# 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?

Let the number of people
at a given year 't' be N

Then,

dNdt=kN where k is the
proportionality constant.

dNN=kdt

∫NN0dNN=∫t0kdt

ln(NN0)=kt

N=N0ekt ...(i)

If N0=2×104 and t=2004−1999

t=5years and

N=2.5×104

Hence

lnNN0=kt

ln(2.5×1042×104)=5k

ln(1.25)=5k

k=15ln(1.25)years−1

Now

t=2009−1999

=10years

Hence

ln(N2×104)=15ln(1.25)t

ln(N2×104)=15ln(1.25)×10

ln(N2×104)=ln(1.25)×2

ln(N2×104)=ln(1.252)

N2×104=1.252

N=2×(1.25)2×104

=3.125×104

=31250