# How long will it take for the population to double if after 3yrs the population increases by 20% ?A population in a city is growing at a rate proportional to the populaiton itself.

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Since the population is modelled using exponential growth, the formula we can use to model it is `p(t)=p_0 b^t`, where `b=1+r` and r is the growth rate, t is time, `p(t)` is the population after time t and `p_0` is the initial population.

Since the population increases by 20% in three years, if we substitute into the model we get `1.2p_0=p_0 b^3`. Now we can cancel the initial population from both sides. This means that `1.2=b^3`, which we can use to solve for b.

Therefore `b=1.2^{1/3}`.

Now that we know how fast the population grows, we can substitute b into the model and see how long it takes to double the population.

`2p_0=p_0 (1.2^{1/3})^t` again we can cancel the initial population

`2=(1.2^{1/3})^t` simplify the exponents

`2=1.2^{t/3}` now we need to take logarithms to solve for t

`log 2=log 1.2^{t/3}` use the power rule

`log 2=t/3 log 1.2` isolate t

`t={3log 2}/{log 1.2}`

`t approx 11.41`

**It takes about 11.41 years for the population to double.**

I took 3 years for 20% increase.

=> population p grows to 1.2p in 3 years.

Let t be the years for the population to double.

=> So in t years the population becomes p*(1.2)^(t/3) whoch is 2p.

=> p*(1.2)^(t/3) = 2p

=> (1.2)^(t/3) = 2

We take logarithms on both sides

=> t/3 log1.2 = log2

=> t = 3*log2/log1.2

=> t = 11.405yrs.

**Therefore it requires 11.405yrs for the population to double.**