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Algebra
Exponentials
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A sample of 18000 bacteria grows exponentially and increases by 50% each hour.

  1. How many bacteria will there be after 24 hours?
  2. How long does it take for the population to double?

  1. Using \(\displaystyle P=P_0e^{kt}\) gives \(\displaystyle P=18000e^\frac{t}{2}\). Evaluating at \(\displaystyle t=24\) gives \(\displaystyle P=2929586246\) bacteria.
  2. The doubling time is \(\displaystyle t=\frac{\ln 2}{k}=1.39\) hours

Download \(\LaTeX\) 

\begin{question}A sample of 18000 bacteria grows exponentially and increases by 50\% each hour.
\begin{enumerate}
    \item (10pts) How many bacteria will there be after 24 hours?
        \soln{9cm}{
            Using $P=P_0e^{kt}$ gives $P=18000e^\frac{t}{2}$. Evaluating at $t=24$ gives $P=2929586246$ bacteria. 
        }
    \item (10pts) How long does it take for the population to double?
        \soln{9cm}{
            The doubling time is $t=\frac{\ln 2}{k}=1.39$ hours 
        }
\end{enumerate}
\end{question}

Download Question and Solution Environment\(\LaTeX\)
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\begin{document}\begin{question}(10pts) The question goes here!
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HTML for Canvas 
<p> <p>A sample of 18000 bacteria grows exponentially and increases by 50&#37; each hour.
<ol type="a">
	<li>How many bacteria will there be after 24 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=18000e^\frac{t}{2} " src="/equation_images/%20%5Cdisplaystyle%20P%3D18000e%5E%5Cfrac%7Bt%7D%7B2%7D%20" alt="LaTeX:  \displaystyle P=18000e^\frac{t}{2} " data-equation-content=" \displaystyle P=18000e^\frac{t}{2} " /> . Evaluating at  <img class="equation_image" title=" \displaystyle t=24 " src="/equation_images/%20%5Cdisplaystyle%20t%3D24%20" alt="LaTeX:  \displaystyle t=24 " data-equation-content=" \displaystyle t=24 " />  gives  <img class="equation_image" title=" \displaystyle P=2929586246 " src="/equation_images/%20%5Cdisplaystyle%20P%3D2929586246%20" alt="LaTeX:  \displaystyle P=2929586246 " data-equation-content=" \displaystyle P=2929586246 " />  bacteria.</li>
	<li>The doubling time is  <img class="equation_image" title=" \displaystyle t=\frac{\ln 2}{k}=1.39 " src="/equation_images/%20%5Cdisplaystyle%20t%3D%5Cfrac%7B%5Cln%202%7D%7Bk%7D%3D1.39%20" alt="LaTeX:  \displaystyle t=\frac{\ln 2}{k}=1.39 " data-equation-content=" \displaystyle t=\frac{\ln 2}{k}=1.39 " />  hours</li>
</ol></p> </p>