Question

which linear relation has a different rate of change from the others?

• the cost of producing shirts is made up of a \\$10 initial fee, plus \\$15 per shirt.
• \\(c = 77 + 15n\\)
• graph showing a line passing through (0,0) and (6,90)
• table with columns \\(n\\) and \\(c\\): (20, 100), (40, 115), (60, 130)

Explanation:

Analyze the verbal description

Using the Constant Rate of Change and Linear Relationships knowledge points
The verbal description states: "The cost of producing shirts is made up of a $10 initial fee, plus $15 per shirt."
The rate of change is the cost per shirt:

$$ \text{Rate of change} = 15 $$

Analyze the equation

Using the Constant Rate of Change and Linear Relationships knowledge points
The given equation is:

$$ C = 77 + 15n $$

In a linear equation of the form \(C = mn + b\), the coefficient \(m\) represents the rate of change:

$$ \text{Rate of change} = 15 $$

Analyze the graph

Using the Constant Rate of Change and Linear Relationships knowledge points
The graph shows a line passing through the origin \((0,0)\) and the point \((4, 60)\).
We calculate the slope (rate of change):

$$ \text{Rate of change} = \frac{60 - 0}{4 - 0} = 15 $$

Analyze the table

Using the Constant Rate of Change and Linear Relationships knowledge points
The table provides the following coordinate pairs \((n, C)\):

$$ (20, 100), \quad (40, 115), \quad (60, 130) $$

We calculate the rate of change between the first two points:

$$ \text{Rate of change} = \frac{115 - 100}{40 - 20} = \frac{15}{20} = 0.75 $$

Identify the different relation

Using the Constant Rate of Change and Linear Relationships knowledge points
Comparing the rates of change:

• Verbal description: \(15\)
• Equation: \(15\)
• Graph: \(15\)
• Table: \(0.75\)

The table has a different rate of change.

Answer:

• (A) The verbal description: "The cost of producing shirts is made up of a $10 initial fee, plus $15 per shirt."
• (B) The equation: \(C = 77 + 15n\)
• (C) The graph showing a line passing through \((0,0)\) and \((4,60)\)
• (D) The table with values \((20, 100)\), \((40, 115)\), and \((60, 130)\) (Correct answer)