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Maximize \(\displaystyle p = 8 x + 22 y\) subject to \(\displaystyle \begin{cases}79 x + 44 y \leq 3476 \\ 69 x + 98 y \leq 6762 \\ x \geq 0, y \geq 0 \end{cases}\)

Drawing a graph gives
Solving the system of equations gives the intersection at \(\displaystyle \left( \frac{21560}{2353}, \ \frac{147177}{2353}\right)\). Making a table gives to test the verticies in \(\displaystyle p=8 x + 22 y\) gives

\begin{tabular}{|c|c|}\hline Point & Function \\[3pt] \hline \(\displaystyle \left( 0, \ 0\right)\) & \(\displaystyle 0\) \\[3pt] \hline \(\displaystyle \left( 44, \ 0\right)\) & \(\displaystyle 352\) \\[3pt] \hline \(\displaystyle \left( \frac{21560}{2353}, \ \frac{147177}{2353}\right)\) & \(\displaystyle \frac{3410374}{2353}\) \\[3pt] \hline \(\displaystyle \left( 0, \ 69\right)\) & \(\displaystyle 1518\) \\[3pt] \hline \end{tabular}
The gives the maximum value of \(\displaystyle 1518\) located at \(\displaystyle \left( 0, \ 69\right)\).

Download \(\LaTeX\) 

\begin{question}Maximize $p = 8 x + 22 y$ subject to $\begin{cases}79 x + 44 y \leq 3476 \\ 69 x + 98 y \leq 6762 \\ x \geq 0, y \geq 0 \end{cases}$
    \soln{9cm}{Drawing a graph gives\newline 
\begin{tikzpicture}[scale=0.1]
	\draw[very thick, color=blue, opacity=0.3, xstep=10, ystep=10] (-10,-10) grid (98,98);
	\draw[latex-latex, very thick] (-12.0,0)--(100.0,0) node[right]{$x$};
	\draw[latex-latex, very thick] (0,-12.0)--(0,100.0) node[above]{$y$};
	\foreach \x in { -10,0,...,98 }{
		\pgfmathsetmacro{\xtick}{\x != 0 ? "\x" : ""}
		\draw (\x, .2) -- (\x,-.2) node[below]{$\xtick$};
		}
	\foreach \y in { -10,0,...,98 }{
		\pgfmathsetmacro{\ytick}{\y != 0 ? "\y" : ""}
		\draw (.2, \y) -- (-.2, \y) node[left]{$\ytick$};
		}
	 \filldraw[fill=gray,opacity=.8] (0, 79) -- (0, 98) -- (98, 98) -- (98, 0) -- (44, 0) -- (0, 79);
	 \filldraw[fill=gray,opacity=.8] (0, 69) -- (0, 98) -- (98, 98) -- (98, 0) -- (98, 0) -- (0, 69);
		\draw[fill=red] (21560/2353,147177/2353) circle (2pt);
\end{tikzpicture}

 Solving the system of equations gives the intersection at $\left( \frac{21560}{2353}, \  \frac{147177}{2353}\right)$. Making a table gives to test the verticies in $p=8 x + 22 y$ gives \begin{center}
\begin{tabular}{|c|c|}\hline
Point & Function \\[3pt] \hline
$\left( 0, \  0\right)$ & $0$ \\[3pt] \hline
$\left( 44, \  0\right)$ & $352$ \\[3pt] \hline
$\left( \frac{21560}{2353}, \  \frac{147177}{2353}\right)$ & $\frac{3410374}{2353}$ \\[3pt] \hline
$\left( 0, \  69\right)$ & $1518$ \\[3pt] \hline
\end{tabular}
\end{center}The gives the maximum value of $1518$ located at $\left( 0, \  69\right)$. }

\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.}

\end{question}\end{document}
HTML for Canvas 
<p> <p>Maximize  <img class="equation_image" title=" \displaystyle p = 8 x + 22 y " src="/equation_images/%20%5Cdisplaystyle%20p%20%3D%208%20x%20%2B%2022%20y%20" alt="LaTeX:  \displaystyle p = 8 x + 22 y " data-equation-content=" \displaystyle p = 8 x + 22 y " />  subject to  <img class="equation_image" title=" \displaystyle \begin{cases}79 x + 44 y \leq 3476 \\ 69 x + 98 y \leq 6762 \\ x \geq 0, y \geq 0 \end{cases} " src="/equation_images/%20%5Cdisplaystyle%20%5Cbegin%7Bcases%7D79%20x%20%2B%2044%20y%20%5Cleq%203476%20%5C%5C%2069%20x%20%2B%2098%20y%20%5Cleq%206762%20%5C%5C%20x%20%5Cgeq%200%2C%20y%20%5Cgeq%200%20%5Cend%7Bcases%7D%20" alt="LaTeX:  \displaystyle \begin{cases}79 x + 44 y \leq 3476 \\ 69 x + 98 y \leq 6762 \\ x \geq 0, y \geq 0 \end{cases} " data-equation-content=" \displaystyle \begin{cases}79 x + 44 y \leq 3476 \\ 69 x + 98 y \leq 6762 \\ x \geq 0, y \geq 0 \end{cases} " /> </p> </p>
HTML for Canvas
<p> <p>Drawing a graph gives<br> 
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 Solving the system of equations gives the intersection at  <img class="equation_image" title=" \displaystyle \left( \frac{21560}{2353}, \  \frac{147177}{2353}\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28%20%5Cfrac%7B21560%7D%7B2353%7D%2C%20%5C%20%20%5Cfrac%7B147177%7D%7B2353%7D%5Cright%29%20" alt="LaTeX:  \displaystyle \left( \frac{21560}{2353}, \  \frac{147177}{2353}\right) " data-equation-content=" \displaystyle \left( \frac{21560}{2353}, \  \frac{147177}{2353}\right) " /> . Making a table gives to test the verticies in  <img class="equation_image" title=" \displaystyle p=8 x + 22 y " src="/equation_images/%20%5Cdisplaystyle%20p%3D8%20x%20%2B%2022%20y%20" alt="LaTeX:  \displaystyle p=8 x + 22 y " data-equation-content=" \displaystyle p=8 x + 22 y " />  gives <center>
\begin{tabular}{|c|c|}\hline
Point & Function \\[3pt] \hline
 <img class="equation_image" title=" \displaystyle \left( 0, \  0\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28%200%2C%20%5C%20%200%5Cright%29%20" alt="LaTeX:  \displaystyle \left( 0, \  0\right) " data-equation-content=" \displaystyle \left( 0, \  0\right) " />  &  <img class="equation_image" title=" \displaystyle 0 " src="/equation_images/%20%5Cdisplaystyle%200%20" alt="LaTeX:  \displaystyle 0 " data-equation-content=" \displaystyle 0 " />  \\[3pt] \hline
 <img class="equation_image" title=" \displaystyle \left( 44, \  0\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28%2044%2C%20%5C%20%200%5Cright%29%20" alt="LaTeX:  \displaystyle \left( 44, \  0\right) " data-equation-content=" \displaystyle \left( 44, \  0\right) " />  &  <img class="equation_image" title=" \displaystyle 352 " src="/equation_images/%20%5Cdisplaystyle%20352%20" alt="LaTeX:  \displaystyle 352 " data-equation-content=" \displaystyle 352 " />  \\[3pt] \hline
 <img class="equation_image" title=" \displaystyle \left( \frac{21560}{2353}, \  \frac{147177}{2353}\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28%20%5Cfrac%7B21560%7D%7B2353%7D%2C%20%5C%20%20%5Cfrac%7B147177%7D%7B2353%7D%5Cright%29%20" alt="LaTeX:  \displaystyle \left( \frac{21560}{2353}, \  \frac{147177}{2353}\right) " data-equation-content=" \displaystyle \left( \frac{21560}{2353}, \  \frac{147177}{2353}\right) " />  &  <img class="equation_image" title=" \displaystyle \frac{3410374}{2353} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B3410374%7D%7B2353%7D%20" alt="LaTeX:  \displaystyle \frac{3410374}{2353} " data-equation-content=" \displaystyle \frac{3410374}{2353} " />  \\[3pt] \hline
 <img class="equation_image" title=" \displaystyle \left( 0, \  69\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28%200%2C%20%5C%20%2069%5Cright%29%20" alt="LaTeX:  \displaystyle \left( 0, \  69\right) " data-equation-content=" \displaystyle \left( 0, \  69\right) " />  &  <img class="equation_image" title=" \displaystyle 1518 " src="/equation_images/%20%5Cdisplaystyle%201518%20" alt="LaTeX:  \displaystyle 1518 " data-equation-content=" \displaystyle 1518 " />  \\[3pt] \hline
\end{tabular}
</center>The gives the maximum value of  <img class="equation_image" title=" \displaystyle 1518 " src="/equation_images/%20%5Cdisplaystyle%201518%20" alt="LaTeX:  \displaystyle 1518 " data-equation-content=" \displaystyle 1518 " />  located at  <img class="equation_image" title=" \displaystyle \left( 0, \  69\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28%200%2C%20%5C%20%2069%5Cright%29%20" alt="LaTeX:  \displaystyle \left( 0, \  69\right) " data-equation-content=" \displaystyle \left( 0, \  69\right) " /> . </p> </p>