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.
\nSolve \( \frac{dp}{dt} - 0.5p = -450 \).
\nIntegrating factor: \( \mu(t) = e^{-0.5t} \).
\nSolution: \( p(t) e^{-0.5t} = 900 e^{-0.5t} + C \), which simplifies to \( p(t) = 900 + Ce^{0.5t} \).
\n\n
Step 2: Apply the initial condition.
\nUse \( p(0) = 850 \).
\nTherefore, \( 850 = 900 + C \), so \( C = -50 \).
\nThus, \( p(t) = 900 - 50e^{0.5t} \).
\n\n
Step 3: Determine when p(t) is zero.
\nFind \( t \) when \( p(t) = 0 \).
\nThis gives us \( 0 = 900 - 50e^{0.5t} \), or \( e^{0.5t} = 18 \).
\n\n
Step 4: Calculate t.
\nSolve for \( t \): \( 0.5t = \ln 18 \), therefore \( t = 2 \ln 18 \).