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The growth rates of a bacteria population is proportional to its size. Initially the population is $10,000$ while after 10 days its size is $25,000$ . What will be population after 20 days?

$(a)\;100(2.5)^2 \\ (b)\;1000(2.5)^2 \\ (c)\;2500 \\ (d)\;250 $
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According to given condition $ \large\frac{dP}{dt}=$$KP$
$\int \large\frac{dP}{P}$$= K \int dt$
$\log P \bigg ]_{10000}^{25000}=K t \bigg]_0^{10}$
$\log 2.5 =10.K$
$K= \large\frac{\log 2.5 }{10}$
$\log P\bigg]_{10000}^{P}=\large\frac{\log 2.5}{10} \times t \bigg]_0^{20}$
$\log \large\frac{P}{10000} =\large\frac{\log 2.5}{10}$$ \times 20$
$\log \large\frac{P}{10000} $$=\log (2.5)^2$
$\large\frac{P}{10000} $$=(2.5)^2$
$P= (2.5) ^2 \times 10000$
Hence b is the correct answer.
answered Feb 6, 2014 by meena.p

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