Question

Mia is deciding between two truck rental companies. Company A charges an initial fee of $40 for the rental plus $1 per mile driven. Company B charges an initial fee of $20 for the rental plus $1.50 per mile driven. Let AA represent the amount Company A would charge if Mia drives xx miles, and let BB represent the amount Company B would charge if Mia drives xx miles. Graph each function and determine the number miles driven, x, commax, that would make the cost of each company the same. include how to mark the graph.

Answer 1

To find the cost of each truck rental company as a function of the number of miles driven \( x \), we can define the cost functions for Company A and Company B as follows:

1. **Company A's cost function (A):**
\[
A(x) = 40 + 1x = 40 + x
\]

2. **Company B's cost function (B):**
\[
B(x) = 20 + 1.5x
\]

### Step 1: Set Up the Equations
To find the number of miles driven where the costs are the same, we need to set the two cost functions equal to each other:
\[
40 + x = 20 + 1.5x
\]

### Step 2: Solve for x
Now, let's solve the equation:
1. Subtract \( x \) from both sides:
\[
40 = 20 + 0.5x
\]

2. Subtract 20 from both sides:
\[
20 = 0.5x
\]

3. Multiply both sides by 2 to solve for \( x \):
\[
40 = x
\]

### Step 3: Determine the Intersection Point
Mia should drive **40 miles** for the costs to be the same for both companies.

### Step 4: Create a Graph
To graph the functions \( A(x) \) and \( B(x) \):

1. **Calculate some points for each function:**
- For Company A:
- \( A(0) = 40 + 0 = 40 \)
- \( A(20) = 40 + 20 = 60 \)
- \( A(40) = 40 + 40 = 80 \)
- \( A(60) = 40 + 60 = 100 \)

- For Company B:
- \( B(0) = 20 + 1.5(0) = 20 \)
- \( B(20) = 20 + 1.5(20) = 50 \)
- \( B(40) = 20 + 1.5(40) = 80 \)
- \( B(60) = 20 + 1.5(60) = 110 \)

2. **Plot the points on a graph:**
- **Company A (A):**
- Points: (0, 40), (20, 60), (40, 80), (60, 100)
- **Company B (B):**
- Points: (0, 20), (20, 50), (40, 80), (60, 110)

3. **Draw the lines:**
- Plot these points on a graph with \( x \) (miles driven) on the horizontal axis and the cost on the vertical axis. Draw a straight line through the points for Company A and Company B.

4. **Mark the intersection:**
- The intersection point of the two lines occurs where \( x = 40 \) and the cost at that point is \( A(40) = B(40) = 80 \). You can label this point on the graph as (40, 80).

### Additional Markings:
- Label the axes: "Miles Driven" for the x-axis and "Cost ($)" for the y-axis.
- Label the lines: "Company A" for the line corresponding to \( A(x) \) and "Company B" for the line corresponding to \( B(x) \).
- Highlight the intersection point with a different color or a dot and label it as "Costs Equal: (40, 80)".

This way, the graph will effectively show the comparison between the two companies and visually indicate the mileage where costs are the same.