Answered

A turnpike driver drove 18 miles from milepost 65 before having car trouble. Assuming the driver is confused and disoriented when calling for help, explain how the driver can determine their possible location.



How can the driver determine their possible location?

AThe driver could have driven 18 miles in either direction from milepost 65. Therefore, their location could be either at the $65-18=47$ milepost or at the $65+18=83$ milepost.
BThe driver drove 18 miles from milepost 65. Therefore, their location is at the $65-18=47$ milepost.
CThe driver drove 18 miles from milepost 65. Therefore, their location is at the $65+18=83$ milepost.
DThere is no way to determine the driver's location.

Verified Answer :

  • Identify the starting position at milepost 65 and the distance traveled of 18 miles.
  • Recognize that the direction of travel is unknown, requiring both addition and subtraction.
  • Calculate the two possible mileposts: \(65 + 18 = 83\) and \(65 - 18 = 47\).
  • Select option A as it correctly represents these two possibilities.
  • The final answer is \(\boxed{\text{A}}\).

Explanation

  1. Problem Analysis and Data Collection

The driver starts at milepost 65 on a turnpike and travels a distance of 18 miles. Because the driver is disoriented, the direction of travel is uncertain. This means they could have driven 18 miles in the direction of increasing milepost numbers or 18 miles in the direction of decreasing milepost numbers.

  1. Calculating Possible Locations

To find the possible locations, we consider both directions of travel from the starting point of 65:

  1. If the driver moved in the direction of increasing mileposts, the new position is calculated by adding the distance traveled:

\[65 + 18 = 83\]

  1. If the driver moved in the direction of decreasing mileposts, the new position is calculated by subtracting the distance traveled:

\[65 - 18 = 47\]

Thus, the driver could be at either milepost 47 or milepost 83. Comparing this to the given options, option A correctly accounts for both possibilities.

  1. Conclusion and Final Answer Selection

The driver's possible locations are milepost 47 and milepost 83. Therefore, option A provides the correct assessment of the situation. \(\boxed{\text{A}}\)

Examples

This problem illustrates the concept of absolute value in distance. If you represent the starting position as a coordinate \(x_0 = 65\) and the distance traveled as \(d = 18\), the final position \(x\) satisfies the equation \(|x - 65| = 18\). This means the distance between the final location and the starting point is exactly 18 miles. Such problems are foundational for understanding how to model spatial relationships, navigation, and relative changes in position, which are common in real-world scenarios like tracking vehicles or planning routes.

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