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?

Answer 1

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.