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 \(0 \le x \le 2000\).

Short answer

The monthly cost \(E\) in piecewise-defined function is\(E\left( x \right) = \left\{ \begin{aligned}10 &+ 0.06x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\mathop{\rm if}\nolimits} \,\,0 \le x \le 1200\\82 &+ 0.07\left( {x - 1200} \right)\,\,{\mathop{\rm if}\nolimits} \,\,x > 1200\end{aligned} \right.\).

Step-by-step solution

Express the monthly cost E as a function of the x amount of electricity used

\(E\left( x \right) = 10 + 0.06x\)

The monthly cost for all usage over 1200 kWh is shown below:

\(\begin{aligned}E\left( x \right) &= 10 + 0.06\left( {1200} \right) + 0.07\left( {x - 1200} \right)\\ &= 82 + 0.07\left( {x - 1200} \right)\end{aligned}\)

The monthly cost\(E\)in piecewise-defined function is shown below:

\(E\left( x \right) = \left\{ \begin{aligned}10 &+ 0.06x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\mathop{\rm if}\nolimits} \,\,0 \le x \le 1200\\82 &+ 0.07\left( {x - 1200} \right)\,\,{\mathop{\rm if}\nolimits} \,\,x > 1200\end{aligned} \right.\)

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

The table of \(E\left( x \right)\) is shown below:

\(x\)	0	1200
\(E\left( x \right)\)	10	82

Sketch the graph \(E\left( x \right)\) for \(0 \le x \le 1200\) as shown below: