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Maximize \(\displaystyle p = 12 x + 10 y\) subject to \(\displaystyle \begin{cases}61 x + 27 y \leq 1647 \\ 16 x + 82 y \leq 1312 \\ x \geq 0, y \geq 0 \end{cases}\)
Drawing a graph gives
Solving the system of equations gives the intersection at \(\displaystyle \left( \frac{9963}{457}, \ \frac{5368}{457}\right)\). Making a table gives to test the verticies in \(\displaystyle p=12 x + 10 y\) gives
\begin{question}Maximize $p = 12 x + 10 y$ subject to $\begin{cases}61 x + 27 y \leq 1647 \\ 16 x + 82 y \leq 1312 \\ x \geq 0, y \geq 0 \end{cases}$
\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 (82,82);
\draw[latex-latex, very thick] (-12.0,0)--(84.0,0) node[right]{$x$};
\draw[latex-latex, very thick] (0,-12.0)--(0,84.0) node[above]{$y$};
\foreach \x in { -10,0,...,82 }{
\pgfmathsetmacro{\xtick}{\x != 0 ? "\x" : ""}
\draw (\x, .2) -- (\x,-.2) node[below]{$\xtick$};
}
\foreach \y in { -10,0,...,82 }{
\pgfmathsetmacro{\ytick}{\y != 0 ? "\y" : ""}
\draw (.2, \y) -- (-.2, \y) node[left]{$\ytick$};
}
\filldraw[fill=gray,opacity=.8] (0, 61) -- (0, 82) -- (82, 82) -- (82, 0) -- (27, 0) -- (0, 61);
\filldraw[fill=gray,opacity=.8] (0, 16) -- (0, 82) -- (82, 82) -- (82, 0) -- (82, 0) -- (0, 16);
\draw[fill=red] (9963/457,5368/457) circle (2pt);
\end{tikzpicture}
Solving the system of equations gives the intersection at $\left( \frac{9963}{457}, \ \frac{5368}{457}\right)$. Making a table gives to test the verticies in $p=12 x + 10 y$ gives \begin{center}
\begin{tabular}{|c|c|}\hline
Point & Function \\[3pt] \hline
$\left( 0, \ 0\right)$ & $0$ \\[3pt] \hline
$\left( 27, \ 0\right)$ & $324$ \\[3pt] \hline
$\left( \frac{9963}{457}, \ \frac{5368}{457}\right)$ & $\frac{173236}{457}$ \\[3pt] \hline
$\left( 0, \ 16\right)$ & $160$ \\[3pt] \hline
\end{tabular}
\end{center}The gives the maximum value of $\frac{173236}{457}$ located at $\left( \frac{9963}{457}, \ \frac{5368}{457}\right)$. }
\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 = 12 x + 10 y " src="/equation_images/%20%5Cdisplaystyle%20p%20%3D%2012%20x%20%2B%2010%20y%20" alt="LaTeX: \displaystyle p = 12 x + 10 y " data-equation-content=" \displaystyle p = 12 x + 10 y " /> subject to <img class="equation_image" title=" \displaystyle \begin{cases}61 x + 27 y \leq 1647 \\ 16 x + 82 y \leq 1312 \\ x \geq 0, y \geq 0 \end{cases} " src="/equation_images/%20%5Cdisplaystyle%20%5Cbegin%7Bcases%7D61%20x%20%2B%2027%20y%20%5Cleq%201647%20%5C%5C%2016%20x%20%2B%2082%20y%20%5Cleq%201312%20%5C%5C%20x%20%5Cgeq%200%2C%20y%20%5Cgeq%200%20%5Cend%7Bcases%7D%20" alt="LaTeX: \displaystyle \begin{cases}61 x + 27 y \leq 1647 \\ 16 x + 82 y \leq 1312 \\ x \geq 0, y \geq 0 \end{cases} " data-equation-content=" \displaystyle \begin{cases}61 x + 27 y \leq 1647 \\ 16 x + 82 y \leq 1312 \\ 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{9963}{457}, \ \frac{5368}{457}\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28%20%5Cfrac%7B9963%7D%7B457%7D%2C%20%5C%20%20%5Cfrac%7B5368%7D%7B457%7D%5Cright%29%20" alt="LaTeX: \displaystyle \left( \frac{9963}{457}, \ \frac{5368}{457}\right) " data-equation-content=" \displaystyle \left( \frac{9963}{457}, \ \frac{5368}{457}\right) " /> . Making a table gives to test the verticies in <img class="equation_image" title=" \displaystyle p=12 x + 10 y " src="/equation_images/%20%5Cdisplaystyle%20p%3D12%20x%20%2B%2010%20y%20" alt="LaTeX: \displaystyle p=12 x + 10 y " data-equation-content=" \displaystyle p=12 x + 10 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( 27, \ 0\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28%2027%2C%20%5C%20%200%5Cright%29%20" alt="LaTeX: \displaystyle \left( 27, \ 0\right) " data-equation-content=" \displaystyle \left( 27, \ 0\right) " /> & <img class="equation_image" title=" \displaystyle 324 " src="/equation_images/%20%5Cdisplaystyle%20324%20" alt="LaTeX: \displaystyle 324 " data-equation-content=" \displaystyle 324 " /> \\[3pt] \hline
<img class="equation_image" title=" \displaystyle \left( \frac{9963}{457}, \ \frac{5368}{457}\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28%20%5Cfrac%7B9963%7D%7B457%7D%2C%20%5C%20%20%5Cfrac%7B5368%7D%7B457%7D%5Cright%29%20" alt="LaTeX: \displaystyle \left( \frac{9963}{457}, \ \frac{5368}{457}\right) " data-equation-content=" \displaystyle \left( \frac{9963}{457}, \ \frac{5368}{457}\right) " /> & <img class="equation_image" title=" \displaystyle \frac{173236}{457} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B173236%7D%7B457%7D%20" alt="LaTeX: \displaystyle \frac{173236}{457} " data-equation-content=" \displaystyle \frac{173236}{457} " /> \\[3pt] \hline
<img class="equation_image" title=" \displaystyle \left( 0, \ 16\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28%200%2C%20%5C%20%2016%5Cright%29%20" alt="LaTeX: \displaystyle \left( 0, \ 16\right) " data-equation-content=" \displaystyle \left( 0, \ 16\right) " /> & <img class="equation_image" title=" \displaystyle 160 " src="/equation_images/%20%5Cdisplaystyle%20160%20" alt="LaTeX: \displaystyle 160 " data-equation-content=" \displaystyle 160 " /> \\[3pt] \hline
\end{tabular}
</center>The gives the maximum value of <img class="equation_image" title=" \displaystyle \frac{173236}{457} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B173236%7D%7B457%7D%20" alt="LaTeX: \displaystyle \frac{173236}{457} " data-equation-content=" \displaystyle \frac{173236}{457} " /> located at <img class="equation_image" title=" \displaystyle \left( \frac{9963}{457}, \ \frac{5368}{457}\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28%20%5Cfrac%7B9963%7D%7B457%7D%2C%20%5C%20%20%5Cfrac%7B5368%7D%7B457%7D%5Cright%29%20" alt="LaTeX: \displaystyle \left( \frac{9963}{457}, \ \frac{5368}{457}\right) " data-equation-content=" \displaystyle \left( \frac{9963}{457}, \ \frac{5368}{457}\right) " /> . </p> </p>