Question

In a certain state the maximum speed permitted on freeways in 65 mi/h and the minimum speed is 40 mi/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 \(F\left( x \right)\) for \(0 \le x \le 100\).

Short answer

The piecewise-defined function is \(F\left( x \right) = \left\{ \begin{aligned}15\left( {40 - x} \right)\,\,\,\,\,{\mathop{\rm if}\nolimits} \,\,0 &\le x < 40\\0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\mathop{\rm if}\nolimits} \,\,40 &\le x \le 65\\15\left( {x - 65} \right)\,\,\,\,\,{\mathop{\rm if}\nolimits} \,\,\,x &> 65\end{aligned} \right.\).

Step-by-step solution

Express the amount of fine F as a function of the driving speed x

\(F\left( x \right) = 15\left( {40 - x} \right)\)

Express the amount of fine \(F\) per hour above the maximum speed as a function of the driving speed x.

\(F\left( x \right) = 15\left( {x - 65} \right)\)

The amount of fine\(F\)in a piecewise-defined function is shown below:

\(F\left( x \right) = \left\{ \begin{aligned}15\left( {40 - x} \right)\,\,\,\,\,{\mathop{\rm if}\nolimits} \,\,0 &\le x < 40\\0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\mathop{\rm if}\nolimits} \,\,40 &\le x \le 65\\15\left( {x - 65} \right)\,\,\,\,\,{\mathop{\rm if}\nolimits} \,\,\,x &> 65\end{aligned} \right.\)

Sketch the graph of the function \(F\left( x \right)\)

The table of the function \(F\left( x \right)\) is shown below:

\(x\)	0	40	65	100
\(F\left( x \right)\)	600	0	0	525

Sketch the graph \(F\left( x \right)\) for \(0 \le x \le 100\) as shown below: