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Difficulty: MediumLinear Equations and Graphing

A taxi company charges a flat fee plus a constant rate per mile driven. The total cost, CC (in dollars), for a ride of dd miles is given by a linear relationship. If a 6-mile ride costs 18.50anda10−mileridecosts18.50 and a 10-mile ride costs 26.50, what is the flat fee, in dollars, charged by the taxi company?

  1. A
    15.50
  2. B
    2.00
  3. 6.50Answer
  4. D
    12.50
  5. E
    25.00

Answer

The flat fee charged by the taxi company is 6.50 dollars.
The correct flat fee is 6.50 dollars. The linear relationship is of the form C=md+bC = md + b, where mm is the cost per mile and bb is the flat fee. The slope is calculated as m=26.50−18.5010−6=2.00m = \frac{26.50 - 18.50}{10 - 6} = 2.00. Substituting the point (6,18.50)(6, 18.50) into the equation gives 18.50=2(6)+b18.50 = 2(6) + b, which simplifies to 18.50=12.00+b18.50 = 12.00 + b, yielding b=6.50b = 6.50.

Step-by-Step Solution

1
Identify the two coordinates from the word problem and calculate the slope (rate per mile).
The points are (6,18.50)(6, 18.50) and (10,26.50)(10, 26.50). The slope m=26.50−18.5010−6=8.004=2.00m = \frac{26.50 - 18.50}{10 - 6} = \frac{8.00}{4} = 2.00.
In a linear relationship, the constant rate of change is represented by the slope of the line.
2
Use the slope-intercept form C=md+bC = md + b and substitute one of the points to solve for the vertical intercept bb (flat fee).
Using (6,18.50)(6, 18.50): 18.50=2.00(6)+b  ⟹  18.50=12.00+b  ⟹  b=6.5018.50 = 2.00(6) + b \implies 18.50 = 12.00 + b \implies b = 6.50.
The flat fee corresponds to the cost when the distance is zero miles, which is the vertical intercept bb.

Key Concept

Determining the equation of a line from two points in a real-world linear application.
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Topics

Linear Equations and Graphing

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