# Mary flew her airplane 100 kilometers north before turning 20 degrees from north clockwise and flying another 700 kilometers how far is she from her starting point

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It is given that Mary flew her airplane 100 kilometers north before turning 20 degrees in the clockwise direction and flying another 700 kilometers.

At the point that she decides to turn, she is 100 away from her starting point. When she turns and travels 700 km, she travels a distance of 700*cos 20 km towards the north and 700*sin 20 km towards the east.

From the starting point her total distance traveled towards the north is 100 + 700*cos 20 and the distance traveled towards the east is 700*sin 20. Her distance from the starting point is `sqrt((100+700*cos 20)^2 + (700*sin 20)^2)`

=>` sqrt(10000 + 490000 + 2*70000*cos 20)`

`~~ sqrt(631556.96)`

`~~794.70 km`

**Mary is approximately 794.7 km from her starting point.**

Let the starting point be (0,0) on x,y coordinates, East the x-direction and North the y-direction.

100km north is point (0,100)

20 degrees clockwise and 700km is further 700.sin(20) EastÂ and 700.cos(20) WestÂ and the new location is (700.sin(20),100+700.cos(20))

Distance from the starting point is sqrt[(700.sin(20))^2+(100+700.cos(20))^2] = sqrt(239.4^2+757.8^2) = 794.7km

Angle from North of the new Location = arctan(239.4/757.8) = 17.53 degrees

**The airplain is at 794.7km from starting point **17.53 degrees clockwise from North