Anaeroplane flies from G to a city h on a bearing of 150degree the distance between G and H is 300km it then flew a bearing of 60degree to J a distance of 450km. Calculate G to J
use the law of cosines.
GH^2 = 300^2 + 450^2 - 2*300*450 cosθ
so find θ, the angle between the two segments, and plug it in.
To calculate the distance between point G and point J, we can use the concept of vector addition.
We start by looking at the triangle formed by the positions of G, H, and J. Let's denote the position of G as (0, 0) on a coordinate plane.
Next, using the given information, we can determine the coordinates of point H relative to G. The bearing of 150 degrees means that the angle between the line GH and the positive x-axis is 150 degrees. This information allows us to calculate the change in x and y coordinates from G to H.
Using trigonometry, we can find the change in x-coordinate (Δx) and the change in y-coordinate (Δy) as follows:
Δx = 300 km * cos(150 degrees)
Δy = 300 km * sin(150 degrees)
Now, using the obtained values of Δx and Δy, we can determine the coordinates of point H:
Coordinates of point H = (0 + Δx, 0 + Δy) = (Δx, Δy)
Moving on to point J, we can use a similar approach to determine the change in x and y coordinates from H to J, given that the bearing is 60 degrees and the distance is 450 km.
Using trigonometry, we can find the change in x-coordinate (Δx) and the change in y-coordinate (Δy) as follows:
Δx = 450 km * cos(60 degrees)
Δy = 450 km * sin(60 degrees)
Now, using the obtained values of Δx and Δy, we can determine the coordinates of point J relative to G:
Coordinates of point J = (Δx + H_x, Δy + H_y)
Finally, we can calculate the distance between G and J using the coordinates of point J:
Distance between G and J = √((J_x - G_x)^2 + (J_y - G_y)^2)
To calculate the distance between point G and point J, we need to use the concept of vector addition.
First, let's break down the given information:
- Anaeroplane flies from G to H on a bearing of 150 degrees.
- The distance between G and H is 300 km.
- Then, it flies from H to J on a bearing of 60 degrees.
- The distance between H and J is 450 km.
To calculate the distance between G and J, we need to find the horizontal and vertical components of the distances traveled in each direction.
Step 1: Finding the horizontal and vertical components:
For the first leg of the trip (G to H):
- Horizontal component (GHx): GH * cos(bearing of 150 degrees)
- Vertical component (GHy): GH * sin(bearing of 150 degrees)
For the second leg of the trip (H to J):
- Horizontal component (HJx): HJ * cos(bearing of 60 degrees)
- Vertical component (HJy): HJ * sin(bearing of 60 degrees)
Step 2: Adding the horizontal and vertical components:
To find the combined horizontal and vertical components, we add up the respective components for each leg of the trip.
GJx = GHx + HJx
GJy = GHy + HJy
Step 3: Calculating the distance between G and J:
The distance between G and J can be calculated using the Pythagorean theorem:
GJ = sqrt((GJx)^2 + (GJy)^2)
Let's calculate it:
GHx = 300 km * cos(150°) ≈ -259.81 km
GHy = 300 km * sin(150°) ≈ 150 km
HJx = 450 km * cos(60°) = 225 km
HJy = 450 km * sin(60°) = 389.42 km
GJx = -259.81 km + 225 km ≈ -34.81 km
GJy = 150 km + 389.42 km ≈ 539.42 km
GJ = sqrt((-34.81 km)^2 + (539.42 km)^2) ≈ 542.43 km
Therefore, the distance between G and J is approximately 542.43 km.