Exercise 1.2.19

Question

The manager of a furniture factory finds that it costs \$2200 to manufacture 100 chairs in one day and \$4800 to produce 300 chairs in one day.
    \begin{enumerate}[label=(\alph*)]
        \item Express the cost as a function of the number of chairs produced, assuming that it is linear. Then sketch the graph.
        \item What is the slope of the graph and what does it represent?
        \item What is the $y$-intercept of the graph and what does it represent?
    \end{enumerate}

Khagan Gulusoy · Accepted Answer

\begin{enumerate}[label=(\alph*)]

\item Recall that for any linear function $f:X 	o \mathbb{R}$ and for two points $(x_1,y_1), \ (x_2,y_2)$ belonging to its graph, this formula is true:
    \begin{equation}\label{ee}
        \forall x\in X:\quad \frac{x-x_1}{x_2-x_1}=\frac{f(x)-y_1}{y_2-y_1},
    \end{equation}
    Let $m$ be the function of money required to manufacture chairs that takes the number of chairs $c$ as its input. Since the graph of the function $m$ passes through points $(100,2200),\ (300,4800)$, using equation ef{ee}, we can write:
    $$\frac{c-100}{300-100}=\frac{m(c)-2200}{4800-2200}.$$
    $$\frac{c-100}{200}=\frac{m(c)-2200}{2600}.$$
    $$m(c)=13c+900.$$

\begin{figure}[H] \centering
    \begin{tikzpicture}
    \begin{axis}[axis lines=left, xmin=0, xmax=410, ymax=6200, ymin=0, ytick={0, 900, 2200, 4800}, xlabel={$c$}, ylabel={$m(c)$}, x label style = {at={(ticklabel cs:0.9)},anchor=south,above=10mm}, y label style={anchor=north, above=3mm}]
         \addplot[domain=0:410, samples=2, mark=none, color=blue, ultra thick] {13*x+900};
         \addplot[mark=*, mark options={scale=1, fill=blue}](100,2200);
         \addplot[mark=*, mark options={scale=1, fill=blue}](300,4800);
         \addplot +[dashed,gray,mark=none] coordinates {(100,0) (100,2200) (0,2200)};
         \addplot +[dashed,gray,mark=none] coordinates {(300,0) (300,4800) (0,4800)};
    \end{axis}    
    \end{tikzpicture}
    \end{figure}

\item %The slope of the graph, namely $13$, represents the rate of change in the cost with respect to the number of chairs. This means that it costs 13\$ more if the factory produces $1$ chair more.
    %The slope of the graph, namely $13$, represents the rate of change in the cost with respect to the number of chairs. This means that it costs 13\$ more for every $1$ chair produced.
    The slope of the graph, namely $13$, represents the rate of change in the cost with respect to the number of chairs. This means that the overall cost increases by 13\$ when the number of chairs produced by the factory increases by 1.

\item The $y$-intercept of $900$ represents the money spent by the factory when there are no chairs produced. In economics, this quantity is often described as \emph{fixed costs} of production.
    
\end{enumerate}

Exercise 1.2.19

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Exercise 1.2.19

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