Q32 / 105 · NTA Abhyas (JEE Main) 2020 · Mathematics · Differential Equations

The population p(t) at a time t of a certain mouse species satisfies the differential equation  dptdt=0.5p(t)-450. If p(0)=850, then the time at which the population becomes zero is

  • A
    12ln18
     
  • B
    ln18
     
  • Correct answer
    C
    2ln18
     
  • D
    ln9
     

Solution

Let, p=p(t)

dp dt = 0.5 p - 450

        = p - 9 0 0 2

⇒∫850p2p-900dp=∫0tdt

⇒2logp-900850p=t

⇒2logp-900-log-50=t

⇒ 2 log p - 900 - 5 0 = t  ...(1)

⇒ | p - 900 - 5 0 |= e t/2

⇒ p = 9 0 0 - 5 0 e t/2

Let, when t=T, p=0 (using (1))

 ⇒T=2ln⁡18

Hence, the correct answer is option: 'C'.

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