Question

How long will it take for the population of a certain country to double is its annual growth rate is 5.3%?

Answer 1

Answer:

It will take approximately 20 years ($\:t\approx \:20$ years) for the population of a certain country to double if its annual growth rate is 5.3%.

Step-by-step explanation:

• Let $P(0)$ be the initial population

• Let $k$ be the growth rate

Then the population after $t$ years is:

$P=P\left(0\right)e^{kt}$

Let $x$ be the current population.

$2x$ would be the double this population.

The growth rate is $k = 0.053$

so the value of $t$ can be determined as:

$xe^{0.053t}=2x$

$e^{0.053t}=2$

$0.053t=ln(2)$

$\mathrm{Multiply\:both\:sides\:by\:}1000$

$0.035t\cdot \:1000=\ln \left(2\right)\cdot \:1000$

$\mathrm{Refine}$

$35t=\ln \left(2\right)\cdot \:1000$

$\mathrm{Divide\:both\:sides\:by\:}35$

$\frac{35t}{35}=\frac{\ln \left(2\right)\cdot \:1000}{35}$

$t=\frac{200\ln \left(2\right)}{7}$

$t=19.80$

$\:t\approx \:20$ years

Therefore, it will take approximately 20 years for the population of a certain country to double if its annual growth rate is 5.3%.