# Use the exponential growth model P(t) = P_0e^(kt). How long will it take for the population of a certain country to double if its annual growth rate is 3.5%?

Aug 1, 2016

20.15 years to 2 dp

#### Explanation:

Let's assume the units of t are years. We want to find k.

$P \left(1\right) = 1.035 {P}_{0} = {P}_{0} {e}^{k}$

${e}^{k} = 1.035$

$\ln {e}^{k} = \ln \left(1.035\right)$

$k \ln \left(e\right) = \ln \left(1.035\right)$

$\implies k = \ln \left(1.035\right) y {r}^{- 1}$

Now need $P \left(t\right) = 2 {P}_{0}$

$2 {P}_{0} = {P}_{0} {e}^{k t}$

${e}^{k t} = 2$

$k t = \ln \left(2\right)$

$t = \ln \frac{2}{k} = \ln \frac{2}{\ln} \left(1.035\right)$

$t = 20.15$ years