The rate of growth of the population of a city at any time t is proportional to the size of the population. For a certain city, it is found that the constant of proportionality is 0.04. Find the population of the city after 25 years, if the initial population is 10,000. [Take e = 2.7182]

#### Solution

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

Then the rate of growth of population is `"dP"/"dt"` which is proportional to P.

∴ `"dP"/"dt" prop "P"`

∴ `"dP"/"dt"` = kP, where k = 0.04

∴ `"dP"/"dt" = (0.04)"P"`

∴ `1/"P" "dP" = (0.04) "dt"`

On integrating, we get

`int 1/"P" "dP" = (0.04) int "dt" + "c"`

∴ log P = (0.04)t +c

Initially, i.e., when t = 0, P = 10000

∴ log 10000 = (0.04) × 0 + c

∴ c = log 10000

∴ log P = (0.04) t + log 10000

∴ log P - log 10000 = (0.04) t

∴ log `("P"/10000) = (0.04) "t"`

When t = 25, then

log `("P"/10000) = (0.04) xx 25 = 1`

∴ log `("P"/10000)` = log e ....[∵ log e = 1]

∴ `"P"/10000 = "e" = 2.7182`

∴ P = 2.7182 × 10000 = 27182

∴ the population of the city after 25 years will be 27,182.