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Question
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.
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