Question

The population $p\left(t\right)$ at time of a certain mouse species satisfies the differential equation $\frac{d\left[p\left(t\right)\right]}{dt}=0.5·p\left(t\right)-450$. If $p\left(0\right)=850$, then the time at which the population becomes $0$ is

• A. $2\ln \left(18\right)$
• B. $\ln \left(19\right)$
• C. $\frac{\ln \left(18\right)}{2}$
• D. $\ln \left(18\right)$

Answer 1

The correct option is A

$2\ln \left(18\right)$

Explanation of the correct option:

Solve the given differential equation to satisfy the given condition

A differential equation $\frac{d\left[p\left(t\right)\right]}{dt}=0.5·p\left(t\right)-450$ is given

Since, the population of mice at $t=0$ is $850$.

Assume that, the population of mice at $t=t$ is $p$.

Rewrite the given differential equation as follows:

$\frac{d\left[p\left(t\right)\right]}{0.5·p\left(t\right)-450}=dt$

Solve the given differential equation as follows:

$$\begin{gathered} {\int }_{850}^p\left(\frac{1}{0.5·p\left(t\right)-450}\right)d\left[p\left(t\right)\right]={\int }_0^{t}dt \\ \Rightarrow 2{\int }_{850}^p\left(\frac{1}{p\left(t\right)-900}\right)d\left[p\left(t\right)\right]={\int }_0^{t}dt \\ \Rightarrow 2{\left[\ln \left|p\left(t\right)-900\right|\right]}_{850}^p={\left[t\right]}_0^t \\ \Rightarrow 2\left[\ln \left|p\left(t\right)-900\right|-\log \left|850-900\right|\right]=t \\ \Rightarrow 2\ln \left|\frac{p\left(t\right)-900}{50}\right|=t \end{gathered}$$

Now, find the time at which the population becomes $0$.

$$\begin{gathered} t=2\ln \left|\frac{0-900}{50}\right| \\ \Rightarrow t=2\ln \left(\frac{900}{50}\right) \\ \Rightarrow t=2\ln \left(18\right) \end{gathered}$$

Since at time $t=2\ln \left(18\right)$ the population becomes $0$.

Therefore, the time at which the population becomes $0$ is $2\ln \left(18\right)$.

Hence, option $A$, $2\ln \left(18\right)$ is the correct answer.