Key Concepts and Formulas

• Separable Differential Equations: A differential equation of the form $\frac{dy}{dx} = f(x)g(y)$ can be separated as $\frac{dy}{g(y)} = f(x)dx$.
• Integration: $\int \frac{1}{x} dx = \ln|x| + C$, where $C$ is the constant of integration.
• Logarithm Properties: $\ln(a) - \ln(b) = \ln(\frac{a}{b})$, $e^{\ln(x)} = x$.

Step-by-Step Solution

Step 1: Rewrite the Differential Equation

The given differential equation is $\frac{dp(t)}{dt} = 0.5p(t) - 450$. We want to rewrite it in a form suitable for separation of variables.

$\frac{dp(t)}{dt} = \frac{1}{2}p(t) - 450$ $\frac{dp(t)}{dt} = \frac{p(t) - 900}{2}$

This expresses the derivative as a function of $p(t)$ only, preparing it for separation.

Step 2: Separate the Variables

Separate the variables by multiplying both sides by $dt$ and dividing by $(p(t) - 900)$:

$\frac{dp(t)}{p(t) - 900} = \frac{1}{2} dt$

Now all terms involving $p(t)$ are on the left and terms involving $t$ are on the right.

Step 3: Integrate Both Sides

Integrate both sides of the equation:

$\int \frac{dp(t)}{p(t) - 900} = \int \frac{1}{2} dt$ $\ln|p(t) - 900| = \frac{1}{2}t + C$

Here, $C$ is the constant of integration.

Step 4: Apply the Initial Condition

We are given $p(0) = 850$. Substitute $t=0$ and $p(0) = 850$ into the equation:

$\ln|850 - 900| = \frac{1}{2}(0) + C$ $\ln|-50| = C$ $\ln(50) = C$

So, the constant of integration is $C = \ln(50)$. Substituting this back into the general solution:

$\ln|p(t) - 900| = \frac{1}{2}t + \ln(50)$

Since $p(0) = 850 < 900$, and $\frac{dp}{dt} = 0.5p - 450$, $p(t)$ will decrease until it reaches zero. Therefore, $p(t) - 900 < 0$, which implies $|p(t) - 900| = 900 - p(t)$.

$\ln(900 - p(t)) = \frac{1}{2}t + \ln(50)$

Step 5: Solve for $p(t)$

Isolate $p(t)$:

$\ln(900 - p(t)) - \ln(50) = \frac{1}{2}t$ $\ln\left(\frac{900 - p(t)}{50}\right) = \frac{1}{2}t$ $\frac{900 - p(t)}{50} = e^{\frac{1}{2}t}$ $900 - p(t) = 50e^{\frac{1}{2}t}$ $p(t) = 900 - 50e^{\frac{1}{2}t}$

Step 6: Find the Time When the Population is Zero

Set $p(t) = 0$ and solve for $t$:

$0 = 900 - 50e^{\frac{1}{2}t}$ $50e^{\frac{1}{2}t} = 900$ $e^{\frac{1}{2}t} = \frac{900}{50} = 18$ $\frac{1}{2}t = \ln(18)$ $t = 2\ln(18)$

The time at which the population becomes zero is $2\ln(18)$.

Common Mistakes & Tips

• Remember the constant of integration, $C$, and solve for it using the initial condition.
• Be careful with signs, especially when dealing with the absolute value in the logarithm.
• Use logarithm properties correctly to simplify the equation.

Summary

We solved the separable differential equation by separating variables, integrating both sides, using the initial condition to find the constant of integration, and then solving for $p(t)$. Finally, we set $p(t)=0$ and solved for $t$ to find the time when the population reaches zero. The time at which the population becomes zero is $2\ln(18)$.

Final Answer

The final answer is $\boxed{2\ln 18}$, which corresponds to option (A).