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
To find the equation of the line that represents the relationship between the number of brochures (x) and the total cost (y), we can use the two-point form of a linear equation.
First, we need to identify two points on the line. Looking at the table, we can choose the points (40, $600.00) and (55, $637.50).
Using the two-point form: y - y1 = ((y2 - y1)/(x2 - x1)) * (x - x1), we can plug in the values to find the equation.
Using the point (40, $600.00):
y - 600 = ((637.50 - 600)/(55 - 40)) * (x - 40)
Simplifying the equation:
y - 600 = (37.50/15) * (x - 40)
y - 600 = 2.50 * (x - 40)
y - 600 = 2.50x - 100
y = 2.50x + 500
Therefore, the equation that represents the relationship between the number of brochures (x) and the total cost (y) is y = 2.50x + 500.
To find the linear relationship between the number of brochures and the total cost of printing, we can use the given data points. Let's create a table showing the relationship:
Number of brochures, x | Total cost, y
-------------------------------------
40 | $600.00
43 | $607.50
45 | $612.50
52 | $630.00
55 | $637.50
Now, let's find the additional fee per brochure printed:
Step 1: Find the difference in the total cost for each pair of consecutive brochures.
- For example, for 40 brochures to 43 brochures, the difference in the total cost is $607.50 - $600.00 = $7.50.
- Similarly, find the differences for other pairs:
- 43 brochures to 45 brochures: $612.50 - $607.50 = $5.00
- 45 brochures to 52 brochures: $630.00 - $612.50 = $17.50
- 52 brochures to 55 brochures: $637.50 - $630.00 = $7.50
Step 2: Find the additional fee per brochure by dividing the difference in the total cost by the difference in the number of brochures for each pair.
- For example, for 40 brochures to 43 brochures, the difference in the number of brochures is 43 - 40 = 3 brochures.
- Dividing the difference in the total cost ($7.50) by the difference in the number of brochures (3) gives an additional fee of $2.50 per brochure.
Step 3: Determine the printing fee by subtracting the additional fee per brochure from the total cost for any given number of brochures.
- For example, for 40 brochures:
- Additional fee per brochure: $2.50
- Total cost: $600.00
- Printing fee = Total cost - Additional fee per brochure * Number of brochures
= $600.00 - $2.50 * 40
= $600.00 - $100.00
= $500.00
Now, using this information, we can find the printing fee and the additional fee per brochure for each number of brochures.
To find the relationship between the number of brochures and the total cost of printing, we can plot the data points on a graph.
The x-axis represents the number of brochures (x), and the y-axis represents the total cost (y).
Plotting the given data points:
Number of brochures (x) | Total cost (y)
-----------------------------------------
40 | $600.00
43 | $607.50
45 | $612.50
52 | $630.00
55 | $637.50
Now, we can connect the data points with a line to visualize the relationship.
After plotting the data points and connecting them with a line, we can observe that the line is a straight line which means the relationship between the number of brochures and the total cost of printing is linear.
To find the equation of the line and determine the additional fee per brochure printed, we need to find the slope (m) and the y-intercept (b) of the line.
The slope (m) of the line can be calculated using the formula:
m = (change in y) / (change in x)
For example, let's choose the first two data points to calculate the slope:
m = (607.50 - 600.00) / (43 - 40) = 7.50 / 3 = 2.50
Now that we know the slope (m), we can calculate the y-intercept (b) using the formula:
b = y - mx
Let's choose one of the data points, for example, (40, $600.00):
b = $600.00 - (2.50 * 40) = $600.00 - $100.00 = $500.00
Therefore, the equation of the line is:
y = mx + b
Substituting the values of m and b:
y = 2.50x + $500.00
This means that the total cost (y) of printing brochures is equal to $2.50 multiplied by the number of brochures (x), plus a fixed cost of $500.00. The additional fee per brochure printed is $2.50.
Now, you can use this equation to find the total cost of printing for any given number of brochures.