To find the rate at which the level of pollutant $P$ is increasing with respect to the number of residents $n$ (in thousands), we need to calculate the derivative of $P$ with respect to $n$.

The given equation is:

$P = 0.25 \sqrt{0.5n^2 + 4n + 22}$

We will use the chain rule to differentiate $P$ with respect to $n$.

Step 1: Differentiate $P$ with respect to $n$

Let $u = 0.5n^2 + 4n + 22$. Then, we have:

$P = 0.25 \sqrt{u}$

Now, differentiate $P$ with respect to $u$:

$\frac{dP}{du} = 0.25 \cdot \frac{1}{2\sqrt{u}} = \frac{0.25}{2\sqrt{u}} = \frac{0.125}{\sqrt{u}}$

Next, differentiate $u$ with respect to $n$:

$\frac{du}{dn} = \frac{d}{dn} (0.5n^2 + 4n + 22) = n + 4$

Step 2: Apply the chain rule

Now, using the chain rule:

$\frac{dP}{dn} = \frac{dP}{du} \cdot \frac{du}{dn}$

Substitute $\frac{dP}{du}$ and $\frac{du}{dn}$:

$\frac{dP}{dn} = \frac{0.125}{\sqrt{u}} \cdot (n + 4)$

Step 3: Substitute $u$ back in terms of $n$

Recall $u = 0.5n^2 + 4n + 22$. Therefore, the derivative becomes:

$\frac{dP}{dn} = \frac{0.125 (n + 4)}{\sqrt{0.5n^2 + 4n + 22}}$

Step 4: Evaluate the derivative at $n = 16$

Substitute $n = 16$ (since 16,000 people correspond to $n = 16$):

1. Compute $u$ when $n = 16$: $u = 0.5(16)^2 + 4(16) + 22 = 0.5(256) + 64 + 22 = 128 + 64 + 22 = 214$

2. Substitute $n = 16$ and $u = 214$ into $\frac{dP}{dn}$: $\frac{dP}{dn} = \frac{0.125(16 + 4)}{\sqrt{214}} = \frac{0.125 \times 20}{\sqrt{214}} = \frac{2.5}{\sqrt{214}}$

3. Compute $\sqrt{214}$: $\sqrt{214} \approx 14.6287$

4. Finally, calculate $\frac{2.5}{14.6287}$: $\frac{2.5}{14.6287} \approx 0.171$

Final Answer

When the population of the community is 16,000 people, the level of pollutant is increasing at approximately 0.171 parts per million per thousand people.

Would you like further details on any part of this solution or have any questions?

Related Questions:

1. How would the rate of increase change if the population is 20,000 people?
2. What is the physical significance of the rate of change of $P$ with respect to $n$?
3. Can you find the value of $P$ itself for a community of 16,000 people?
4. How does the pollutant level change as the population approaches zero?
5. How would the equation change if the units of the population were given in hundreds instead of thousands?

Tip

When differentiating square roots, always remember to simplify the expression as much as possible before taking the derivative to avoid complex calculations.