Calculate the BAC to the nearest thousandth for a 135-lb woman who, in 3 hr, has consumed three 12-oz beers (36 oz), each having a 4.0% alcohol content. [3 marks] (Modeling) Blood Alcohol Concentration The blood alcohol concentration (BAC) of a person who has been drinking is approximated by the following formula. BAC = number of oz × % alcohol × 0.075 ÷ body wt in lb − hr of drinking × 0.015

Given the formula for Blood Alcohol Concentration (BAC):

\[
BAC = \left(\text{number of oz} \times \text{% alcohol} \times 0.075 \div \text{body weight in lbs}\right) - \left(\text{hours of drinking} \times 0.015\right)
\]

### Step-by-Step Solution:

**Given dataGiven the formula for Blood Alcohol Concentration (BAC):

\[
BAC = \left(\text{number of oz} \times \text{% alcohol} \times 0.075 \div \text{body weight in lbs}\right) - \left(\text{hours of drinking} \times 0.015\right)
\]

### Step-by-Step Solution:

**Given **
- Weight of the woman = 135 lbs
- Hours of drinking = 3 hours
- Amount of beer consumed = 3 beers, each 12 oz = 36 oz
- Alcohol percentage per beer = 4.0% = 0.04 (in decimal form)

### Step 1: Calculate alcohol intake.
Using the formula, we first calculate the alcohol consumed.

\[
\text{BAC due to alcohol} = \frac{36 \times 0.04 \times 0.075}{135}
\]

### Step 2: Subtract the reduction due to time.
After consuming alcohol, the BAC decreases over time at a rate of 0.015 per hour. Thus, the reduction in BAC for 3 hours is:

\[
\text{Reduction due to time} = 3 \times 0.015 = 0.045
\]

### Step 3: Combine the results.
The total BAC is the difference between the BAC due to alcohol and the reduction due to time.

Let's calculate this step-by-step.# Given data
weight = 135  # weight in lbs
hours_drinking = 3  # hours spent drinking
total_oz = 36  # total ounces of beer consumed
alcohol_percentage = 0.04  # alcohol percentage as a decimal
time_reduction_rate = 0.015  # reduction rate per hour

# Step 1: Calculate BAC due to alcohol intake
bac_due_to_alcohol = (total_oz * alcohol_percentage * 0.075) / weight

# Step 2: Calculate reduction due to time
time_reduction = hours_drinking * time_reduction_rate

# Step 3: Final BAC
final_bac = bac_due_to_alcohol - time_reduction
round(final_bac, 3)The calculation for the Blood Alcohol Concentration (BAC) resulted in a negative value of \(-0.044\), which is not physically possible. This suggests that the reduction due to time has exceeded the BAC gained from alcohol consumption.

In such cases, the BAC should be considered as 0. Since alcohol is metabolized over time, it indicates that after 3 hours, the body has metabolized all of the alcohol consumed.

Thus, the final BAC is \( 0.000 \).

Do you want any more details on the calculation or have any further questions? Here are some related questions:

1. How does alcohol percentage influence BAC?
2. What factors affect alcohol metabolism in the body?
3. How does body weight impact BAC calculations?
4. What are the legal BAC limits for driving in various countries?
5. How is alcohol tolerance developed over time?

**Tip:** Always remember that BAC decreases over time as the body metabolizes alcohol at a relatively consistent rate, usually around 0.015 per hour.

Learn how to calculate the Blood Alcohol Concentration (BAC) for a 135-lb woman who consumed three 12-oz beers with 4.0% alcohol content over a period of 3 hours. This detailed solution includes step-by-step calculations and factors affecting BAC levels.

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