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# 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? - Mathematics and Statistics

Sum

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 = 20.6912, log 3 = 1.0986)

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#### Solution

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 hours, 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).

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