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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 :

JEE · Math · previous-year question

  1. A.$${\log _e}18$$
  2. B.$${1 \over 2}{\log _e}18$$
  3. C.2$${\log _e}18$$correct
  4. D.$${\log _e}9$$

Answer

C. 2$${\log _e}18$$

Explanation

$${{dp} \over {dt}} = {{p - 900} \over 2}$$ $$\int\limits_{850}^0 {{{dp} \over {p - 900}} = \int\limits_0^t {{{dt} \over 2}} } $$ $$ \Rightarrow $$ $$\ln |p - 900|_{850}^0 = {t \over 2}$$ $$ \Rightarrow $$ $$\ln 900 - \ln 50 = {t \over 2}$$ $$ \Rightarrow $$ $${t \over 2} = \ln 18$$ $$ \Rightarrow t = 2\ln 18$$

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