The population P = P(t) at time t of a certain species follows the differential equation (dP)/(dt) = 0 . 5P - 450 . If P(0) = 850 , then the time at which population becomes zero is:

Correct answer: B. 2 _ e 18 (dP)/(dt) = 0 . 5P - 450 ⇒ ∫ _ 0 ^ t (dp)/(P - 900) = ∫ _ 0 ^ t (dt)/(2) ⇒ [In|P(t) - 900|] _ 0 ^ t = [(t)/(2)] _ 0 ^ t ⇒ In|P(t) - 900| - In|P(0) - 900| = (t)/(2) ⇒ In|P(t) - 900| - In|50| = (t)/(2) for P(t) = 0 ⇒ In|(900)/(50)| = (t)/(2) ⇒ t = 2In18

JEE Main 202124 FebruaryMorning Shift

Exam: JEE Main
Year: 2021
Date: 24 February
Shift: Morning Shift
Subject: Mathematics
Chapter: Differential Equations

JEE Main 2021 · 24 February · Morning Shift · Mathematics · Differential Equations

The population P = P(t) at time t of a certain species follows the...

The population

P = P\left(t\right)

at time

t

of a certain species follows the differential equation

\dfrac{dP}{dt} = 0 . 5P - 450

. If

P\left(0\right) = 850

, then the time at which population becomes zero is:

Options

A $\log_{e}9$
B $2\log_{e}18$
C $\log_{e}18$
D $\dfrac{1}{2}\log_{e}18$

Correct answer

B. $2\log_{e}18$

Step-by-step solution

\dfrac{dP}{dt} = 0 . 5P - 450

$⇒ {∫}_{0}^{t}\dfrac{dp}{P - 900} = {∫}_{0}^{t}\dfrac{dt}{2}$

$⇒ {\left[In\left|P\left(t\right) - 900\right|\right]}_{0}^{t} = {\left[\dfrac{t}{2}\right]}_{0}^{t}$

$⇒ In\left|P\left(t\right) - 900\right| - In\left|P\left(0\right) - 900\right| = \dfrac{t}{2}$

$⇒ In\left|P\left(t\right) - 900\right| - In\left|50\right| = \dfrac{t}{2}$

for $P\left(t\right) = 0$

$⇒ In\left|\dfrac{900}{50}\right| = \dfrac{t}{2} ⇒ t = 2In18$

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