Question

The population of a village increases continuously at the rate proportional to the number of its inhabitants present at any time. If the population of the village was 20.000 in 1999 and 25000 in the year 2004, what will be the population of the village in 2009 ?

Solution

Let y denote the population at any t.
From the given condition,

dy over dt space equals space straight k space straight y. space space

where k is constant of proportionality.
Separating the variables and integrating,

integral 1 over straight y dy space equals space straight k integral space dt

therefore space space space space space space space space space space log space straight y space equals space straight k space straight t space space plus space straight c

...(1)
Let

straight y subscript 0 space equals space 20000 space

be the population at t = 0.

therefore space space space space space space space space space log space straight y subscript 0 space equals space 0 plus space straight c space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space space space space straight c space equals space log space straight y subscript 0 therefore space space from space left parenthesis 1 right parenthesis comma space space log space straight y space equals space straight k space straight t space plus space log space straight y subscript 0 space space space space space space space rightwards double arrow space space log space straight y space minus space log space straight y subscript 0 minus space straight k space straight t therefore space space space space space space log space open parentheses straight y over straight y subscript 0 close parentheses space equals space straight k space straight t space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis

Now y = 25000, when t = 5

therefore space space space log space open parentheses 25000 over 20000 close parentheses space equals space 5 space straight k space space space space space space rightwards double arrow space straight k space equals space 1 fifth log space open parentheses 5 over 4 close parentheses therefore space space from space left parenthesis 2 right parenthesis comma space space log space open parentheses straight y over straight y subscript 0 close parentheses space equals space 1 fifth straight t space log space open parentheses 5 over 4 close parentheses

Let

straight y subscript 1

be the population in 2004 i.e. after 10 years

therefore space space space space log space open parentheses straight y subscript 1 over straight y subscript 0 close parentheses space equals space 1 fifth cross times 10 cross times log space open parentheses 5 over 4 close parentheses therefore space log space open parentheses straight y subscript 1 over straight y subscript 0 close parentheses space equals space 2 space log space open parentheses 5 over 4 close parentheses space space space space rightwards double arrow space space space space log open parentheses straight y subscript 1 over straight y subscript 0 close parentheses space equals space log space 25 over 16 therefore space space space space space space space straight y subscript 1 over straight y subscript 0 equals space 25 over 16 space space space space rightwards double arrow space space space space space straight y subscript 1 space equals space 25 over 16 cross times 20000 space equals space 31250

therefore

required population in 2004 is 31250.

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