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.

Question

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Number of Miles Driven
Total Cost of Rental (Dollars)
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y
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Number of Miles Driven
Total Cost of Rental (Dollars)
Answer
Attempt 1 out of 2

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Answer 1

To find the point where the costs from Company A and Company B are the same based on the number of miles driven, we can first set up the functions for each company's rental cost:

1. Company A charges: \[ A(x) = 40 + 1x \] where \(A(x)\) is the total cost from Company A after driving \(x\) miles.

2. Company B charges: \[ B(x) = 20 + 1.5x \] where \(B(x)\) is the total cost from Company B after driving \(x\) miles.

Next, we want to find the number of miles, \(x\), where \(A(x) = B(x)\): \[ 40 + x = 20 + 1.5x \]

Now, let's solve for \(x\):

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: \[ x = 40 \]

So, the two companies' total costs will be the same when Mia drives 40 miles.

Summary of Costs:

When \(x = 40\):

• Cost from Company A: \[ A(40) = 40 + 1(40) = 40 + 40 = 80 \]

• Cost from Company B: \[ B(40) = 20 + 1.5(40) = 20 + 60 = 80 \]

Both costs are $80 when 40 miles are driven.

Graphing the Functions:

To graph the functions:

• Company A: \(y = 40 + x\)
• Company B: \(y = 20 + 1.5x\)

For graphing:

• The y-intercept for Company A is 40 and the slope is 1.
• The y-intercept for Company B is 20 and the slope is 1.5.

When you plot these on a graph, you will see that the lines intersect at the point (40, 80). This is where both companies would charge the same amount, confirming \(x = 40\) miles.