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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?

User Mahlon
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1 Answer

5 votes

Answer:


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!

User Dzianis Yafimau
by
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