Question

1. modeling with mathematics two newspapers charge a fee for placing an advertisement in their paper plus a fee based on the number of lines in the advertisement. the table shows the total costs for different length advertisements at the daily times. the total cost y (in dollars) for an advertisement that is x lines long at the greenville journal is represented by the equation y = 2x + 20. which newspaper charges less per line? how many lines must be in an advertisement for the total costs to be the same? (see example 2.)

daily times

number of lines, x	total cost, y
5	30
6	33
7	36
8	39

Explanation:

Step1: Find the cost - per - line for the Daily Times

The cost for the Daily Times is a linear relationship. We use the formula for the slope of a line $m=\frac{y_2 - y_1}{x_2 - x_1}$. Let's take two points $(x_1,y_1)=(4,27)$ and $(x_2,y_2)=(5,30)$. Then $m_{Daily\ Times}=\frac{30 - 27}{5 - 4}=3$.

Step2: Find the cost - per - line for the Greenville Journal

The cost function for the Greenville Journal is $y = 2x+20$, and the slope of the line (cost per line) is $m_{Greenville\ Journal}=2$.

Step3: Compare the cost - per - line

Since $2<3$, the Greenville Journal charges less per line.

Step4: Set the two cost functions equal to find when the total costs are the same

Let the cost function for the Daily Times be $y = 3x + b$. Using the point $(4,27)$: $27=3\times4 + b$, so $b = 27-12 = 15$, and the cost function for the Daily Times is $y = 3x+15$. Set $3x + 15=2x + 20$.
Subtract $2x$ from both sides: $3x-2x+15=2x-2x + 20$, which gives $x+15 = 20$.
Subtract 15 from both sides: $x=20 - 15=5$.

Answer:

The Greenville Journal charges less per line. The total costs are the same when there are 5 lines in the advertisement.