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.

for problems 2 and 3, identify whether the relationship is proportional. if so, define your variables and write an equation. if not, change a value to make it a proportional relationship.

1. each day of doggie daycare costs $25.

Explanation:

a.

Step1: Recall proportional - relationship definition

A proportional relationship has the form $y = kx$, where $k$ is a constant and when $x = 0$, $y=0$. Let $x$ be the number of phones and $y$ be the total monthly cost. The cost function is $y=20x + 40$.

Step2: Check the condition for $x = 0$

When $x = 0$ (no phones on the plan), $y=40$. In a proportional relationship, when $x = 0$, $y$ should be $0$. Since $y
eq0$ when $x = 0$, the relationship is not proportional.

Step1: Create a table

Number of Phones ($x$)	Total Monthly Cost ($y$)
1	$y=20\times1+40 = 60$
2	$y=20\times2 + 40=80$
3	$y=20\times3+40 = 100$

The ratios $\frac{y}{x}$ are: $\frac{40}{0}$ (undefined), $\frac{60}{1}=60$, $\frac{80}{2}=40$, $\frac{100}{3}\approx33.33$. Since the ratios are not constant, it is not a proportional relationship.
If we were to graph this, the equation $y = 20x+40$ is a linear - equation with a $y$ - intercept of $40$ and a slope of $20$. A proportional relationship would be a straight line passing through the origin $(0,0)$, but this line does not pass through the origin.

Step1: Identify the variables and relationship

Let $x$ be the number of days of doggie daycare and $y$ be the total cost. The relationship is $y = 25x$. When $x = 0$, $y=0$, and the ratio $\frac{y}{x}=\frac{25x}{x}=25$ (for $x
eq0$). Since the relationship is of the form $y = kx$ ($k = 25$) and passes through the origin, it is a proportional relationship.

Answer:

No, the relationship is not proportional because when the number of phones ($x$) is $0$, the total monthly cost ($y$) is $40$ instead of $0$ as required for a proportional relationship.

b.