Question

The population $p\left(t\right)$ at time $t$ of a certain mouse species satisfies the differential equation $\frac{dp\left(t\right)}{dt}=0.5p\left(t\right)-450$. If $p\left(0\right)=850$, then the come at which the population become zero as:

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

Answer 1

The correct option is A $2\log 18$

The given differential equation $\frac{dp\left(t\right)}{dt}=0.5p\left(t\right)-450$
$\Rightarrow \frac{dp\left(t\right)}{dt}=\frac{p\left(t\right)-900}{2}$
$\Rightarrow 2\int \frac{dp\left(t\right)}{p\left(t\right)-900}=\int dt$
$\Rightarrow 2ln|p\left(t\right)-900=t+c$
At $t=0,2ln50=0+c\Rightarrow c=2ln50$
$\therefore 2ln|p\left(t\right)-900|=t+2ln50$
$t=2(ln900-ln50)=2ln\left(\frac{900}{50}\right)$
$t=2ln18$ .