199$ million (the population in $2002$), and $t$ will be $8$ years (the number of years between $1994$ and $2002$):

$R i g h t a r r o w P = A {e}^{k t}$

$R i g h t a r r o w 199 = 195 {e}^{k \times 8}$

$R i g h t a r r o w \frac{199}{195} = {e}^{8 k}$

Applying $\ln$ to both sides of the equation:

$R i g h t a r r o w \ln \left(\frac{199}{195}\right) = \ln \left({e}^{8 k}\right)$

Using the laws of logarithms:

$R i g h t a r r o w 8 k \ln \left(e\right) = \ln \left(199\right) - \ln \left(195\right)$

$R i g h t a r r o w 8 k = \ln \left(199\right) - \ln \left(195\right)$

$R i g h t a r r o w k = \frac{\ln \left(199\right) - \ln \left(195\right)}{8}$

$\therefore k = 0.00253815825$

Now, let's use this information to determine $P$ in $2016$.

$A$ will still be $195$ million, but $t$ will be $22$ years.

$R i g h t a r r o w P = 195 {e}^{0.00253815825 \times 22}$

$R i g h t a r r o w P = 195 {e}^{0.0558394815}$

$R i g h t a r r o w P = 195 \times 1.057427933$

$\therefore P = 206.19844694$

Therefore, the population in $2016$ will be around $207$ million.