Exercise 1.4.29

Question

A bacteria culture starts with 500 bacteria and doubles in size every half hour.

\begin{enumerate}[label=(\alph*)]
\item How many bacteria are there after 3 hours?

\item  How many bacteria are there after $t$ hours?

\item  How many bacteria are there after 40 minutes?

\item  Graph the population function and estimate the time for the population to reach 100,000 .
\end{enumerate}

Khagan Gulusoy · Accepted Answer

\begin{enumerate}[label=(\alph*)]
\item After 3 hours, the population of the bacterial will increase by 64 times (since $64=2^6$):

\begin{figure}[H]\centering
	ikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt

\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]
%uncomment if require: \path (0,300); %set diagram left start at 0, and has height of 300

%Shape: Brace [id:dp4945336320999514] 
\draw   (146,148) .. controls (145.95,143.33) and (143.6,141.02) .. (138.93,141.07) -- (108.43,141.39) .. controls (101.76,141.46) and (98.4,139.17) .. (98.35,134.5) .. controls (98.4,139.17) and (95.1,141.53) .. (88.43,141.6)(91.43,141.57) -- (57.93,141.92) .. controls (53.26,141.97) and (50.95,144.33) .. (51,148.99) ;
%Shape: Brace [id:dp22092305340024876] 
\draw   (242,146) .. controls (241.95,141.33) and (239.6,139.02) .. (234.93,139.07) -- (204.43,139.39) .. controls (197.76,139.46) and (194.4,137.17) .. (194.35,132.5) .. controls (194.4,137.17) and (191.1,139.53) .. (184.43,139.6)(187.43,139.57) -- (153.93,139.92) .. controls (149.26,139.97) and (146.95,142.33) .. (147,146.99) ;
%Shape: Brace [id:dp8377340467497068] 
\draw   (339,146) .. controls (338.95,141.33) and (336.6,139.02) .. (331.93,139.07) -- (301.43,139.39) .. controls (294.76,139.46) and (291.4,137.17) .. (291.35,132.5) .. controls (291.4,137.17) and (288.1,139.53) .. (281.43,139.6)(284.43,139.57) -- (250.93,139.92) .. controls (246.26,139.97) and (243.95,142.33) .. (244,146.99) ;
%Shape: Brace [id:dp7278876703953902] 
\draw   (435,144) .. controls (434.95,139.33) and (432.6,137.02) .. (427.93,137.07) -- (397.43,137.39) .. controls (390.76,137.46) and (387.4,135.17) .. (387.35,130.5) .. controls (387.4,135.17) and (384.1,137.53) .. (377.43,137.6)(380.43,137.57) -- (346.93,137.92) .. controls (342.26,137.97) and (339.95,140.33) .. (340,144.99) ;
%Shape: Brace [id:dp8821191051796045] 
\draw   (531,144) .. controls (530.95,139.33) and (528.6,137.02) .. (523.93,137.07) -- (493.43,137.39) .. controls (486.76,137.46) and (483.4,135.17) .. (483.35,130.5) .. controls (483.4,135.17) and (480.1,137.53) .. (473.43,137.6)(476.43,137.57) -- (442.93,137.92) .. controls (438.26,137.97) and (435.95,140.33) .. (436,144.99) ;
%Shape: Brace [id:dp1979547045929273] 
\draw   (627,142) .. controls (626.95,137.33) and (624.6,135.02) .. (619.93,135.07) -- (589.43,135.39) .. controls (582.76,135.46) and (579.4,133.17) .. (579.35,128.5) .. controls (579.4,133.17) and (576.1,135.53) .. (569.43,135.6)(572.43,135.57) -- (538.93,135.92) .. controls (534.26,135.97) and (531.95,138.33) .. (532,142.99) ;
%Shape: Brace [id:dp26383549680542673] 
\draw   (625,100) .. controls (624.99,95.33) and (622.65,93.01) .. (617.98,93.02) -- (347.98,93.97) .. controls (341.31,93.99) and (337.97,91.67) .. (337.95,87) .. controls (337.97,91.67) and (334.65,94.01) .. (327.98,94.04)(330.98,94.03) -- (57.98,94.98) .. controls (53.31,94.99) and (50.99,97.33) .. (51,102) ;

% Text Node
\draw (40,149.4) node [anchor=north west][inner sep=0.75pt]    {$p_{0}$};
% Text Node
\draw (39,108.4) node [anchor=north west][inner sep=0.75pt]    {$t_{0}$};
% Text Node
\draw (16,178) node [anchor=north west][inner sep=0.75pt]   [align=left] {	extcolor[rgb]{0.56,0.07,1}{ \ \ \ initial}\	extcolor[rgb]{0.56,0.07,1}{population}};
% Text Node
\draw (65,112) node [anchor=north west][inner sep=0.75pt]   [align=left] {	extbf{1/2 hour}};
% Text Node
\draw (133,148.4) node [anchor=north west][inner sep=0.75pt]    {$2p_{0}$};
% Text Node
\draw (228,147.4) node [anchor=north west][inner sep=0.75pt]    {$4p_{0}$};
% Text Node
\draw (160,113) node [anchor=north west][inner sep=0.75pt]   [align=left] {	extbf{1/2 hour}};
% Text Node
\draw (233,147.4) node [anchor=north west][inner sep=0.75pt]    {$$};
% Text Node
\draw (258,110) node [anchor=north west][inner sep=0.75pt]   [align=left] {	extbf{1/2 hour}};
% Text Node
\draw (326,146.4) node [anchor=north west][inner sep=0.75pt]    {$8p_{0}$};
% Text Node
\draw (421,145.4) node [anchor=north west][inner sep=0.75pt]    {$16p_{0}$};
% Text Node
\draw (353,111) node [anchor=north west][inner sep=0.75pt]   [align=left] {	extbf{1/2 hour}};
% Text Node
\draw (450,108) node [anchor=north west][inner sep=0.75pt]   [align=left] {	extbf{1/2 hour}};
% Text Node
\draw (518,144.4) node [anchor=north west][inner sep=0.75pt]    {$32p_{0}$};
% Text Node
\draw (613,143.4) node [anchor=north west][inner sep=0.75pt]    {$64p_{0}$};
% Text Node
\draw (545,109) node [anchor=north west][inner sep=0.75pt]   [align=left] {	extbf{1/2 hour}};
% Text Node
\draw (306,62) node [anchor=north west][inner sep=0.75pt]   [align=left] {	extbf{3 hours}};
% Text Node
\draw (573,167) node [anchor=north west][inner sep=0.75pt]   [align=left] {	extcolor[rgb]{0.56,0.07,1}{population after}\	extcolor[rgb]{0.56,0.07,1}{ \ \ \ three hours}};

\end{tikzpicture}
\end{figure}

Therefore, the bacteria population after 3 hours will be:

$$
500\cdot 64=32000 	ext {. }
$$

\item  After $t$ hours, the population of the bacteria will increase by

$$
2^{t:(1 / 2)}=2^{2 t} 	ext {, }
$$

and therefore, the population of the bacteria will be

$$
\forall t \in \mathbb{R}_{+}: \quad p\item =500 \cdot 2^{2 t}
$$

\item Since 40 minutes $=2 / 3$ hour, the population of the bacteria will be:

$$
500 \cdot 2^{2 \cdot 2 / 3}=500 \cdot 2^{4 / 3}=500 \cdot \sqrt[3]{2^{4}}=500 \cdot 2 \sqrt[3]{2}=1000 \sqrt[3]{2} 	ext {. }
$$

\item  We sketch the graph of the function $p$ as follows:

\begin{figure}[ht]\centering
\begin{tikzpicture}
\begin{axis}[
    axis lines=middle,
    xlabel=$t$,
    ylabel=$p(t)$,
    xmin=-5,
    xmax=5,
    ymin=0,
    ymax=300000,
    width=10cm, % Adjust the width of the plot
    height=6cm, % Adjust the height of the plot
    legend pos=north west, % Position of the legend
    scaled y ticks=false
]

% Define the function p(t)
\addplot[blue,smooth,thick,domain=-5:5,samples=100] {500 * 2^(2*x)};

% Draw a horizontal dashed line at y=100000
\draw[dashed] (axis cs:0,100000) -- (axis cs:3.8,100000);
\draw[dashed] (axis cs:3.8,0) -- (axis cs:3.8,100000);

% Add a legend entry for p(t)
\addlegendentry{$p(t) = 500 \cdot 2^{2t}$}

\end{axis}
\end{tikzpicture}

\end{figure}

As we can see from the graph, the population reaches 100,000 when

$$t \approx 4 	ext{ hours.}$$
\end{enumerate}

Exercise 1.4.29

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Exercise 1.4.29

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