Question

The population of a certain community is known to increase at a rate proportional to the number of people present at time t. If the population has doubled in 5 years, how long will it take to triple?

Answer 1

`t=7.85 years`

Explanation:

We can write this rate as an ordinary differential equation.

`(dP)/(dt)=aP`

Where a is proportional constant, P the population variable, and t the time.

`(dP)/(P)=adt`

Integrating each side of the equation.

`\int (dP)/(P)=\int adt`

`ln(P)=at+c`

`P=e^(at+c)`

`P=e^(c)e^(at)=Ce^(at)`

To find C we need to use the initial condiction, it means evaluae P at t=0.

`P_(0)=Ce^(0)=C`

`P=P_(0)e^(at)`

Now, we use the sentence the population has doubled in 5 years.

`2P_(0)=P_(0)e^(a5)`

We can find "a" in this condition.

`2=e^(a5)`

`ln(2)=a5`

`a=(ln(2))/(5)`

`a=0.14`

Finally, let's find how long will it take to triple.

`3P_(0)=P_(0)e^(0.14t)`

`3=e^(0.14t)`

`t=(ln(3))/(0.14)`

`t=7.85 years`

I hope it helps you!