A custom printing company uses a linear relationship to determine the total cost of printi… — ACT

A custom printing company uses a linear relationship to determine the total cost of printing t-shirts. The table below displays the total cost, $C$ , in dollars, for printing $x$ t-shirts:

Number of Shirts ( $x$ )	Total Cost ( $C$ )
15	\$162.50
25	\$237.50
35	\$312.50

If the total cost consists of a one-time setup fee plus a constant charge per t-shirt, what is the one-time setup fee?

A
\$7.50
B
\$10.83
C
\$12.50
\$50.00Answer
E
\$112.50

Answer

The one-time setup fee is \$50.00.

The correct answer of \

50.00 is found by first determining the constant rate of change (slope) from the table, which is \$7.50 per shirt. Substituting this rate and the data point for 15 shirts into the linear model

C = mx + b

gives

162.50 = 7.50(15) + b

. Solving for the y-intercept

yields

50.00, which represents the one-time setup fee.

Step-by-Step Solution

Calculate the constant charge per t-shirt (the slope,

m

) by finding the change in cost divided by the change in the number of shirts between two data points.

m = \frac{237.50 - 162.50}{25 - 15} = \frac{75.00}{10} = 7.50

Since the relationship is linear, the rate of change (cost per shirt) remains constant.

Write the linear equation representing the total cost:

C = 7.50x + b

, where

b

represents the one-time setup fee.

C = 7.50x + b

This sets up the slope-intercept form of the linear relationship.

Substitute one of the points from the table, such as

(15, 162.50)

, into the equation to solve for

b

162.50 = 7.50(15) + b

162.50 = 112.50 + b

b = 50.00

Using a known coordinate point allows us to isolate and solve for the unknown y-intercept (the setup fee).

Key Concept

Linear Equations and Graphing

Question

Answer

Step-by-Step Solution

Key Concept

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