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Question

The population of a city increases at a rate proportional to the number of inhabitants present at any time t. If the population of the city was 200000 in 1990 and 250000 in 2000, what will be the population in 2010?

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Solution

Let the population at any time t be P.

Given: dPdt α P

dPdt=βPdPP=βdtlogP=βt+log C ...1Now, At t=1990, P=200000 and at t=2000, P=250000 log 200000=1990β+log C ...2 log 250000=2000β+log C ...3Subtracting 3 from 2, we getlog 200000-log 250000=10ββ=110log54Putting β=110log 54 in 2, we getlog 200000=1990×110log54+log Clog 200000=199log54+log C log C=log 200000-199log54 Putting β=110log 54, log C=log 200000-199 log54 and t=2010 in 1, we getlogP=110×2010log 54+log 200000-199 log54logP=201 log 54+log 200000-199log54logP=log 54201- log54199+log 200000logP=log 5420145199+log 200000logP=log 542+log 200000logP=log2516×200000logP=log 312500P=312500

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