Exercise 1.1.32

Question

Researchers measured the blood alcohol concentration (BAC) of eight adult male subjects after rapid consumption of 30 mL of ethanol (corresponding to two standard alcoholic drinks). The table shows the data they obtained by averaging the BAC (in g/dL) of the eight men.
    \begin{enumerate}[label=(\alph*)]
        \item Use the readings to sketch a graph of the BAC as a function of $t$.
        \item Use your graph to describe how the effect of alcohol varies with time.
    \end{enumerate}
    \begin{table}[H]\centering
\begin{tabular}{|l|l|l|l|l|}
\hline
$t$, hours & BAC   &  & $t$, hours & BAC   \ \hline
0          & 0     &  & 1.75       & 0.022 \ \hline
0.2        & 0.025 &  & 2.0        & 0.018 \ \hline
0.5        & 0.041 &  & 2.25       & 0.015 \ \hline
0.75       & 0.040 &  & 2.5        & 0.012 \ \hline
1.0        & 0.033 &  & 3.0        & 0.007 \ \hline
1.5        & 0.024 &  & 4.0        & 0.001 \ \hline
\end{tabular}
\end{table}
\end{exercise}
\footnotesize{Source: Adapted from P. Wilkinson et al., “Pharmacokinetics of Ethanol after Oral Administration in the Fasting State,” Journal of Pharmacokinetics and Biopharmaceutics 5 (1977): 207–24.}

Khagan Gulusoy · Accepted Answer

\begin{enumerate}[label=(\alph*)]
    \item We sketch the data as follows.
    \begin{figure}[H] \centering
    \begin{tikzpicture}
        \begin{axis}[height=8cm, width=8cm, axis lines=left, xmin=0, xmax=4, ymin=0, ymax=0.05, xtick={0,0.5,...,4}, ytick={0,0.01,...,0.04}, scaled y ticks=false, yticklabel style={/pgf/number format/fixed, /pgf/number format/precision=5}, xlabel={$t,$ hours}, ylabel={$BAC$}]
            \addplot[mark=*, only marks] coordinates {(0, 0) (0.2,0.025) (0.5,0.041) (0.75,0.040) (1.0,0.033) (1.25,0.029) (1.5,0.024) (1.75, 0.022) (2.0,0.018) (2.25, 0.015) (2.5, 0.012) (3.0, 0.007) (3.5, 0.003) (4.0, 0.001)};

\addplot[domain=0:1]{0.0181818*x^4 + 0.0257576*x^3 - 0.168455*x^2 + 0.157515*x + 0}; % interpolate points 1 to 5
            \addplot[domain=1:2.25]{0.145067*x^5 - 1.184*x^4 + 3.78933*x^3 - 5.934*x^2 + 4.5246*x - 1.308}; % interpolate points 5 to 10 
            \addplot[domain=2.25:4]{0.00106667*x^4 - 0.0125333*x^3 + 0.0569333*x^2 - 0.127467*x + 0.129}; % interpolate points 10 to 15        
        \end{axis}
    \end{tikzpicture}
    \end{figure}
    \item 
    The concentration of alcohol in blood increases rapidly immediately after the ethanol administration. After approximately half an hour, alcohol concentration starts to decrease gradually. It takes more than 3 hours for the alcohol to disappear from the blood altogether.
\end{enumerate}

Exercise 1.1.32

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$t$ , hours	BAC	$t$ , hours	BAC

0	0	1.75	0.022

0.2	0.025	2.0	0.018

0.5	0.041	2.25	0.015

0.75	0.040	2.5	0.012

1.0	0.033	3.0	0.007

1.5	0.024	4.0	0.001

Exercise 1.1.32

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