Q. Let y be the number of people in a village at time t. Assume that the rate of change of the population is proportional to the number of people in the village at any time and further assume that the population never increase in time. Then, the population of the village at any fixed $t$ is given by
A) $y = e^{kt} + c,$ for some constants$\ c \leq 0$ and $k \geq \$0
B) $y = ce^{kt},$ for some constants $c \geq 0$ and $k \leq 0$
C) $y = e^{ct} + k,\$for some constants $c \leq 0$ and $k \geq 0$
D) $y = k\ e^{ct}\ ,$ for some constants $c \geq 0$ and $k \leq 0$
Explanation
(b)
Given, $\frac{dy}{dt} \propto y,where$ y is the position of village
$\Rightarrow \frac{1}{y}dy = k\ dt$
$\Rightarrow \ \ \ \log{y = \log{c + kt}\ \ \ }$ [on integrating]
$\Rightarrow \ \ \ \ \log\frac{y}{c} = kt \Rightarrow y = ce^{kt}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.