Question

Alice and her friend Emma leave on different flights from the same airport. Alice's flight flies 100 miles due south, then turns 70° toward west and flies 50 miles. Emma's flight flies 100 miles due north, then turns 50° toward east and flies 50 miles. Which fight among you is farther from the airport? Explan your rreasoning
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Answer 1

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Answer:

  Emma's flight is farther from the airport

Step-by-step explanation:

Even without doing any detailed calculation, you can see on a graph that the larger turn Alice's flight makes keeps it closer to the airport.

Emma's flight is farther from the airport.

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The distance can be calculated using the Law of Cosines. The angle of interest is the supplement of the turn angle. Since the leg distances are the same in each case, the only variable affecting the distance from the airport is the angle measure. Again, the larger the turn, the shorter the distance from the airport. If T is the turn angle, the distance is ...

  d² = 100² +50² -2·100·50·cos(180° -T)

  d² = 12500 +10000·cos(T) . . . . . . . using cos(180°-T) = -cos(T)

  d = 100√(1.25+cos(T))

For Alice's 70° turn, the distance is ...

  d = 100√(1.25+0.342) ≈ 126.18 mi

For Emma's 50° turn, the distance is ...

  d = 100√(1.25 +0.643) ≈ 137.58 mi

Emma's flight is farther from the airport.

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The law of cosines formula is ...

  c² = a² +b² -2ab·cos(C)

where triangle sides are lengths a, b, c and angle C is opposite side c.