Question

a population in a certain city is growing at a rate proportional to the population itself. after 3years the population increases by 20%. how long will it take for the population to double?

Answer 1

Call the initial population P.

After 3 years, ie when t=3, the population increased by 20%, so it's now 120% of the original population (P plus 20% of P), or 1.2 times the original:

Use this formula
\[\Large Population = P e^{r t}\]
Population is now 1.2P, t=3, so
\[\LARGE 1.2 P = P e^{3r}\]
See if you can solve this for r, the rate of growth (you'll need to take natural logs).

Answer 2

Once you have r, you'll need to find t for the population to double (that means 2P)
\[\LARGE 2P = P e^{r t}\]