The polynomial function ​A(t)equals0.003407t Superscript 3minus0.03764t Superscript 2plus0.1068tplus0.007 gives the approximate blood alcohol concentration in a person t hours after drinking some alcohol on an empty​ stomach, for t in the interval​ [0,5].
a. Graph​ A(t) on 0less than or equalstless than or equals5.
b. Using the graph from part​ a, estimate the time of maximum alcohol concentration.
c. Use the graph from part a to estimate the period in which the concentration of the alcohol in the person exceeds 0.05.

### b. Estimate the time of maximum alcohol concentration:
From the graph, the peak occurs roughly around \( t \approx 1.5 \) hours. This suggests that the maximum alcohol concentration is reached approximately 1.5 hours after drinking.

### c. Estimate the period in which the alcohol concentration exceeds 0.05:
Based on the graph, the blood alcohol concentration \( A(t) > 0.05 \) appears to occur from about \( t \approx 0.5 \) hours to \( t \approx 3.7 \) hours. Therefore, the alcohol concentration exceeds 0.05 for approximately the interval \( [0.5, 3.7] \) hours.

Let me know if you need more details on these estimates or the calculations involved!

Here are some related questions to expand understanding:

1. How would the maximum alcohol concentration change if the coefficients of the polynomial were slightly altered?
2. What is the significance of finding the period where alcohol concentration exceeds 0.05?
3. Can you derive the exact time of maximum alcohol concentration by finding the derivative of \( A(t) \)?
4. How could we calculate the total area under the curve for \( A(t) \) in the interval \( [0, 5] \)?
5. What would happen to the graph if the alcohol was consumed on a full stomach instead of an empty stomach?

**Tip:** The time of maximum concentration can be found precisely by setting the derivative of the function equal to zero and solving for \( t \).

The polynomial function A(t) = 0.003407t^3 - 0.03764t^2 + 0.1068t + 0.007 gives the approximate blood alcohol concentration in a person t hours after drinking some alcohol on an empty stomach, for t in the interval [0,5]. a. Graph A(t) on 0 ≤ t ≤ 5. b. Using the graph from part a, estimate the time of maximum alcohol concentration. c. Use the graph from part a to estimate the period in which the concentration of the alcohol in the person exceeds 0.05.

Explore how to graph the blood alcohol concentration over time using the cubic polynomial function A(t) = 0.003407t^3 - 0.03764t^2 + 0.1068t + 0.007. Learn how to estimate the maximum alcohol concentration time and the interval when the concentration exceeds 0.05. Suitable for high school and college students studying polynomial functions.

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