Question

At time t the population P(t) of a city is increasing at a rate proportional to the population size. In year 0 the population of the city is 10000. In year 5 the population of the city is 20000. Let Q1 be the population of the city in year 20. Let Q2 be the year in which the population of the city reaches 1000000. What will the population of the city be at the beginning of the year 2010? In what year will the population reach one million?

Answer 1

Find the amount Q1

`P(T)=10000 e^{(ln(2))/(5)20}=160000`

In what year will the population reach one million?

`Q_2= (ln(100))/((ln(2))/(5))=33.219` years

What will the population of the city be at the beginning of the year 2010?Assuming that the starting point is th year 2000, we just need to replace T =10 in the model in order to find the population, like this:

`P(T)=10000 e^{(ln(2))/(5)10}=40000`

Explanation:

Notation

T= represent the time in years

P(T)= represent the population at the time T

K= represent the proportional constant for the model

Solution to the problem

So for this case we need a proportional model given by:

`(DP)/(DT)=KP`

And we can express the model like this

`(DP)/(P)=K DT`

And integrating on both sides we got:

`ln P= KT +C`

And we can exponentiate both sides:

`P(T)=P_o e^(KT)`

We have two initial conditions on this case, given by:

`P(0)=10000, P(5)=20000`

We can use the first condition in order to find
`P_o`, like this:

`10000=P_o e^(0)`, so then we have that
`P_o =10000`

And we can use the second condition to find the constant K, like this:

`20000=10000 e^(5K)`,a dn solving for K we have:

`ln(2)= 5K` and
`K=(ln(2))/(5)`

So then our model would be given by:

`P(T)=10000 e^{(ln(2))/(5)T}`

Find the amount Q1

`P(T)=10000 e^{(ln(2))/(5)20}=160000`

In what year will the population reach one million?

We can find the time Q2 where the population is 1000000, like this:

`1000000=10000 e^{(ln(2))/(5)T}`

`100 =e^{(ln(2))/(5)Q_2}`

And using natural log on both sides we have:

`ln(100)=(ln(2))/(5)Q_2`

And
`Q_2= (ln(100))/((ln(2))/(5))=33.219` years

What will the population of the city be at the beginning of the year 2010?Assuming that the starting point is th year 2000, we just need to replace T =10 in the model in order to find the population, like this:

`P(T)=10000 e^{(ln(2))/(5)10}=40000`