Question

Find the population of a city at any time t, given that the rate of increase of population is proportional to the population at that instant and that in a period of 40 years, the population increased from 30,000 to 40,000.

Answer 1

Let P be the population of the city at time t.

Then `"dP"/"dt"`, the rate of increase of population, is proportional to P.

∴ `"dP"/"dt" prop "P"`

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

∴ `"dP"/"P"` = k dt

On integrating, we get

`int 1/"P" "dP" = "k" int "dt" + "c"`

∴ log P = kt + c

Initially, i.e. when t = 0, P = 30000

∴ log 30000 = k × 0 + c       ∴ c = log 30000

∴ log P = kt + log 30000

∴ log P - log 30000 = kt

∴ `log("P"/30000)` = kt          .....(1)     

Now, when t = 40, P = 40000

∴ `log (40000/30000) = "k" xx 40`

∴ k = `1/40 log (4/3)`

∴ (1) becomes, `log ("P"/30000) = "t"/40 log (4/3) = log (4/3)^("t"/40)`

∴ `"P"/30000 = (4/3)^("t"/40)`

∴ P = 30000 `(4/3)^("t"/40)`

∴ the population of the city at time t = 30000 `(4/3)^("t"/40)`