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Question
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 4,00,000?
Solution
Let ‘x’ be the population at time ‘t’ years.
∴ `dx/dt prop x`
∴ `dx/dt = kx`, where k is the constant of proportionality.
∴ `dx/x = kdt`
Integrating on both sides, we get
`int dx/x = int kdt`
∴ logx = kt + c …(i)
When t = 0, x = 50,000
∴ log(50,000) = k(0) + c
∴ c = log(50,000)
∴ logx = kt + log(50,000) ...(ii) [From (i)]
When t = 25, x = 1,00,000, we have
log(1,00,000) = 25k + log(50000)
∴ log2 = 25k
∴ k = `1/25 log 2` …(iii)
When x = 4,00,000, we get
`log(4,00,000) = [1/25 log (2)]t + log(50,000)` ...[From (ii) and (iii)]
∴ `log[400000/50000] = t/25 log2`
∴ log8 = `t/25 log2`
∴ 3log2 = `t/25 log2`
∴ 3 = `t/25`
∴ t = 75 years.
Thus, the population will be 4,00,000 after 75 – 25 = 50 years from present date.
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