Question

The population \( p(t) \) at time \( t \) of a certain mouse species follows the differential equation \( \frac{dp(t)}{dt} = 0.5p(t) - 450 \). If \( p(0) = 850 \), then the time at which the population becomes zero is:

• A. \( \log 9 \)
• B. \( \frac{1}{2} \log 18 \)
• C. \( \log 18 \)
• D. \( 2 \log 18 \)

Hint:

Carefully solve the linear first-order differential equation and use the initial condition.

Answer 1

The Correct Option is D

Step 1: Solve the differential equation \( \frac{dp}{dt} - 0.5p = -450 \).
Integrating factor \( \mu(t) = e^{-0.5t} \).
Solution: \( p(t) e^{-0.5t} = 900 e^{-0.5t} + C \Rightarrow p(t) = 900 + Ce^{0.5t} \).

Step 2: Use \( p(0) = 850 \).
\( 850 = 900 + C \Rightarrow C = -50 \).
\( p(t) = 900 - 50e^{0.5t} \).

Step 3: Find \( t \) when \( p(t) = 0 \).
\( 0 = 900 - 50e^{0.5t} \Rightarrow e^{0.5t} = 18 \).

Step 4: Solve for \( t \).
\( 0.5t = \ln 18 \Rightarrow t = 2 \ln 18 \).