Question

A cell phone plan has a basic charge of $35 a month. The plan includes 400 free minutes and charges 10 cents for each additional minute of usage. Write the monthly cost as a function of the number of minutes used and graph as a function of for\({\bf{0}} \le {\bf{x}} \le {\bf{600}}\)

Short answer

The monthly cost as a function of the number of minutes for the phone plan is\({\bf{C}}\left( {\bf{x}} \right){\bf{ = }}\left\{ \begin{array}{l}{\bf{35,x}} \le {\bf{400}}\\{\bf{35 + 0}}{\bf{.1}}\left( {{\bf{x - 400}}} \right){\bf{,x}} \ge {\bf{400}}\end{array} \right.\)and the graph of the cost function is shown in figure (1).

Step-by-step solution

Determine the equation for the volume of a cube

The monthly basic charge for the phone plan is

The number of free minutes is

The charge for minutes above free minutes is

The total number of minutes for the phone plan is

The monthly cost of the phone plan as a function of the number of minutes is given as:

\(C\left( {x \ge {x_0}} \right) = {C_{{x_0}}} + r\left( {x - {x_0}} \right)\)

Substitute all the values in the above equation.

\(\begin{array}{l}C\left( {x \ge 400} \right) = \left( {\$ 35} \right) + \left( {\$ 0.1/\min } \right)\left( {x - 400\;\min } \right)\\C\left( {x \ge 400} \right) = 35 + 0.1\left( {x - 400} \right)\end{array}\)

The cost for a monthly phone plan for less than free minutes is given as:

\(\begin{array}{l}C\left( {x \le {x_0}} \right) = \$ 35\\C\left( {x \le 400} \right) = 35\end{array}\)

The monthly cost of a phone plan as a function of the number of minutes is:

Determine the graph of the cost function

The graph of the cost as a function of the number of minutes is given below:

Figure (1)

Therefore, the monthly cost as a function of the number of minutes for the phone plan is \(C\left( x \right) = \left\{ \begin{array}{l}35,x \le 400\\35 + 0.1\left( {x - 400} \right),x \ge 400\end{array} \right.\) and a graph of the cost function is shown in figure (1) in which the red line shows the function for less than or equal to 400 and the blue line shows a function for greater than or equal to 400 but less than or equal to 600.