Question

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

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

Hint:

For solving first-order linear differential equations, use separation of variables and then integrate to find the general solution.

Answer 1

The Correct Option is A

The given differential equation is: \[ \frac{d p(t)}{dt} = \frac{1}{2} p(t) - 450. \] Rewriting the equation: \[ \frac{d p(t)}{dt} = \frac{p(t) - 900}{2}. \] Next, integrate both sides: \[ 2 \int \frac{d p(t)}{p(t) - 900} = \int - dt. \] This results in: \[ 2 \ln|p(t) - 900| = -t + C. \] Using the initial condition \( p(0) = 850 \): \[ 2 \ln(50) = C. \] Now, solving for \( p(t) \) when \( p(t) = 0 \): \[ p(t) = 900 - 50e^{-t/2}. \] Set \( p(t) = 0 \), solving for \( t \) gives: \[ t = 2 \ln 18. \]