Question

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

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

Hint:

In solving differential equations, always separate the variables and integrate carefully. The solution will give you the behavior of the population over time.

Answer 1

The Correct Option is C

Step 1: Solve the differential equation.
We have the equation \( \frac{dP}{dt} = 0.5P(t) - 450 \). To solve it, we separate the variables and integrate.
Step 2: Apply the initial condition.
Using the initial condition \( P(0) = 850 \), we solve for the constant of integration. After solving, we find that the time when the population becomes zero is \( 2 \log 18 \).
Step 3: Conclusion.
The correct answer is (C) \( 2 \log 18 \).