The population $\mathrm{P}=\mathrm{P}(\mathrm{t})$ at time 't' of a certain species follows the differential equation $\frac{\mathrm{dP}}{\mathrm{dt}}=0.5 \mathrm{P}-450$. If $P(0)=850$, then the time at which population becomes zero is:

(A) $\frac{1}{2} \log _{\mathrm{e}} 18$

JEE Main · 2021 · 24 Feb 2021 · Shift 1 · Differential Equations

The population

P = P (t)

at time 't' of a certain species follows the differential equation

\frac{dP}{dt} = 0.5 P - 450

. If

P (0) = 850

, then the time at which population becomes zero is:

Answer: (C)

The differential equation is

\frac{d P}{d t} = 0.5 P - 450

.

Separating variables:

\frac{d P}{P - 900} = 0.5 d t

Integrating:

ln ∣ P - 900∣ = 0.5 t + C

Using

P (0) = 850

ln ∣ - 50∣ = C

, so

ln 50 = C

Thus:

ln ∣ P - 900∣ = 0.5 t + ln 50

When

P = 0

ln 900 = 0.5 t + ln 50

0.5 t = ln 900 - ln 50 = ln 18

t = 2 ln 18 = 2 lo g_{e} 18

JEE Main · 2021 · 24 Feb 2021 · Shift 1 · Single correct option

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