Starting with an initial value of $$ P(0)=30 $$, the population of a prairie dog community grows at a rate of $$ P^{\prime}(t)=20-\frac{t}{2} $$ (in units of prairie dogs/month), for $$ 0 \leq t \leq 40 $$. a. What is the population 13 months later? b. Find the population $$ P(t) $$ for $$ 0 \leq t \leq 40 $$. a. After 13 months, the population is $$ \square $$ prairie dogs. (Type a whole number. Round to the nearest prairie dog as needed.)

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We begin with the differential equation for the growth rate of the population:
$$
P'(t)=20-\frac{t}{2},
$$
and the initial condition:
$$
P(0)=30.
$$

### Part (a)

To find the population after 13 months, we integrate the growth rate from $$ t=0 $$ to $$ t=13 $$ and add the initial population:
$$
P(13)=P(0)+\int_{0}^{13}\left(20-\frac{t}{2}\right)dt.
$$

First, we compute the integral:
$$
\int_{0}^{13}\left(20-\frac{t}{2}\right)dt = \int_{0}^{13}20\,dt-\int_{0}^{13}\frac{t}{2}\,dt.
$$

Evaluate the first integral:
$$
\int_{0}^{13}20\,dt=20t\Big|_{0}^{13}=20(13)=260.
$$

Next, evaluate the second integral:
$$
\int_{0}^{13}\frac{t}{2}\,dt=\frac{1}{2}\int_{0}^{13}t\,dt=\frac{1}{2}\left(\frac{t^2}{2}\Big|_{0}^{13}\right)=\frac{1}{2}\left(\frac{169}{2}\right)=\frac{169}{4}=42.25.
$$

Subtracting, we have:
$$
\int_{0}^{13}\left(20-\frac{t}{2}\right)dt=260-42.25=217.75.
$$

Thus, the population at $$ t=13 $$ is:
$$
P(13)=30+217.75=247.75.
$$

Rounding to the nearest whole number, we have:
$$
P(13)=248 \quad \text{prairie dogs}.
$$

### Part (b)

To find the general population function $$ P(t) $$, we integrate $$ P'(t) $$:
$$
P(t)=\int\left(20-\frac{t}{2}\right)dt.
$$

Integrate term-by-term:
$$
\int 20\,dt=20t,
$$
$$
\int \frac{t}{2}dt=\frac{1}{2}\cdot\frac{t^2}{2}=\frac{t^2}{4}.
$$

So, the antiderivative is:
$$
P(t)=20t-\frac{t^2}{4}+C,
$$
where $$ C $$ is the constant of integration.

Using the initial condition $$ P(0)=30 $$:
$$
30=20(0)-\frac{0^2}{4}+C \quad \Longrightarrow \quad C=30.
$$

Thus, the population function is:
$$
P(t)=20t-\frac{t^2}{4}+30, \quad 0\leq t\leq40.
$$

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a. After 13 months, the population is $$248$$ prairie dogs.
a. After 13 months, the population is 248 prairie dogs.

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