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Algebra
Exponentials
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A sample of 200 bacteria grows exponentially and increases by 7% each hour.

  1. How many bacteria will there be after 36 hours?
  2. How long does it take for the population to double?

  1. Using \(\displaystyle P=P_0e^{kt}\) gives \(\displaystyle P=200e^\frac{7 t}{100}\). Evaluating at \(\displaystyle t=36\) gives \(\displaystyle P=2486\) bacteria.
  2. The doubling time is \(\displaystyle t=\frac{\ln 2}{k}=9.9\) hours

Download \(\LaTeX\) 

\begin{question}A sample of 200 bacteria grows exponentially and increases by 7\% each hour.
\begin{enumerate}
    \item (10pts) How many bacteria will there be after 36 hours?
        \soln{9cm}{
            Using $P=P_0e^{kt}$ gives $P=200e^\frac{7 t}{100}$. Evaluating at $t=36$ gives $P=2486$ bacteria. 
        }
    \item (10pts) How long does it take for the population to double?
        \soln{9cm}{
            The doubling time is $t=\frac{\ln 2}{k}=9.9$ 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 200 bacteria grows exponentially and increases by 7&#37; each hour.
<ol type="a">
	<li>How many bacteria will there be after 36 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=200e^\frac{7 t}{100} " src="/equation_images/%20%5Cdisplaystyle%20P%3D200e%5E%5Cfrac%7B7%20t%7D%7B100%7D%20" alt="LaTeX:  \displaystyle P=200e^\frac{7 t}{100} " data-equation-content=" \displaystyle P=200e^\frac{7 t}{100} " /> . Evaluating at  <img class="equation_image" title=" \displaystyle t=36 " src="/equation_images/%20%5Cdisplaystyle%20t%3D36%20" alt="LaTeX:  \displaystyle t=36 " data-equation-content=" \displaystyle t=36 " />  gives  <img class="equation_image" title=" \displaystyle P=2486 " src="/equation_images/%20%5Cdisplaystyle%20P%3D2486%20" alt="LaTeX:  \displaystyle P=2486 " data-equation-content=" \displaystyle P=2486 " />  bacteria.</li>
	<li>The doubling time is  <img class="equation_image" title=" \displaystyle t=\frac{\ln 2}{k}=9.9 " src="/equation_images/%20%5Cdisplaystyle%20t%3D%5Cfrac%7B%5Cln%202%7D%7Bk%7D%3D9.9%20" alt="LaTeX:  \displaystyle t=\frac{\ln 2}{k}=9.9 " data-equation-content=" \displaystyle t=\frac{\ln 2}{k}=9.9 " />  hours</li>
</ol></p> </p>