Two puzzles for me (but probably not for others)

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I am confident that I need either the law of sines and/or the law of cosines, but for both of these I cannot seem to translate the problem into the correct triangles to solve both.

1. After leaving an airport, a plane flies for 1.5 hours at a speed of 200 km/hr on a course of 200 degrees. Then, on a course of 340 degrees, the plane flies for 2 hours at a speed of 250 km/hr. At this time, how far from the airport is the plane?

2. Two hikers follow a trail that splits into two forks. Each hiker takes a different fork. The forks diverge at an angle of 67 degrees and both hikers walk at a speed of 3.5 miles / hour. How far apart are the hikers after 1 hour?

Phew.

Thanks.

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Pardon my MSPAINT skills.

There are two ways of solving these questions: one uses the sine or cosine rule, one uses simpler trig.

Sine/Cosine rule:
1. You need to find the third length of the blue triangle (you know the other two: distance = speed * time). If you look at the bottom of the “illustration” you see a straight line. Angles on one side always add up to 180 degrees, and you can find out the angles in the yellow and green triangles (they’re both right-angled triangles so you have two of the three angles already).
Once you’ve found the angle for the bottom of the blue triangle you’ll have two side lengths and one angle. So now: sine rule or cosine rule?…

Simple trig (it’s up to you to see how you can do this for #1):
2. I interpret “diverge” to mean that one fork will diverge from the original (straight) path by 67 degrees, not that both diverge away from each other by 67 degrees.
Either way, the two blue triangles have the same angles and, since the hikers are walking at the same speed, the same side lengths too. This also means that the two are right-angled triangles. You know one angle and one side length so use sine/cosine/tangent to get another side length…

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GREAT. Thanks.

I had the first set of triangles drawn (for the airplane issue), but just went blind or brain-dead trying to see the relationship.
The second one is really easy once you see your explanation.

Thanks, again.

Kevin

Pardon my MSPAINT skills.

There are two ways of solving these questions: one uses the sine or cosine rule, one uses simpler trig.

Sine/Cosine rule:
1. You need to find the third length of the blue triangle (you know the other two: distance = speed * time). If you look at the bottom of the “illustration” you see a straight line. Angles on one side always add up to 180 degrees, and you can find out the angles in the yellow and green triangles (they’re both right-angled triangles so you have two of the three angles already).
Once you’ve found the angle for the bottom of the blue triangle you’ll have two side lengths and one angle. So now: sine rule or cosine rule?…

Simple trig (it’s up to you to see how you can do this for #1):
2. I interpret “diverge” to mean that one fork will diverge from the original (straight) path by 67 degrees, not that both diverge away from each other by 67 degrees.
Either way, the two blue triangles have the same angles and, since the hikers are walking at the same speed, the same side lengths too. This also means that the two are right-angled triangles. You know one angle and one side length so use sine/cosine/tangent to get another side length…

Reply | Quote