Martha is printing brochures for Barton Middle School's STEM Night. The printing company charges a printing fee plus an additional fee per brochure printed. The table shows the linear relationship
between the number of brochures and the total cost of printing, including the printing fee.
Number of brochures, x
40
43
45
52
55
Total cost, y
$600.00
$607.50
$612.50
$630.00
$637.50
Which statement is true?
A. The additional fee per brochure is $7.50.
B. The printing fee is $2.50.
C. The additional fee per brochure is $2.50.
D. The printing fee is $250.
In order to determine the additional fee per brochure and the printing fee, we need to find the equation of the linear relationship between the number of brochures and the total cost.
Using the given data points, we can use the two-point form of a linear equation to find the equation of the line:
(y - y1) = [(y2 - y1)/(x2 - x1)](x - x1)
Using the points (40, $600.00) and (52, $630.00):
(y - 600.00) = [(630.00 - 600.00)/(52 - 40)](x - 40)
(y - 600.00) = [30.00/12](x - 40)
(y - 600.00) = 2.50(x - 40)
y - 600.00 = 2.50x - 100.00
y = 2.50x + 500.00
From the equation, we can see that the coefficient of x (2.50) represents the additional fee per brochure and the constant term (500.00) represents the printing fee.
Therefore, the correct statement is:
C. The additional fee per brochure is $2.50.
To find the additional fee per brochure and the printing fee, we can use the given data points and solve the linear relationship between the number of brochures and the total cost.
Let's start by calculating the difference in the total cost of printing for each pair of data points:
For the first pair:
Total cost difference = $607.50 - $600.00 = $7.50
For the second pair:
Total cost difference = $612.50 - $607.50 = $5.00
For the third pair:
Total cost difference = $630.00 - $612.50 = $17.50
For the fourth pair:
Total cost difference = $637.50 - $630.00 = $7.50
Now, let's examine the differences in total cost and the corresponding differences in the number of brochures:
For the first pair:
Number of brochures difference = 43 - 40 = 3
For the second pair:
Number of brochures difference = 45 - 43 = 2
For the third pair:
Number of brochures difference = 52 - 45 = 7
For the fourth pair:
Number of brochures difference = 55 - 52 = 3
The additional fee per brochure can be found by dividing the total cost difference by the number of brochures difference. Let's calculate:
For the first pair:
Additional fee per brochure = $7.50 / 3 = $2.50
For the second pair:
Additional fee per brochure = $5.00 / 2 = $2.50
For the third pair:
Additional fee per brochure = $17.50 / 7 ≈ $2.50
For the fourth pair:
Additional fee per brochure = $7.50 / 3 = $2.50
Since the additional fee per brochure is consistently $2.50 for each pair of data points, we can conclude that the statement:
C. The additional fee per brochure is $2.50.
is true.
To find the answer, we need to analyze the given table and determine the relationship between the number of brochures and the total cost.
Let's start by looking at the change in the total cost as the number of brochures increases. We can calculate the difference between consecutive values of the total cost:
(607.50 - 600.00) = 7.50
(612.50 - 607.50) = 5.00
(630.00 - 612.50) = 17.50
(637.50 - 630.00) = 7.50
Now, let's examine the difference in the number of brochures:
43 - 40 = 3
45 - 43 = 2
52 - 45 = 7
55 - 52 = 3
Based on our analysis, we can determine that the additional fee per brochure is not constant, as it changes from 5.00 to 17.50 and then back to 7.50. Therefore, statement A (The additional fee per brochure is $7.50) cannot be true.
Next, let's consider the printing fee. By comparing the change in the total cost with the smallest change in the number of brochures (2), we can calculate the printing fee:
(612.50 - 607.50) = 5.00
From this, we can see that the printing fee is $5.00.
Therefore, statement B (The printing fee is $2.50) cannot be true, as we have determined the printing fee to be $5.00.
Now, let's examine the additional fee per brochure again. By comparing the change in the total cost with the same change in the number of brochures (3), we can calculate the additional fee per brochure:
(607.50 - 600.00) = 7.50
This calculation shows that the additional fee per brochure is $7.50.
Therefore, statement A (The additional fee per brochure is $7.50) is true.
Looking at statement C, it claims that the additional fee per brochure is $2.50, which we have determined is not correct based on our analysis.
Finally, statement D claims that the printing fee is $250.00, which is also incorrect based on our calculations.
In conclusion, the correct statement is:
A. The additional fee per brochure is $7.50.