Part 1:
The Cost function C(x) can be expressed as a linear function y = mx + b, where m is the slope, and b is the y-intercept.
Let x be the number of minutes of calling time, y be the cost for the month.
Given (x,y)
(280,60) → 280 minutes of calling time, with a cost of $60
(630,95) → 630 minutes of calling time, with a cost of $95
First find the slope of the function C(x)
[tex]\begin{gathered} \text{The slope is determined by} \\ m=\frac{y_2-y_1}{x_2-x_1} \\ \text{If }_{}(x_1,y_1)=\mleft(280,60\mright),\text{ and }(x_2,y_2)=\mleft(630,95\mright)\text{ then the slope is} \\ \\ m=\frac{y_2-y_1}{x_2-x_1} \\ m=\frac{95-60}{630-280} \\ m=\frac{35}{350} \\ m=\frac{1}{10} \end{gathered}[/tex]Now that we have the slope of the Cost function, we can now solve for its y-intercept. We will use the point (280,60) to solve, but using (630,95) will work just as well.
[tex]\begin{gathered} y=mx+b \\ \text{IF }(x,y)=(280,60)\text{ and }m=\frac{1}{10},\text{ THEN} \\ \\ y=mx+b \\ 60=(\frac{1}{10})(280)+b \\ 60=\frac{280}{10}+b \\ 60=28+b \\ 60-28=b \\ 32=b \\ b=32 \end{gathered}[/tex]Putting it together the cost function is
[tex]C(x)=\frac{1}{10}x+32[/tex]Part 2:
If jane used 710 minutes of calling time, how much was her bill.
Substitute x = 710 to the cost function and we get
[tex]\begin{gathered} C(x)=\frac{1}{10}x+32 \\ C(x)=\frac{1}{10}(710)+32 \\ C(x)=\frac{710}{10}+32 \\ C(x)=71+32 \\ C(x)=103 \end{gathered}[/tex]Therefore, her bill for the month of August is $103.
