Question

The population of a town increases at a rate proportional to the population at that time.if the population increases from 40000 to 60000 in 40 year's, what will be the population in another 20 years

Answer 1

Answer:

20000 to 30000× 20

most probably

answer is 400000 or 600000

Answer 2

In another 20 years the population of the town will be 73485

Step-by-step explanation:

Given

At any time, the rate of change of population is proportional to the population P

Thus,

$\frac{dP}{dt}\propto P$

$\implies \frac{dP}{dt}=kP$   (where k is a constant)

$\implies \frac{dP}{P}=kdt$

$\implies \int_{40000}^P\frac{dP}{P}=k\int_0^tdt$

$\implies \ln P\Bigr|_{40000}^P=kt$

$\implies (\ln P-\ln 40000)=kt$

Given, at t = 40, P = 60000

Therefore,

$k\times 40=\ln 60000-\ln 40000$

$\implies 40k=\ln (\frac{60000}{40000})$

$\implies k=\frac{1}{40}(\ln (\frac{3}{2}))$

Now, if t = 40+2=60 years

Then

$(\ln P-\ln 40000)=60\times\frac{1}{40}\ln (\frac{3}{2})$

$\implies \ln (\frac{P}{40000})=\frac{3}{2}\ln (\frac{3}{2})$

$\implies \ln (\frac{P}{40000})=\ln (\frac{3}{2})^{3/2}$

$\implies \frac{P}{40000}=(\frac{3}{2})^{3/2}$

$\implies P=40000\times\frac{3}{2}\sqrt{\frac{3}{2}}$

$\implies P=60000\sqrt{\frac{3}{2}}$

$\implies P=73485$

Therefore, in another 20 years the population of the town will be 73485

Hope this answer is helpful.

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