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 provided differential equation is:\[\frac{d p(t)}{dt} = \frac{1}{2} p(t) - 450.\]The equation is rewritten as:\[\frac{d p(t)}{dt} = \frac{p(t) - 900}{2}.\]Integrating both sides yields:\[2 \int \frac{d p(t)}{p(t) - 900} = \int - dt.\]The integration results in:\[2 \ln|p(t) - 900| = -t + C.\]Applying the initial condition \( p(0) = 850 \):\[2 \ln(50) = C.\]Solving for \( p(t) \) when \( p(t) = 0 \):\[p(t) = 900 - 50e^{-t/2}.\]Setting \( p(t) = 0 \) and solving for \( t \) gives:\[t = 2 \ln 18.\]