Village A, B, C, D, are such that B is 4km due east of A, c is 3km due south of B and D is 4km s50degreew from c. Calculate the distance and bearing ofA from D.
add up the x- and y-components of the vectors, then convert the result back to distance and bearing.
Send my the diagram and the solution.
To calculate the distance and bearing of A from D, we can break down the steps as follows:
Step 1: Find the coordinates of each village.
Let's assume that the coordinates of village A are (0,0).
Since village B is 4km due east of A, its coordinates would be (4,0).
Village C is 3km due south of B, so its coordinates would be (4,-3).
Lastly, D is 4km at an angle of 50 degrees from C. To find its coordinates, we can calculate the horizontal and vertical distances as follows:
Horizontal distance = 4km * cos(50 degrees) = 4km * 0.64279 = 2.57116km (approximately)
Vertical distance = 4km * sin(50 degrees) = 4km * 0.76604 = 3.06418km (approximately)
Adding these distances to the coordinates of C, we get the coordinates of D as (4 + 2.57116, -3 + 3.06418) = (6.57116, 0.06418) (approximately).
Step 2: Calculate the distance between A and D.
Using the coordinates of A and D, we can apply the distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Distance = sqrt((6.57116 - 0)^2 + (0.06418 - 0)^2)
Distance = sqrt(43.14 + 0.00413)
Distance ≈ sqrt(43.14) ≈ 6.57 km (approximately).
Step 3: Calculate the bearing of A from D.
To calculate the bearing, we consider the angle between the straight line connecting A and D with respect to the north direction.
Using the coordinates of A and D, we can apply the bearing formula:
Bearing = atan2(x2 - x1, y2 - y1)
Bearing = atan2(6.57116 - 0, 0.06418 - 0)
Bearing = atan2(6.57116, 0.06418)
Bearing ≈ 89.08 degrees (approximately).
Therefore, the distance from A to D is approximately 6.57 km and the bearing of A from D is approximately 89.08 degrees.
To calculate the distance and bearing of village A from village D, we can use the concept of vectors and trigonometry.
First, let's visualize the positions of the villages on a coordinate grid. Assume the starting point is at the origin (0, 0).
Since village B is 4km due east of A, we can assign the coordinates A(0, 0) and B(4, 0).
Next, since village C is 3km due south of B, we can assign the coordinates C(4, -3).
Now, we need to find the coordinates of village D. We know that D is situated 4km s50degreew from C. To represent this, we can draw a line segment from C of length 4km at an angle of 50 degrees counterclockwise from the positive x-axis.
To find the x-coordinate of D, we use cosine of the angle: Dx = Cx + (4 * cos(50)).
To find the y-coordinate of D, we use sine of the angle: Dy = Cy + (4 * sin(50)).
Calculating the x-coordinate of D:
Dx = 4 + (4 * cos(50)) = 4 + (4 * 0.6428) = 4 + 2.5712 ≈ 6.5712.
Calculating the y-coordinate of D:
Dy = -3 + (4 * sin(50)) = -3 + (4 * 0.7660) = -3 + 3.064 ≈ 0.064.
Therefore, the coordinates of village D are approximately D(6.5712, 0.064).
To calculate the distance between village A and D, we use the distance formula:
Distance = sqrt((Dx - Ax)^2 + (Dy - Ay)^2)
Distance = sqrt((6.5712 - 0)^2 + (0.064 - 0)^2)
Distance = sqrt(6.5712^2 + 0.064^2)
Distance = sqrt(43.1879 + 0.0041)
Distance ≈ sqrt(43.192)
Distance ≈ 6.573 km (rounded to 3 decimal places).
To calculate the bearing of village D from A, we use the concept of direction angles:
Bearing = atan2(Dy - Ay, Dx - Ax) * (180/π)
Bearing = atan2(0.064 - 0, 6.5712 - 0) * (180/π)
Bearing = atan2(0.064, 6.5712) * (180/π)
Bearing ≈ 0.5501 * (180/π)
Bearing ≈ 31.542 degrees (rounded to 3 decimal places).
Therefore, the distance and bearing of village A from village D are approximately 6.573 km and 31.542 degrees, respectively.