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 time at which the population becomes zero is

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

Answer 1

The correct option is C $2\ln 18$

We have,

$\begin{array}{c}\frac{d\left(p\left(t\right)\right)}{dt}=\frac{1}{2}p\left(t\right)-450\\ \frac{d\left(p\left(t\right)\right)}{dt}=\frac{p\left(t\right)-900}{2}\\ 2\int \frac{d\left(p\left(t\right)\right)}{p\left(t\right)-900}=\int dt\\ 2In\left|p\left(t\right)-900\right|=t+c\\ t=0\\ \Rightarrow 2In50=0+c\\ \therefore 2In900=t+2In50\\ t=2\left(In900-In50\right)\\ =2In\left(\frac{900}{50}\right)\\ =2In18\end{array}$

Hence, option $C$ is correct answer.