Question

the amount of alcohol in a person’s blood is known as blood alcohol concentration (bac), measured in grams of alcohol per deciliter of blood. a bac of 0.08, meaning 0.08%, indicates that a person has 8 parts alcohol per 10,000 parts blood. it is illegal to drive with a bac of 0.08 or higher. use the formula (\frac{600n}{w(0.6n + 169)}) where w is the weight of a 147-pound person and n is the number of drinks per hour, and a calculator, to compute the bac for integers from n=1 to n=10. how many drinks can the person consume in an hour without exceeding the legal measure of drunk driving? according to this model, the person can consume (square) drinks in an hour without exceeding the legal measure of drunk driving

Explanation:

The formula for BAC is given by $\frac{600n}{w(0.6n + 169)}$, where $w = 147$ (weight of the person in pounds) and $n$ is the number of drinks per hour. We need to find the largest integer $n$ (from $n = 1$ to $n = 10$) such that the BAC is less than $0.08$.

Step 1: Substitute $w = 147$ into the formula

The formula becomes $\frac{600n}{147(0.6n + 169)}$.

Step 2: Calculate BAC for $n = 1$

Substitute $n = 1$ into the formula:
\[

$$\begin{align*} \frac{600\times1}{147(0.6\times1 + 169)}&=\frac{600}{147(0.6 + 169)}\\ &=\frac{600}{147\times169.6}\\ &\approx\frac{600}{24931.2}\\ &\approx0.024 \end{align*}$$

\]
This is less than $0.08$.

Step 3: Calculate BAC for $n = 2$

Substitute $n = 2$ into the formula:
\[

$$\begin{align*} \frac{600\times2}{147(0.6\times2 + 169)}&=\frac{1200}{147(1.2 + 169)}\\ &=\frac{1200}{147\times170.2}\\ &\approx\frac{1200}{25019.4}\\ &\approx0.048 \end{align*}$$

\]
This is less than $0.08$.

Step 4: Calculate BAC for $n = 3$

Substitute $n = 3$ into the formula:
\[

$$\begin{align*} \frac{600\times3}{147(0.6\times3 + 169)}&=\frac{1800}{147(1.8 + 169)}\\ &=\frac{1800}{147\times170.8}\\ &\approx\frac{1800}{25107.6}\\ &\approx0.072 \end{align*}$$

\]
This is less than $0.08$.

Step 5: Calculate BAC for $n = 4$

Substitute $n = 4$ into the formula:
\[

$$\begin{align*} \frac{600\times4}{147(0.6\times4 + 169)}&=\frac{2400}{147(2.4 + 169)}\\ &=\frac{2400}{147\times171.4}\\ &\approx\frac{2400}{25195.8}\\ &\approx0.095 \end{align*}$$

\]
This is greater than $0.08$.

Since for $n = 3$ the BAC is approximately $0.072$ (less than $0.08$) and for $n = 4$ it is approximately $0.095$ (greater than $0.08$), the largest number of drinks the person can consume in an hour without exceeding the legal limit is $3$.

Answer:

3