We are given that Jane used 200 minutes and the cost was $75, and also she used 680 minutes and the cost was $195. To determine a function of the cost "C" as a function of the minutes "x" we will assume that the behavior of this function is that of a line. Therefore, the function must have the following form:
[tex]C(x)=mx+b[/tex]Where "m" is the slope and "b" the y-intercept. We will determine the slope using the following formula:
[tex]m=\frac{C_2-C_1}{x_2-x_1}[/tex]Where:
[tex](x_1,C_1),(x_2,C_2)[/tex]Are points in the line. The given points are:
[tex]\begin{gathered} (x_1_{},C_1)=(200,75) \\ (x_2,C_2)=(680,195) \end{gathered}[/tex]Substituting in the formula for the slope we get:
[tex]m=\frac{195-75}{680-200}[/tex]Solving the operations we get:
[tex]m=\frac{120}{480}=\frac{1}{4}[/tex]Now we substitute in the formula for the line:
[tex]C(x)=\frac{1}{4}x+b[/tex]Now we determine the value if "b" by substituting the first point. This means that when C = 200, x = 75.
[tex]200=\frac{1}{4}(75)+b[/tex]Solving the product:
[tex]200=18.75+b[/tex]Now we subtract 18.75 from both sides:
[tex]\begin{gathered} 200-18.75=b \\ 181.25=b \end{gathered}[/tex]Therefore, the formula of the cost is:
[tex]C(x)=\frac{1}{4}x+181.25[/tex]Part B. We are asked to determine the cost is there is a consumption of 323 minutes. To do that we will substitute in the formula for "C" the value of x = 323.
[tex]C(323)=\frac{1}{4}(323)+181.25[/tex]Solving the operations we get:
[tex]C(323)=262[/tex]Therefore, the cost is $262.
