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Maximize \(\displaystyle p = 15 x + 6 y\) subject to \(\displaystyle \begin{cases}56 x + 13 y \leq 728 \\ 55 x + 89 y \leq 4895 \\ x \geq 0, y \geq 0 \end{cases}\)
Drawing a graph gives
Solving the system of equations gives the intersection at \(\displaystyle \left( \frac{1157}{4269}, \ \frac{234080}{4269}\right)\). Making a table gives to test the verticies in \(\displaystyle p=15 x + 6 y\) gives
\begin{question}Maximize $p = 15 x + 6 y$ subject to $\begin{cases}56 x + 13 y \leq 728 \\ 55 x + 89 y \leq 4895 \\ x \geq 0, y \geq 0 \end{cases}$
\soln{9cm}{Drawing a graph gives\newline
\begin{tikzpicture}[scale=0.1]
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\filldraw[fill=gray,opacity=.8] (0, 56) -- (0, 89) -- (89, 89) -- (89, 0) -- (13, 0) -- (0, 56);
\filldraw[fill=gray,opacity=.8] (0, 55) -- (0, 89) -- (89, 89) -- (89, 0) -- (89, 0) -- (0, 55);
\draw[fill=red] (1157/4269,234080/4269) circle (2pt);
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Solving the system of equations gives the intersection at $\left( \frac{1157}{4269}, \ \frac{234080}{4269}\right)$. Making a table gives to test the verticies in $p=15 x + 6 y$ gives \begin{center}
\begin{tabular}{|c|c|}\hline
Point & Function \\[3pt] \hline
$\left( 0, \ 0\right)$ & $0$ \\[3pt] \hline
$\left( 13, \ 0\right)$ & $195$ \\[3pt] \hline
$\left( \frac{1157}{4269}, \ \frac{234080}{4269}\right)$ & $\frac{473945}{1423}$ \\[3pt] \hline
$\left( 0, \ 55\right)$ & $330$ \\[3pt] \hline
\end{tabular}
\end{center}The gives the maximum value of $\frac{473945}{1423}$ located at $\left( \frac{1157}{4269}, \ \frac{234080}{4269}\right)$. }
\end{question}
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\begin{document}\begin{question}(10pts) The question goes here!
\soln{9cm}{The solution goes here.}
\end{question}\end{document}<p> <p>Maximize <img class="equation_image" title=" \displaystyle p = 15 x + 6 y " src="/equation_images/%20%5Cdisplaystyle%20p%20%3D%2015%20x%20%2B%206%20y%20" alt="LaTeX: \displaystyle p = 15 x + 6 y " data-equation-content=" \displaystyle p = 15 x + 6 y " /> subject to <img class="equation_image" title=" \displaystyle \begin{cases}56 x + 13 y \leq 728 \\ 55 x + 89 y \leq 4895 \\ x \geq 0, y \geq 0 \end{cases} " src="/equation_images/%20%5Cdisplaystyle%20%5Cbegin%7Bcases%7D56%20x%20%2B%2013%20y%20%5Cleq%20728%20%5C%5C%2055%20x%20%2B%2089%20y%20%5Cleq%204895%20%5C%5C%20x%20%5Cgeq%200%2C%20y%20%5Cgeq%200%20%5Cend%7Bcases%7D%20" alt="LaTeX: \displaystyle \begin{cases}56 x + 13 y \leq 728 \\ 55 x + 89 y \leq 4895 \\ x \geq 0, y \geq 0 \end{cases} " data-equation-content=" \displaystyle \begin{cases}56 x + 13 y \leq 728 \\ 55 x + 89 y \leq 4895 \\ x \geq 0, y \geq 0 \end{cases} " /> </p> </p><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{1157}{4269}, \ \frac{234080}{4269}\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28%20%5Cfrac%7B1157%7D%7B4269%7D%2C%20%5C%20%20%5Cfrac%7B234080%7D%7B4269%7D%5Cright%29%20" alt="LaTeX: \displaystyle \left( \frac{1157}{4269}, \ \frac{234080}{4269}\right) " data-equation-content=" \displaystyle \left( \frac{1157}{4269}, \ \frac{234080}{4269}\right) " /> . Making a table gives to test the verticies in <img class="equation_image" title=" \displaystyle p=15 x + 6 y " src="/equation_images/%20%5Cdisplaystyle%20p%3D15%20x%20%2B%206%20y%20" alt="LaTeX: \displaystyle p=15 x + 6 y " data-equation-content=" \displaystyle p=15 x + 6 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( 13, \ 0\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28%2013%2C%20%5C%20%200%5Cright%29%20" alt="LaTeX: \displaystyle \left( 13, \ 0\right) " data-equation-content=" \displaystyle \left( 13, \ 0\right) " /> & <img class="equation_image" title=" \displaystyle 195 " src="/equation_images/%20%5Cdisplaystyle%20195%20" alt="LaTeX: \displaystyle 195 " data-equation-content=" \displaystyle 195 " /> \\[3pt] \hline
<img class="equation_image" title=" \displaystyle \left( \frac{1157}{4269}, \ \frac{234080}{4269}\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28%20%5Cfrac%7B1157%7D%7B4269%7D%2C%20%5C%20%20%5Cfrac%7B234080%7D%7B4269%7D%5Cright%29%20" alt="LaTeX: \displaystyle \left( \frac{1157}{4269}, \ \frac{234080}{4269}\right) " data-equation-content=" \displaystyle \left( \frac{1157}{4269}, \ \frac{234080}{4269}\right) " /> & <img class="equation_image" title=" \displaystyle \frac{473945}{1423} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B473945%7D%7B1423%7D%20" alt="LaTeX: \displaystyle \frac{473945}{1423} " data-equation-content=" \displaystyle \frac{473945}{1423} " /> \\[3pt] \hline
<img class="equation_image" title=" \displaystyle \left( 0, \ 55\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28%200%2C%20%5C%20%2055%5Cright%29%20" alt="LaTeX: \displaystyle \left( 0, \ 55\right) " data-equation-content=" \displaystyle \left( 0, \ 55\right) " /> & <img class="equation_image" title=" \displaystyle 330 " src="/equation_images/%20%5Cdisplaystyle%20330%20" alt="LaTeX: \displaystyle 330 " data-equation-content=" \displaystyle 330 " /> \\[3pt] \hline
\end{tabular}
</center>The gives the maximum value of <img class="equation_image" title=" \displaystyle \frac{473945}{1423} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B473945%7D%7B1423%7D%20" alt="LaTeX: \displaystyle \frac{473945}{1423} " data-equation-content=" \displaystyle \frac{473945}{1423} " /> located at <img class="equation_image" title=" \displaystyle \left( \frac{1157}{4269}, \ \frac{234080}{4269}\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28%20%5Cfrac%7B1157%7D%7B4269%7D%2C%20%5C%20%20%5Cfrac%7B234080%7D%7B4269%7D%5Cright%29%20" alt="LaTeX: \displaystyle \left( \frac{1157}{4269}, \ \frac{234080}{4269}\right) " data-equation-content=" \displaystyle \left( \frac{1157}{4269}, \ \frac{234080}{4269}\right) " /> . </p> </p>