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.
There are no human answers yet.
Answered by GPT-4o mini
To determine the mileage at which the costs from Company A and Company B are the same, we can define the equations for each company based on the given information.

Let:
- \( A(x) \) be the cost from Company A.
- \( B(x) \) be the cost from Company B.
- \( x \) be the number of miles driven.

The cost functions are:
\[
A(x) = 40 + 1x
\]
\[
B(x) = 20 + 1.5x
\]

### Step 1: Set the equations equal to find x
To find the number of miles where the costs are equal, we set \( A(x) = B(x) \):

\[
40 + 1x = 20 + 1.5x
\]

### Step 2: Solve for x
We will isolate \( x \) on one side of the equation.

1. Subtract \( 1x \) 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 \( x \):
\[
40 = x
\]

So, \( x = 40 \) miles. This is the number of miles driven at which both companies will charge the same amount.

### Step 3: Calculate the cost for each company at \( x = 40 \)
You can plug \( x = 40 \) back into both functions to find the cost:

For Company A:
\[
A(40) = 40 + 1 \times 40 = 40 + 40 = 80
\]

For Company B:
\[
B(40) = 20 + 1.5 \times 40 = 20 + 60 = 80
\]

### Conclusion
At \( x = 40 \) miles, both companies charge $80.

### Step 4: Graphing the two functions
1. **For Company A:** The line starts at (0, 40) and has a slope of 1.
2. **For Company B:** The line starts at (0, 20) and has a slope of 1.5.

Both equations can be plotted on a graph with the x-axis representing miles and the y-axis representing cost. The point where these two lines intersect is the point \( (40, 80) \).