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 ?

Explanation

Let P be the population at time t years. Then ` (dP)/(dt) ` the rate of increase of population is proportional to P.
` (dp)/(dt) prop p `
` (dP)/(dt) = kP` when k is a constant
` (dP)/(P) = kdt`
On integrating we get
` int (dp)/P = k int dt +c `
`log P = kt +c`
Initially i.e., when t =0 let `P = P_(0)`
`log P_(0) = k xx 0 +c`
`log P = kt + log P_(0)`
` log P - log P_(0) = kt `
`log (P/(P_(0))) = kt `
since the population doubles in 60 years, when t = 60 `P = 2P_(0)`
` log ((2P_(0))/(P_(0))) = 60 k , " " therefore k = 1/ 60 log 2`
(1) becomes `log (P/(P_(0))) = t/60 log 2`
When population becomes triple i.e, when ` P =3P_(0) ` we get
` log ((3P_(0))/(P_(0)) = t/60 log 2`
`log3 = t/60 log 2 `
`log 3 = t/60 log 2`
`t = 60 (log 3/log 2) `
` = 60 ((1.0986)/(0.6912))`
= 60 xx 1.5894 = 95 . 364 = 95 . 4 ` years
The population becomes triple in 95.4 years ( approximately )