JEE Main & Advanced JEE Main Solved Paper-2014

  • question_answer
    Let the population of rabbits surviving at a time t be governed by the differential equation\[\frac{dp(t)}{dt}=\frac{1}{2}p(t)-200.\] If \[p(0)=100,\]then p(t) equals :   JEE Main  Solved  Paper-2014

    A) \[400-300\,{{\text{e}}^{\text{t/2}}}\]    

    B) \[300-200\,{{\text{e}}^{\text{-t/2}}}\]

    C) \[600-500\,{{\text{e}}^{\text{t/2}}}\]    

    D) \[400-300\,{{\text{e}}^{\text{-t/2}}}\]

    Correct Answer: A

    Solution :

    Rearranging the equation we get, \[\frac{dp(t)}{p(t)-400}=\frac{1}{2}dt\]                                  ?(1) Integrating (1) on both sides we get\[p(t)=400+k{{e}^{t/2}},\]where k is a constant of integration. Using p(0) = 100, we get k = −300 \[\therefore \]the relation is\[p(t)=400-300\text{ }{{e}^{t/2}}\]

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