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Step-by-Step Solution
Step 1: Write down the differential equation
The problem states that the population of rabbits, denoted by $p(t)$, satisfies the differential equation
$$ \frac{dp(t)}{dt} = \frac{1}{2} \, p(t) - 200. $$
Step 2: Separate the variables
Rearrange the differential equation to separate $p(t)$ and $t$ on opposite sides:
$$ \frac{dp(t)}{\frac{1}{2} p(t) - 200} = dt. $$
Step 3: Integrate both sides
We now integrate both sides. Let us first rewrite the left side in a convenient form for substitution:
$$ \int \frac{dp(t)}{\frac{1}{2} p(t) - 200} = \int dt. $$
Let
$$ s = \frac{1}{2}p(t) - 200. $$
Then
$$ \frac{ds}{dt} = \frac{1}{2} \frac{dp(t)}{dt} \quad \Longrightarrow \quad dp(t) = 2\,ds. $$
Substituting these into the integral gives:
$$ \int \frac{1}{\frac{1}{2}p(t) - 200} \, dp(t) = \int \frac{2}{s}\, ds. $$
Hence, the integral becomes:
$$ \int \frac{2}{s}\, ds = \int 1 \, dt. $$
Step 4: Integrate and simplify
Perform the integrations:
$$ 2 \ln|s| = t + C, $$
where $C$ is the constant of integration.
Rewrite in terms of $p(t)$:
$$ 2 \ln\left|\frac{1}{2}p(t) - 200\right| = t + C. $$
Exponentiating both sides, we get:
$$ \left|\frac{1}{2}p(t) - 200\right| = e^{\frac{t}{2} + \frac{C}{2}}. $$
We can write $e^{\frac{C}{2}}$ as a new constant $k$ (since $C/2$ is just another constant). Hence:
$$ \frac{1}{2}p(t) - 200 = \pm k e^{\frac{t}{2}}. $$
For simplicity, assume it is a positive or negative constant that we will determine using the initial condition. Let us write it as:
$$ \frac{p(t)}{2} - 200 = K\, e^{\frac{t}{2}},
\quad \text{where } K=\pm k. $$
Thus,
$$ p(t) = 400 + 2K\, e^{\frac{t}{2}}. $$
Step 5: Use the initial condition to determine the constant
We are given $p(0) = 100$. Substitute $t=0$ and $p(0) = 100$ into the general solution:
$$ 100 = 400 + 2K\, e^{\frac{0}{2}} = 400 + 2K. $$
So:
$$ 100 - 400 = 2K \quad \Longrightarrow \quad -300 = 2K \quad \Longrightarrow \quad K = -150. $$
Step 6: Write the final solution
Substitute $K = -150$ back into the general solution:
$$ p(t) = 400 + 2(-150)\, e^{\frac{t}{2}}. $$
$$ p(t) = 400 - 300\, e^{\frac{t}{2}}. $$
This matches the correct answer given in the options.
