Answer :
To solve the problem, we need to determine the relationship between the number of movies rented each month and the total cost using the data provided in the table. Specifically, we need to find the slope of the graph representing this relationship and interpret its meaning.
### Step-by-Step Solution:
1. Identify the data points:
- Number of movies (x-values): 12, 15, 9, 22
- Total cost (y-values): [tex]$35.00, $[/tex]42.50, [tex]$27.50, $[/tex]60.00
2. Choose two points to calculate the slope:
Let's choose the first two data points:
- Point 1: [tex]\((12, 35.00)\)[/tex]
- Point 2: [tex]\((15, 42.50)\)[/tex]
3. Calculate the slope (m):
The slope [tex]\(m\)[/tex] of a line through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Using the points [tex]\((12, 35.00)\)[/tex] and [tex]\((15, 42.50)\)[/tex]:
[tex]\[ m = \frac{42.50 - 35.00}{15 - 12} = \frac{42.50 - 35.00}{3} = \frac{7.50}{3} = 2.50 \][/tex]
4. Interpretation of the slope:
The slope [tex]\(2.50\)[/tex] represents the rate of change of the total cost with respect to the number of movies rented. In this context, it means that for each additional movie rented, the total cost increases by [tex]$2.50. Thus, the slope represents the cost per movie rented. 5. Determine the membership fee (fixed cost): We assume that the total cost includes both a membership fee and the cost of renting movies. The total cost can be expressed as: \[ \text{Total Cost} = \text{Membership Fee} + (\text{Cost per Movie} \times \text{Number of Movies}) \] Using one of the points and the slope, we can solve for the membership fee. Let's use the point \((12, 35.00)\): \[ 35.00 = \text{Membership Fee} + (2.50 \times 12) \] \[ 35.00 = \text{Membership Fee} + 30.00 \] \[ \text{Membership Fee} = 35.00 - 30.00 = 5.00 \] ### Conclusion: - The slope of the graph is \(2.50\), and it represents the cost of each movie rented. - The membership fee is \$[/tex]5.00.
Thus, the correct option is:
- The slope of the graph is 2.50 , and it represents the cost of each movie.
### Step-by-Step Solution:
1. Identify the data points:
- Number of movies (x-values): 12, 15, 9, 22
- Total cost (y-values): [tex]$35.00, $[/tex]42.50, [tex]$27.50, $[/tex]60.00
2. Choose two points to calculate the slope:
Let's choose the first two data points:
- Point 1: [tex]\((12, 35.00)\)[/tex]
- Point 2: [tex]\((15, 42.50)\)[/tex]
3. Calculate the slope (m):
The slope [tex]\(m\)[/tex] of a line through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Using the points [tex]\((12, 35.00)\)[/tex] and [tex]\((15, 42.50)\)[/tex]:
[tex]\[ m = \frac{42.50 - 35.00}{15 - 12} = \frac{42.50 - 35.00}{3} = \frac{7.50}{3} = 2.50 \][/tex]
4. Interpretation of the slope:
The slope [tex]\(2.50\)[/tex] represents the rate of change of the total cost with respect to the number of movies rented. In this context, it means that for each additional movie rented, the total cost increases by [tex]$2.50. Thus, the slope represents the cost per movie rented. 5. Determine the membership fee (fixed cost): We assume that the total cost includes both a membership fee and the cost of renting movies. The total cost can be expressed as: \[ \text{Total Cost} = \text{Membership Fee} + (\text{Cost per Movie} \times \text{Number of Movies}) \] Using one of the points and the slope, we can solve for the membership fee. Let's use the point \((12, 35.00)\): \[ 35.00 = \text{Membership Fee} + (2.50 \times 12) \] \[ 35.00 = \text{Membership Fee} + 30.00 \] \[ \text{Membership Fee} = 35.00 - 30.00 = 5.00 \] ### Conclusion: - The slope of the graph is \(2.50\), and it represents the cost of each movie rented. - The membership fee is \$[/tex]5.00.
Thus, the correct option is:
- The slope of the graph is 2.50 , and it represents the cost of each movie.
