Exercise 1.1.73

Question

In a certain state the maximum speed permitted on freeways is 65 m/h and the minimum speed is 40 m/h. The fine for violating these limits is \$15 for every mile per hour above the maximum speed or below the minimum speed. Express the amount of the fine $F$ as a function of the driving speed $x$ and graph $F(x)$ for $0 \leq  x \leq 100.$

Khagan Gulusoy · Accepted Answer

Since the fine for violating these limits is \$15 for every mile per hour above the maximum speed or below the minimum speed, we define the function $F$ as follows:
$$\forall x\in [0,+\infty):\quad F(x)=\begin{cases}
    (40-x)\cdot 15\quad &	ext{if } 0 \leq x<40\
    0\quad &	ext{if } 40\leq x\leq 65\
    (x-65)\cdot 15\quad &	ext{if } x>65
\end{cases}$$
Now we represent $F$ numerically, i.e. by the table of values:
\begin{table}[H]
\begin{tabular}{|*{5}{c|}||*{4}{c|}||*{5}{c|}} 
 \hline 
 $x$ & 0 & 10 & 20 & 30 & 45 & 50 & 60 & 70 & 80 & 90 & 100\ \hline 
 $y$ & 600 & 450 & 300 & 150 & 0 & 0 & 0 &75 & 225 & 375 & 525 \ \hline
 \end{tabular}
 \end{table}
 We can then easily sketch the graph of the function $F$ as follows:
   \begin{figure}[H] \centering
\begin{tikzpicture}
    \begin{axis}[axis lines=left, xmin=0, xmax=100, ymin=0, ymax=600, 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=0:40, samples=2, mark=none, color=violet, ultra thick] {600-15*x}; 
         \addplot[domain=40:65, samples=2, mark=none, color=violet, ultra thick] {0};
         \addplot[domain=65:100, samples=2, mark=none, color=violet, ultra thick] {15*x-975};
    \end{axis}
\end{tikzpicture}
\end{figure}

Exercise 1.1.73

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Exercise 1.1.73

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