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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.

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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 \( (40, 80) \).