# A country's population in 1995 was 164 million. In 2001 it was 169 million. How do you estimate the population in 2015 using exponential growth formula?

Dec 23, 2017

$181.268$ million to 3 dp

#### Explanation:

$\textcolor{b l u e}{\text{Initial condition}}$

The difference in years from 1995 to 2001 is 6 years
The change in population count in millions was 164 to 169

Let the unknown growth percentage be x% giving:

$164 {\left(1 + \frac{x}{100}\right)}^{6} = 169$

${\left(\frac{100 + x}{100}\right)}^{6} = \frac{169}{164}$

Taking logs

$6 \ln \left(100 + x\right) - 6 \ln \left(100\right) = \ln \left(169\right) - \ln \left(164\right)$

$\ln \left(100 + x\right) = \frac{\ln \left(169\right) - \ln \left(164\right)}{6} + \ln \left(100\right)$

Set $y = 100 + x$ giving

$\ln \left(y\right) \approx 4.610175 \ldots .$

$\implies {e}^{4.610175 \ldots} \approx y$
$\implies y \approx 100.501792 \ldots$

Thus $x \approx 0.501792 \ldots$
~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{New condition}}$

Time span from ${t}_{o} = 2015 - 1995 = 20$ years

Thus population estimate at 2015 is:

$164 {\left(\frac{100.501792 . .}{100}\right)}^{20} \approx 181.268$ million to 3 dp