Exercise 1.1.74

Question

An electricity company charges its customers a base rate of \$10 a month, plus 6 cents per kilowatt-hour (kWh) for the first 1200 kWh and 7 cents per kWh for all usage over 1200 kWh. Express the monthly cost E as a function of the amount $x$ of electricity used. Then graph the function $E$ for $0 \leq x \leq 2000.$

Khagan Gulusoy · Accepted Answer

If the customer uses less than or equal to 1200 kWh of electricity, the customer pays $10+0.06x.$ If the usage is over 1200 Wkh electricity, he(she) pays $10+0.06\cdot 1200+0.07(x-1200)=10+72+0.07(x-1200)=82+0.07(x-1200).$ Thus, we have:
$$\forall x\in [0,+\infty):\quad E(x)=\begin{cases}
    10+0.06x, \quad &	ext{if } 0\leq x\leq 1200\
    82+0.07(x-1200),\quad &	ext{if } 1200< x
\end{cases}$$
Now we represent $E$ numerically, i.e. by the table of values:
\begin{table}[H] \centering 
 \begin{tabular}{|*{9}{c|}} 
 \hline 
 $x$ & 0 & 300 & 600 & 900 & 1200 & 1300 & 1600 & 1900 \ \hline 
 $y$ & 10 & 28 & 46 & 64 & 82 & 89 & 110 & 131 \ \hline
 \end{tabular} 
 \end{table}
we can then easily sketch the graph of the function $E$ as follows:
   \begin{figure}[H] \centering
\begin{tikzpicture}
    \begin{axis}[axis lines=left, xmin=0, xmax=2000, ymin=0, ymax=138, xlabel={$x$}, ylabel={$F(x)$}, x label style = {at={(ticklabel cs:0.9)},anchor=south,above=12mm}, y label style={anchor=north, above=3mm}]
         \addplot[domain=0:1200, samples=2, mark=none, color=blue!30!red, ultra thick] {10+0.06*x}; 
         \addplot[domain=1200:2000, samples=50, mark=none, color=blue, ultra thick] {82+0.07*(x-1200)};
         \addplot +[dashed,gray,mark=none] coordinates {(1200,0) (1200,82)};
    \end{axis}
\end{tikzpicture}
\end{figure}

Exercise 1.1.74

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Exercise 1.1.74

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