Bearings are always measured clockwise from north.
A = start point
B = point when a plane was flying 400 km and starts to fly under a bearing of 70°
C = point after 600 km of fly
Angle betwen AB and BC = θ
A bearing of 70° mean θ = 180° - 70° = 110°
In this case you have a triangle:
AB = 400 km , BC = 600 km , θ = 110°
a)
Use law of cosines:
AC = AB² + BC² - 2 ∙ AB ∙ BC ∙ cos θ
AC² = 400² + 600² - 2 ∙ 400 ∙ 600 ∙ cos 110°
AC² = 160000 + 360000 - 480000 ∙ ( - 0.3420201433 )
AC² = 520000 + 480000 ∙ 0.3420201433
AC² = 520000 +164169.6688
AC² = 684169.6688
AC = √684169.6688
AC = 827.1454943 km
b)
The plane must bearing by angle α ( angle in point A )
Now law of sines:
sin α / sin θ = BC / AC
sin α / sin 110° = 600 / AC
sin α / 0.939692621 = 600 / 827.1454943
cross multiply
827.1454943 ∙ sin α = 600 ∙ 0.939692621
827.1454943 ∙ sin α = 563.8155726
sin α = 563.8155726 / 827.1454943
sin α = 0.681640142
α = arc sin ( 0.681640142 ) = 42.97194285°= 42° 58´ 19"
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Remark:
arc sin = sin⁻¹
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c)
In start point the plane it's not far from the north because it's flying to north.
E = far from the east
In point B angle between horizontal line and line BC is 90° - 70° = 20°
cos 20° = E / BC
far from the east:
E = BC ∙ cos 20° = 600 ∙ cos 20° = 600 ∙ 0.9396926208 = 563.81557248 km
A plane flies 400km North and then a further 600km or a bearing of 70 degree.
a) How far is the plane from the starting point.
b) On what bearing must the plane fly so that it returns to its original point.
c) How far north and how far east is the plane from the starting point.
4 answers
I don't understand it
I don't understand it
I apologize for the confusion. Let's go through the problem step by step.
a) To find how far the plane is from the starting point, we can use the Pythagorean theorem. The plane flew 400km North, so it is 400km away from the starting point in the North direction. Then, it flew a further 600km on a bearing of 70 degrees.
To visualize this, imagine a right angle triangle:
- The 400km length forms the vertical side of the triangle.
- The 600km length forms the horizontal side of the triangle.
- The plane's position is the endpoint of the hypotenuse.
To find the length of the hypotenuse (the distance from the starting point to the plane's position), we can use the formula: c^2 = a^2 + b^2, where c is the hypotenuse and a and b are the lengths of the other two sides.
c^2 = 400^2 + 600^2
c^2 = 160000 + 360000
c^2 = 520000
c = sqrt(520000)
c ≈ 721.11 km
Therefore, the plane is approximately 721.11 km away from the starting point.
b) To find the bearing the plane must fly to return to its original point, we need to find the angle θ between the line connecting the starting point to the final position and the North direction.
Since the plane flew 600km on a bearing of 70 degrees, the angle θ is the complementary angle to 70 degrees, which is 180 - 70 = 110 degrees.
To return to the original point, the plane needs to fly in the opposite direction, which is 180 degrees from the current bearing. So, the plane needs to fly on a bearing of 180 + 110 = 290 degrees.
Therefore, the plane must fly on a bearing of 290 degrees to return to its original point.
c) To find how far north and how far east the plane is from the starting point, we can use trigonometry.
The plane flew 400km North, so it is 400km north of the starting point.
To find the east distance, we can use the angle of 70 degrees that the plane flew. We can break down the 600km distance into its north and east components using the sine and cosine functions.
The east component (E) is given by:
E = 600km * cos(70 degrees)
E ≈ 197.99 km
Therefore, the plane is approximately 400km north and 197.99 km east from the starting point.
a) To find how far the plane is from the starting point, we can use the Pythagorean theorem. The plane flew 400km North, so it is 400km away from the starting point in the North direction. Then, it flew a further 600km on a bearing of 70 degrees.
To visualize this, imagine a right angle triangle:
- The 400km length forms the vertical side of the triangle.
- The 600km length forms the horizontal side of the triangle.
- The plane's position is the endpoint of the hypotenuse.
To find the length of the hypotenuse (the distance from the starting point to the plane's position), we can use the formula: c^2 = a^2 + b^2, where c is the hypotenuse and a and b are the lengths of the other two sides.
c^2 = 400^2 + 600^2
c^2 = 160000 + 360000
c^2 = 520000
c = sqrt(520000)
c ≈ 721.11 km
Therefore, the plane is approximately 721.11 km away from the starting point.
b) To find the bearing the plane must fly to return to its original point, we need to find the angle θ between the line connecting the starting point to the final position and the North direction.
Since the plane flew 600km on a bearing of 70 degrees, the angle θ is the complementary angle to 70 degrees, which is 180 - 70 = 110 degrees.
To return to the original point, the plane needs to fly in the opposite direction, which is 180 degrees from the current bearing. So, the plane needs to fly on a bearing of 180 + 110 = 290 degrees.
Therefore, the plane must fly on a bearing of 290 degrees to return to its original point.
c) To find how far north and how far east the plane is from the starting point, we can use trigonometry.
The plane flew 400km North, so it is 400km north of the starting point.
To find the east distance, we can use the angle of 70 degrees that the plane flew. We can break down the 600km distance into its north and east components using the sine and cosine functions.
The east component (E) is given by:
E = 600km * cos(70 degrees)
E ≈ 197.99 km
Therefore, the plane is approximately 400km north and 197.99 km east from the starting point.
1 answer