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Algebra
Exponentials
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A sample of 190 bacteria grows exponentially and increases by 28% each hour.

  1. How many bacteria will there be after 12 hours?
  2. How long does it take for the population to double?

  1. Using \(\displaystyle P=P_0e^{kt}\) gives \(\displaystyle P=190e^\frac{7 t}{25}\). Evaluating at \(\displaystyle t=12\) gives \(\displaystyle P=5470\) bacteria.
  2. The doubling time is \(\displaystyle t=\frac{\ln 2}{k}=2.48\) hours

Download \(\LaTeX\) 

\begin{question}A sample of 190 bacteria grows exponentially and increases by 28\% each hour.
\begin{enumerate}
    \item (10pts) How many bacteria will there be after 12 hours?
        \soln{9cm}{
            Using $P=P_0e^{kt}$ gives $P=190e^\frac{7 t}{25}$. Evaluating at $t=12$ gives $P=5470$ bacteria. 
        }
    \item (10pts) How long does it take for the population to double?
        \soln{9cm}{
            The doubling time is $t=\frac{\ln 2}{k}=2.48$ hours 
        }
\end{enumerate}
\end{question}

Download Question and Solution Environment\(\LaTeX\)
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\begin{document}\begin{question}(10pts) The question goes here!
    \soln{9cm}{The solution goes here.}

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HTML for Canvas 
<p> <p>A sample of 190 bacteria grows exponentially and increases by 28&#37; each hour.
<ol type="a">
	<li>How many bacteria will there be after 12 hours?</li>
	<li>How long does it take for the population to double?</li>

</ol>
</p> </p>
HTML for Canvas
<p> <p>
<ol type="a">
	<li>Using  <img class="equation_image" title=" \displaystyle P=P_0e^{kt} " src="/equation_images/%20%5Cdisplaystyle%20P%3DP_0e%5E%7Bkt%7D%20" alt="LaTeX:  \displaystyle P=P_0e^{kt} " data-equation-content=" \displaystyle P=P_0e^{kt} " />  gives  <img class="equation_image" title=" \displaystyle P=190e^\frac{7 t}{25} " src="/equation_images/%20%5Cdisplaystyle%20P%3D190e%5E%5Cfrac%7B7%20t%7D%7B25%7D%20" alt="LaTeX:  \displaystyle P=190e^\frac{7 t}{25} " data-equation-content=" \displaystyle P=190e^\frac{7 t}{25} " /> . Evaluating at  <img class="equation_image" title=" \displaystyle t=12 " src="/equation_images/%20%5Cdisplaystyle%20t%3D12%20" alt="LaTeX:  \displaystyle t=12 " data-equation-content=" \displaystyle t=12 " />  gives  <img class="equation_image" title=" \displaystyle P=5470 " src="/equation_images/%20%5Cdisplaystyle%20P%3D5470%20" alt="LaTeX:  \displaystyle P=5470 " data-equation-content=" \displaystyle P=5470 " />  bacteria.</li>
	<li>The doubling time is  <img class="equation_image" title=" \displaystyle t=\frac{\ln 2}{k}=2.48 " src="/equation_images/%20%5Cdisplaystyle%20t%3D%5Cfrac%7B%5Cln%202%7D%7Bk%7D%3D2.48%20" alt="LaTeX:  \displaystyle t=\frac{\ln 2}{k}=2.48 " data-equation-content=" \displaystyle t=\frac{\ln 2}{k}=2.48 " />  hours</li>
</ol></p> </p>