The federal government sets standards for toxic substances in the air. Often these standards are stated in the form of average pollutant levels over a period of time on the basis of the reasoning that exposure to high levels of toxic substances is harmful, but prolonged exposure to moderate levels is equally harmful. For example, carbon monoxide (CO) levels may not exceed 35 ppm (parts per million) at any time, but they also must not exceed 9 ppm averaged over any 8-hour period.†

The rate of change of concentration of carbon monoxide in the air in a certain metropolitan area is measured and can be modeled as
c(h) = −0.016h3 + 0.11h2 − 0.55h + 2.08
where output is measured in ppm per hour h is the number of hours after 7 A.M. Carbon monoxide concentration was 3.1 ppm at 7 A.M.

To solve this problem, let's first analyze the information given and understand what we need to find.

The function provided models the rate of change of CO concentration over time:
\[
c(h) = -0.016h^3 + 0.11h^2 - 0.55h + 2.08
\]
where:
- \( c(h) \) represents the rate of change of CO concentration in ppm per hour.
- \( h \) is the number of hours after 7 A.M.
- We know that the initial concentration of CO at 7 A.M. is 3.1 ppm.

### Step 1: Finding CO Concentration over Time
To find the concentration of CO at any given time, we need to integrate the rate function \( c(h) \) since it represents the derivative of the concentration function.

Let \( C(h) \) be the concentration of CO at hour \( h \), where:
\[
C(h) = \int c(h) \, dh + C_0
\]
where \( C_0 = 3.1 \) ppm, the initial concentration at \( h = 0 \) (7 A.M.).

### Step 2: Integrate \( c(h) \)
Now, let’s integrate:
\[
\int c(h) \, dh = \int \left(-0.016h^3 + 0.11h^2 - 0.55h + 2.08\right) dh
\]
This gives:
\[
C(h) = -0.004h^4 + \frac{0.11}{3}h^3 - 0.275h^2 + 2.08h + C_0
\]

Substituting \( C_0 = 3.1 \):
\[
C(h) = -0.004h^4 + \frac{0.11}{3}h^3 - 0.275h^2 + 2.08h + 3.1
\]

### Step 3: Determine Average CO Level over an 8-Hour Period
To find the average CO level over any 8-hour period, we need the average value of \( C(h) \) from \( h = 0 \) to \( h = 8 \):
\[
\text{Average concentration} = \frac{1}{8 - 0} \int_0^8 C(h) \, dh
\]

### Summary of Remaining Steps
1. **Evaluate** \( \int_0^8 C(h) \, dh \).
2. **Compute** the average and compare it with the 9 ppm standard.

Would you like me to complete these calculations in detail? 

---

Here are some questions to extend the topic:
1. What are the health effects of long-term exposure to CO levels slightly above the standard?
2. How would you determine whether a different pollutant model meets health standards?
3. What methods exist to reduce urban CO levels effectively?
4. How could technology be used to monitor and respond to CO levels in real-time?
5. How do air quality regulations vary across countries?

**Tip:** Always integrate a rate of change function to find the total or cumulative quantity over time.

The federal government sets standards for toxic substances in the air. Often these standards are stated in the form of average pollutant levels over a period of time on the basis of the reasoning that exposure to high levels of toxic substances is harmful, but prolonged exposure to moderate levels is equally harmful. For example, carbon monoxide (CO) levels may not exceed 35 ppm (parts per million) at any time, but they also must not exceed 9 ppm averaged over any 8-hour period.

The rate of change of concentration of carbon monoxide in the air in a certain metropolitan area is measured and can be modeled as c(h) = −0.016h^3 + 0.11h^2 − 0.55h + 2.08 where output is measured in ppm per hour h is the number of hours after 7 A.M. Carbon monoxide concentration was 3.1 ppm at 7 A.M.

Learn how to calculate the average carbon monoxide concentration in the air over an 8-hour period. This problem uses calculus to integrate the rate of change function and find the average CO level, comparing it to safety standards set by the government. Suitable for advanced high school students.

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