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# The rate of growth of the population of a city at any time t is proportional to the size of the population - Mathematics and Statistics

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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]

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#### 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.

Concept: Application of Differential Equations
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