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 = P_{o} t when t = 0,

log (P_{o}) = 0 + c

c = log (P_{o})

Then, equation (i) becomes,

log (P) = λt + log (P_{o})

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

And also form question, given P = 2P_{o} 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 t_{1} be the time to become population 5,00,000 from 1,00,000,

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

By cross multiplication we get,

t_{1} = \(\dfrac{ 25 log 5}{log 2}\)

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

= 58

Therefore, the required time is 58 years.

Answered by Aaryan | 1 year agoFind the particular solution satisfying the given condition \( (\dfrac{dy}{dx}) = (x – 1)\dfrac{dy}{dx} = 2x^3y\)

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