Question

The population $P=P\left(t\right)$ at time $‘t^{\prime }$ of a certain species follows the differential equation $\frac{dP}{dt}=0.5P-450.$ If $P\left(0\right)=850,$ then the time at which population becomes zero is :

• A. $\frac{1}{2}{\log }_{e}18$
• B. $2{\log }_{e}18$
• C. ${\log }_{e}9$
• D. ${\log }_{e}18$

Answer 1

The correct option is B $2{\log }_{e}18$

$\frac{dp}{dt}=\frac{p-900}{2}$
$\Rightarrow {\int }_{850}^0\frac{dp}{p-900}={\int }_0^t\frac{dt}{2}$
$\Rightarrow \ln |p-900|{|}_{850}^0=\frac{t}{2}$
$\Rightarrow \ln |900|-\ln |50|=\frac{t}{2}$
$\Rightarrow \frac{t}{2}=\ln |18|$
$\Rightarrow t=2\ln 18$