Question

1. a cell phone company charges a flat fee of $40.00 per month and an additional $20.00 per month for each phone on the plan.

a. is the relationship between the total monthly cost and the number of phones on the plan a proportional relationship? explain your thinking.
b. justify your thinking by creating a table or a graph.

Explanation:

Part (a)

Step1: Recall Proportional Relationship

A proportional relationship has the form \( y = kx \) (no constant term), where \( k \) is the constant of proportionality. The graph passes through the origin \((0,0)\).

Step2: Define Variables and Cost Equation

Let \( x \) = number of phones, \( y \) = total monthly cost.
Cost equation: \( y = 20x + 40 \).

Step3: Check for Proportionality

For \( x = 0 \) (no phones), \( y = 20(0) + 40 = 40
eq 0 \).
Since the equation has a constant term (\( 40 \)) and does not pass through the origin, it is not a proportional relationship.

Step1: Create a Table

Choose values for \( x \) (number of phones) and calculate \( y \) (total cost) using \( y = 20x + 40 \).

Number of Phones (\( x \))	Total Monthly Cost (\( y \))
1	\( 20(1) + 40 = 60 \)
2	\( 20(2) + 40 = 80 \)
3	\( 20(3) + 40 = 100 \)

Step2: Analyze the Table

In a proportional relationship, \( \frac{y}{x} \) is constant for all \( x
eq 0 \).

• For \( x = 1 \), \( \frac{60}{1} = 60 \)
• For \( x = 2 \), \( \frac{80}{2} = 40 \)
• For \( x = 3 \), \( \frac{100}{3} \approx 33.33 \)

Since \( \frac{y}{x} \) is not constant, and when \( x = 0 \), \( y = 40
eq 0 \), the relationship is not proportional. (A graph would also be a non - vertical line with a y - intercept of 40, not passing through the origin.)

Answer:

The relationship is not proportional. A proportional relationship has the form \( y = kx \) (passes through \((0,0)\)), but here \( y = 20x + 40 \), so when \( x = 0 \), \( y = 40
eq 0 \).

Part (b)