Exercise 1.4.27

Question

A researcher is trying to determine the doubling time for a population of the bacterium Giardia lamblia. He starts a culture in a nutrient solution and estimates the bacteria count every four hours. His data are shown in the table.

\begin{figure}[ht]\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline Time (hours) & 0 & 4 & 8 & 12 & 16 & 20 & 24 \
\hline $\begin{array}{c}	ext { Bacteria count } \
(\mathrm{CFU} / \mathrm{mL})\end{array}$ & 37 & 47 & 63 & 78 & 105 & 130 & 173 \
\hline
\end{tabular}
\end{table}
\begin{enumerate}[label=(\alph*)]
\item Make a scatter plot of the data.

\item  Use a calculator or computer to find an exponential curve $f(t) =a \cdot b^{t}$ that models the bacteria population $t$ hours later.

\item  Graph the model from part (b) together with the scatter plot in part (a). Use the graph to estimate how long it takes for the bacteria count to double. 
\end{enumerate}

Khagan Gulusoy · Accepted Answer

\begin{figure}[ht]
\begin{tikzpicture}
% Axis
\begin{axis}[
    xlabel=$t$,
    ylabel=$f(t)$,
    domain=0:20,
    xmin=0, xmax=25,
    ymin=0,
    ymax=600,
    grid=both,
    axis lines= middle,
    ymax = 200
]

% Function f(t)
\addplot[color=blue,smooth, thick, domain=0:25] {36.7826 * exp(0.064227 * x)} node[above left,pos=0.6] {$f(t) = 36.7826 e^{0.064227t}$};

% Vertical line at t = 10.8
\draw[dashed] (axis cs:10.8,0) -- (axis cs:10.8,74);
\draw[dashed] (axis cs:0,74) -- (axis cs:10.8,74);

\addplot[only marks, mark=*, color=red] coordinates {
    (0,37)
    (4,47)
    (8,63)
    (12,78)
    (16,105)
    (20,130)
    (24,173)
};

\end{axis}
\end{tikzpicture}
\end{figure}

The function $f$ that models the bacteria population $t$ hours later is:

$$\forall t \in[0,+\infty): \quad f(t)=36.7826 e^{0.064227} 	ext {. }$$

Since for $t=0$ we have $f(t)=37$, we should find $t$ such that $f(t)=37.2=74$. As we can see from the graph, $t \approx 11$. Thus, it takes 11 hours for this particular bacteria to double.

Exercise 1.4.27

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Time (hours)	0	4	8	12	16	20	24

$\begin{matrix} Bacteria count \\ (CFU ∕ mL) \end{matrix}$	37	47	63	78	105	130	173

Exercise 1.4.27

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