The rate of growth of population is proportional to the number present. If the population doubled in the last 25 years and the present population is 1 lac, when will the city have population 5,00,000 ?

Asked by Sakshi | 1 year ago |  137

##### Solution :-

From the question it is given that,

The present population is 1,00,000

Then, the population of a city doubled in the past 25 years,

So, let us assume P be the surface area of balloon,

($$\dfrac{dP}{dt}$$) ∝ P

Then,

$$\dfrac{dP}{dt}$$= λP

$$\dfrac{dP}{dt}$$= λ dt

Integrating on both side we get,

$$\dfrac{dP}{dt}$$= λ ∫dt

Log P = λt + c … [equation (i)]

From question, P = Po t when t = 0,

log (Po) = 0 + c

c = log (Po)

Then, equation (i) becomes,

log (P) = λt + log (Po)

log $$\dfrac{P}{P_o}$$ = λt … [equation (ii)]

And also form question, given P = 2Po when t = 25

log$$\dfrac{2P_o}{2P_o}$$ = 25λ

log 2 = 25λ

By cross multiplication we get,

λ =$$\dfrac{ log2}{25}$$

So, now equation (ii) becomes,

log $$\dfrac{P}{P_o}$$ = ($$\dfrac{ log2}{25}$$)t

let us assume that t1 be the time to become population 5,00,000 from 1,00,000,

Then, $$log \dfrac{5,00,000}{1,00,000}$$ = ($$\dfrac{ log2}{25}$$)t1

By cross multiplication we get,

t1 = $$\dfrac{ 25 log 5}{log 2}$$

= 25$$\dfrac{ (1.609)}{(0.6931)}$$

= 58

Therefore, the required time is 58 years.

Answered by Aaryan | 1 year ago

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