A plane flies 400km west then 100km north .find its distance and bearing from its starting point
d^2 = 400^2 + 100^2
..
d = 412.3 km
If graphed, we are in quadrant II
tan(angle) = 100/400
angle = 14° from the horizontal
"bearing" is 270+14 = 284°
How is it 14 for the angle
Student
Well, if the plane flew 400km west and then 100km north, I'm afraid it must be a little lost. It's not even close to where it started! But let's humor it for a moment.
To find the distance and bearing from the starting point, we can use good old Pythagoras. The distance traveled can be found by treating the westward and northward movements as the two sides of a right-angled triangle.
So, we have 400km as the westward side and 100km as the northward side. Let's find the hypotenuse, which represents the straight-line distance from the starting point.
Using the Pythagorean theorem (a^2 + b^2 = c^2), we can calculate the hypotenuse:
400^2 + 100^2 = c^2
160000 + 10000 = c^2
170000 = c^2
c ≈ 412.31 km
So, the distance from the starting point is approximately 412.31 km. However, since the plane is quite lost, the bearing from its starting point might be a little pointless to calculate. It's probably best to focus on helping it find its way back!
To find the distance and bearing of the plane from its starting point, you can use the concept of vector addition.
1. Draw a diagram: Start by drawing a rough diagram representing the plane's movements. Label the starting point as point A, the first destination (400km west) as point B, and the final destination (100km north) as point C.
2. Calculate the displacement vectors: Notice that the plane's movement can be broken down into two displacement vectors. The first displacement vector is 400km west from point A to point B, and the second displacement vector is 100km north from point B to point C.
3. Adding the displacement vectors: To find the final displacement vector, simply add the two displacement vectors together.
- The 400km west displacement vector can be represented as (-400, 0) since it only changes the x-coordinate.
- The 100km north displacement vector can be represented as (0, 100) since it only changes the y-coordinate.
- Adding these two vectors together: (-400, 0) + (0, 100) = (-400, 100)
So, the final displacement vector is (-400, 100).
4. Calculating the distance: To find the distance, we can use the Pythagorean theorem. The distance is equal to the magnitude of the displacement vector.
- Distance = sqrt((-400)^2 + 100^2)
= sqrt(160000 + 10000)
= sqrt(170000)
= 412.31 km (rounded to two decimal places)
5. Calculating the bearing: The bearing is the angle between the displacement vector and the positive x-axis.
- To find the bearing, we can use trigonometric functions.
- Tangent of the bearing angle = (opposite / adjacent)
- Tangent of the bearing angle = 100 / 400
= 0.25
- Therefore, the bearing angle θ is the inverse tangent of 0.25.
- Bearing angle, θ = arctan(0.25)
= 14.04 degrees (rounded to two decimal places)
6. Identifying the direction: Since the plane moved west, the bearing angle will be relative to the positive x-axis. In the northwest quadrant, angles are measured clockwise from the positive x-axis.
- The bearing angle is 14.04 degrees, which puts the plane in the first quadrant.
- The bearing direction can be noted as N14.04°W or E344.04°N.
Therefore, the plane is approximately 412.31 kilometers away from its starting point and its bearing is N14.04°W (or E344.04°N).