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Solution

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

Then,

$dtdN =kN$ where $k$ is the
proportionality constant.

$NdN =kdt$

$∫_{N_{0}}NdN =∫_{0}kdt$

$ln(N_{0}N )=kt$

$N=N_{0}e_{kt}$ ...(i)

If $N_{0}=2×10_{4}$ and $t=2004−1999$

$t=5years$ and

$N=2.5×10_{4}$

Hence

$lnN_{0}N =kt$

$ln(2×10_{4}2.5×10_{4} )=5k$

$ln(1.25)=5k$

$k=51 ln(1.25)years_{−1}$

Now

$t=2009−1999$

$=10years$

Hence

$ln(2×10_{4}N )=51 ln(1.25)t$

$ln(2×10_{4}N )=51 ln(1.25)×10$

$ln(2×10_{4}N )=ln(1.25)×2$

$ln(2×10_{4}N )=ln(1.25_{2})$

$2×10_{4}N =1.25_{2}$

$N=2×(1.25)_{2}×10_{4}$

$=3.125×10_{4}$

$=31250$

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