Question

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

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

Hint:

For linear ODEs:  Use integrating factor. Apply initial condition at end. 

Answer 1

The Correct Option is C

Concept: Solve linear differential equation. Step 1: Solve homogeneous form.} \[ \frac{dp}{dt}-0.5p=-450 \] Integrating factor: \[ e^{-0.5t} \] Step 2: General solution.} \[ p(t)=Ce^{0.5t}+900 \] Step 3: Use initial condition.} \[ 850=C+900 \Rightarrow C=-50 \] \[ p(t)=900-50e^{0.5t} \] Step 4: Set \( p(t)=0 \).} \[ 900=50e^{0.5t} \Rightarrow e^{0.5t}=18 \] \[ t=\log 18 \]