A plane flies north at 122 mph in a west wind of 17 mph, what is the actual plane speed and direction? (1st quadrant is North of east or East of north, and so on...)

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The slope of the plane's course is found by calculating that for every 122 miles the plane goes north it goes -17 miles East (i.e. 17 miles West). This gives us a slope of `-122/17=-7 3/17` We then assume that the airport was the Origin of the Graph and we can plot that the plane is traveling along the ray on the graph below. (NNW)

The speed of the Plane along this ray can be calculated by noting that the ray, the planes hourly travel North and hourly travel west are the respective hypotenuse and legs of a right triangle. According to the Pythagorean Theorem, the square of the hypotenuse of any right triangle is equal to the sum of the squares of the two legs. If we use the plane's speed and wind speed as the legs, then the plane would travel the length of the hypotenuse in one hour. The length of the hypotenuse would be `sqrt(122^2+17^2)=sqrt(14,884+289)=sqrt(15,173)~~123.18` So the plane is traveling NNW at 123.18 mph.

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