Question

If the population of a country doubles in 60 years, in how many years will it be triple (treble) under the assumption that the rate of increase is proportional to the number of inhabitants?
(Given log 2 = 0.6912, log 3 = 1.0986)

Answer 1

Let P be the population at time t years. Then `"dP"/"dt"`, the rate of increase of population is proportional to P.

∴ `"dP"/"dt" ∝  "P"`

∴ `"dP"/"dt"` = kP, where k is a constant

∴ `"dP"/"P"` = k dt

On integrating, we get

`int "dP"/"P" = "k" int "dt" + "c"`

∴ log P = kt + c

Initially, i.e. when t = 0, let P = P₀

∴ log P₀ = k × 0 + c

∴ c = log P₀ 

∴ log P = kt + log P₀ 

∴ log P - log P₀ = kt

∴ `log ("P"/"P"_0)`= kt    ...(1)

Since the population doubles in 60 years, i.e. when t = 60, P = 2P₀ 

∴ `log ((2"P"_0)/"P"_0)` = 60k

∴ k = `1/60` log 2

∴ (1) becomes, `log ("P"/"P"_0) = "t"/60` log 2

When population becomes triple, i.e. when P = 3P₀ , we get

`log ((3"P"_0)/"P"_0) = "t"/60` log 2

∴ `log 3 = ("t"/60)` log 2

∴ t = `60 ((log 3)/(log 2)) = 60 (1.0986/0.6912)`

= 60 × 1.5894 = 95.364 ≈ 95.4 years

∴ the population becomes triple in 95.4 years (approximately).