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Difficulty: HardPythagorean Theorem Word Problems

A rescue helicopter flies 12 miles due south and then 35 miles due west to reach a medical emergency. After the pickup, the helicopter flies directly back to its base in a straight line. What is the total distance the helicopter traveled during this entire trip?

  1. A
    37 miles
  2. B
    47 miles
  3. 84 milesAnswer
  4. D
    94 miles

Answer

84 miles
The total distance is the sum of the outbound journey and the return journey. The outbound journey consists of the two legs of the flight: 12 + 35 = 47 miles. The return journey is a straight line that forms the hypotenuse of a right triangle. Using the Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2), the return distance is 122+352=1369=37\sqrt{12^2 + 35^2} = \sqrt{1369} = 37 miles. The total distance traveled is 47 + 37 = 84 miles.

Step-by-Step Solution

1
Calculate the distance traveled during the outbound trip.
12+35=4712 + 35 = 47 miles
The helicopter travels along two perpendicular paths (south and west) to reach the destination.
2
Set up the Pythagorean theorem to find the straight-line return distance.
c2=122+352c^2 = 12^2 + 35^2
The south and west paths form a right angle, making the direct return path the hypotenuse of a right triangle.
3
Solve for the return distance (the hypotenuse).
c=144+1225=1369=37c = \sqrt{144 + 1225} = \sqrt{1369} = 37 miles
Taking the square root of the sum of the squares gives the length of the straight-line return trip.
4
Add the outbound distance and the return distance.
47+37=8447 + 37 = 84 miles
The question asks for the total distance traveled during the entire round trip.

Key Concept

Applying the Pythagorean theorem to find a missing distance in a multi-step navigation problem.
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Topics

Pythagorean Theorem Word Problems

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