A Plane flies 400km west then 100km north find it's distance and bearing from it starting point with a suitable digram
To find the distance and bearing of the plane from its starting point, we can use the Pythagorean theorem and trigonometry.
First, let's draw a diagram:
```
P A
*--------->
| |
| |
| |
| |
v ^
B C
```
In this diagram, the starting point of the plane is point A, and it has flown 400km west to point B and then 100km north to point C. Point P represents the position we want to find: the distance and bearing of the plane from A.
To find the distance PA, we can use the Pythagorean theorem:
```
PA^2 = PB^2 + AB^2
PA^2 = 400^2 + 100^2
PA^2 = 160000 + 10000
PA^2 = 170000
PA = √170000
PA ≈ 412.3 km
```
So the plane is about 412.3km from its starting point.
To find the bearing of the plane from its starting point, we can use trigonometry. We want to find angle x, which is the angle between the line AB (which points due west) and the line PA.
```
tan(x) = BC/AB
tan(x) = 100/400
tan(x) = 0.25
x ≈ 14.0 degrees
```
So the bearing of the plane from its starting point is 14.0 degrees (measured clockwise from due east).
Therefore, we can conclude that the plane is about 412.3km away from its starting point, and is bearing 14.0 degrees (measured clockwise from due east) from that point.
To find the distance and bearing from the starting point, we can use the Pythagorean theorem to calculate the distance traveled and trigonometry to find the bearing.
Let's start by sketching a diagram:
^
|
100km (N)
|
*----x----> 400km (W)
Starting point
In this diagram, "x" represents the final position of the plane.
To find the distance traveled:
We can use the Pythagorean theorem, which states that the square of the hypotenuse (the longest side) of a right-angled triangle is equal to the sum of the squares of the other two sides.
In this case, the horizontal distance is 400km (west), and the vertical distance is 100km (north). Therefore, the distance traveled (hypotenuse) would be:
Distance = √(400^2 + 100^2)
= √(160000 + 10000)
= √170000
≈ 412.3km
So, the distance traveled by the plane is approximately 412.3km.
To find the bearing:
The bearing of an object is the angle measured clockwise from the north direction. To find the bearing, we can use the tangent function.
In this case, the angle we need to find is the angle opposite to the vertical distance (100km), which is the angle between the line connecting the starting and final points and the north direction.
Bearing = arctan(400/100)
= arctan(4)
≈ 76.04°
So, the bearing of the plane from the starting point is approximately 76.04°.