JEE Main 2021 (Online) 24th February Morning ShiftQuestion 1
The population $P=P(t)$ at time '$t$' of a certain species follows the differential equation $\frac{dP}{dt}=0.5P-450$. If $P(0)=850$, then the time at which population becomes zero is :
Differential Equation
Solution
Concept. $\frac{dP}{dt}=0.5(P-900)$, so $P-900$ grows exponentially. Formulas used. (1) $\dfrac{dz}{dt}=kz\Rightarrow z=z_0e^{kt}$Step 1 - Solve. With $z=P-900$, by (1): $$P-900=(850-900)\,e^{t/2}=-50\,e^{t/2}.$$ Step 2 - $P=0$. $-900=-50\,e^{t/2}$, so $e^{t/2}=18$ and $t=2\ln18$. Answer: (D)