Villages ABCD are such that B is 4km due east of A C is 3km due to South of B and D is 4km S50W from C calculate the distance and bearing of A from aD
To calculate the distance and bearing of A from D, we can use the concept of vector addition.
Step 1: Draw a diagram with the given information:
A
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B
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C
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D
Step 2: Calculate the vector AD by breaking it down into its components:
AD = AC + CD
Step 3: Calculate the vector AC:
AC = -AB + BC
Given AB = 4 km due east, we can represent it as (-4i, 0j).
And BC = -3 km due south, we can represent it as (0i, -3j).
AC = (-4i, 0j) + (0i, -3j) = (-4i, -3j)
Step 4: Calculate the vector CD:
To find CD, we need to calculate the x and y components of CD:
Given CD = 4 km at S50W, we can decompose it as follows:
Component along the x-axis = 4 km * cos(50) = 4 km * cos(50°)
Component along the y-axis = 4 km * sin(50) = 4 km * sin(50°)
CD = (Component along the x-axis, Component along the y-axis)
CD = (4 km * cos(50°), 4 km * sin(50°))
Step 5: Calculate the vector AD:
AD = AC + CD = (-4i, -3j) + (4 km * cos(50°), 4 km * sin(50°))
Step 6: Calculate the magnitude of AD (distance from A to D):
The magnitude of a vector AD = sqrt((ADx)^2 + (ADy)^2)
AD = sqrt((-4)^2 + (-3)^2 + (4 km * cos(50°))^2 + (4 km * sin(50°))^2)
Step 7: Calculate the bearing of A from D:
The bearing of A from D can be calculated using the inverse tangent function:
bearing = arctan(ADy / ADx)
bearing = arctan((4 km * sin(50°)) / (4 km * cos(50°)))
Now you can calculate the distance and bearing of A from D using the given information.
To calculate the distance and bearing of A from D, we need to break down the given information and use some basic trigonometry.
First, let's draw a diagram to visualize the situation:
B (4km, East)
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A (Unknown Position)| | D (Unknown Position)
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C (3km, South of B)
Now, let's analyze the information we have:
1. B is 4km due east of A: This means the distance between A and B in terms of longitude is 4km. Since we don't have any information about the latitude, we'll assume the latitude is the same for both points A and B.
2. C is 3km due south of B: This means the distance between B and C in terms of latitude is 3km. Again, we'll assume the longitude is the same for points B and C.
3. D is 4km S50W from C: This tells us the distance and direction from C to D. S50W means 50 degrees west of south. We can convert this to a bearing relative to the reference line south (180 degrees) by subtracting 50 degrees from 180 degrees. So the bearing from C to D is 180 - 50 = 130 degrees.
Now, let's calculate the coordinates of C and D using the information we have:
Latitude of C = Latitude of B - 3km
Longitude of C = Longitude of B (since B is east of A)
Latitude of D = Latitude of C
Longitude of D = Longitude C - 4km (in the direction of S50W)
Finally, we can calculate the distance and bearing between A and D using the coordinates:
Distance = √[(Latitude of D - Latitude of A)^2 + (Longitude of D - Longitude of A)^2]
Bearing = arctan((Longitude of D - Longitude of A) / (Latitude of D - Latitude of A))
By substituting the calculated values, you can find the exact distance and bearing value from A to D.
Draw the diagram and add up the vectors.
If A is at (0,0) then
AD = AB+BC+CD = <4,0> + <0,-3> + <-4cos40°,-4sin40°> = <0.9358,-5.571>
and the bearing of A from D is SθE such that
tanθ = -5.571/0.9358