JEE MAIN - Mathematics (2021 - 24th February Morning Shift - No. 3)
The population P = P(t) at time 't' of a certain species follows the differential equation
$${{dP} \over {dt}}$$ = 0.5P – 450. If P(0) = 850, then the time at which population becomes zero is :
$${{dP} \over {dt}}$$ = 0.5P – 450. If P(0) = 850, then the time at which population becomes zero is :
$${\log _e}18$$
$${1 \over 2}{\log _e}18$$
2$${\log _e}18$$
$${\log _e}9$$
Explanation
$${{dp} \over {dt}} = {{p - 900} \over 2}$$
$$\int\limits_{850}^0 {{{dp} \over {p - 900}} = \int\limits_0^t {{{dt} \over 2}} } $$
$$ \Rightarrow $$ $$\ln |p - 900|_{850}^0 = {t \over 2}$$
$$ \Rightarrow $$ $$\ln 900 - \ln 50 = {t \over 2}$$
$$ \Rightarrow $$ $${t \over 2} = \ln 18$$
$$ \Rightarrow t = 2\ln 18$$
$$\int\limits_{850}^0 {{{dp} \over {p - 900}} = \int\limits_0^t {{{dt} \over 2}} } $$
$$ \Rightarrow $$ $$\ln |p - 900|_{850}^0 = {t \over 2}$$
$$ \Rightarrow $$ $$\ln 900 - \ln 50 = {t \over 2}$$
$$ \Rightarrow $$ $${t \over 2} = \ln 18$$
$$ \Rightarrow t = 2\ln 18$$
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