Question

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

• A. $2 \,\ell n \,18$
• B. $\ell n \,9$
• C. $\frac{1}{2} \,\ell n \,18$
• D. $\ell n \,18$

Answer 1

Answer: (A)

$2 \,\ell n \,18$

Answer 2

$2 \frac{dp\left(t\right)}{900 - p\left(t\right)}= - dt$ $- 2\,\ell n \left(900 - p\left(t\right)\right) = - t + c$ when $t = 0, p\left(0\right) = 850$ $- 2\ell n\left(50\right) = c$ $\therefore\quad2\ell n\left(\frac{50}{900-p\left(t\right)}\right) = -t$ $900 - p\left(t\right) = 50 e^{t/2}$ $p\left(t\right) = 900 - 50 e^{t/2}$ let $p\left(t_{1}\right) = 0$ $0 = 900 - 50\,e^{\frac{t_{1}}{2}}$ $\therefore\quad t_{1} = 2 \,\ell n \,18$