The population P=P(t) at time


The population $P=P(t)$ at time ' $t^{\prime}$ of a certain species follows the differential equation $\frac{d P}{d t}=0.5 \mathrm{P}-450$. If $\mathrm{P}(0)=850$, then the time at which population becomes zero is :

  1. (1) $\frac{1}{2} \log _{e} 18$

  2. (2) $2 \log _{\mathrm{e}} 18$

  3. (3) $\log _{\mathrm{e}} 9$

  4. (4) $\log _{\mathrm{e}} 18$

Correct Option: , 2


$\frac{d p}{d t}=\frac{p-900}{2}$

$\int_{850}^{0} \frac{d p}{p-900}=\int_{0}^{t} \frac{d t}{2}$

$\ell n \mid P-900 \|_{850}^{0}=\frac{t}{2}$

$\ell n|900|-\ell n|50|=\frac{t}{2}$

$\frac{t}{2}=\ln |18|$

$\Rightarrow t=2 \ln 18$

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