Villages 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 S 50 W from C.calculate the distance and bearing of A from D.
Put A at origin
Dx = 4 - 4 sin 50 = .9358
Dy = -3 -4 cos 50 = -5.571
from origin to D (w will reverse later)
tan angle east of south = .9358/5.571
angle east 0f south = 9.54 degrees east of south
Reversed for bearing from D to A that is 9.54 degrees west of north
or 360 - 9.54 = 350.5 degrees true on chart
distance = sqrt(.9358^2 + 5.571^2)
= 5.65 km
The DIAGRAM
To calculate the distance and bearing of Village A from Village D, we can break down the problem into steps.
Step 1: Calculate the position of Village C
- Start by drawing a diagram with Villages A, B, C, and D.
- From the information given, we know that Village B is 4km due east of A. So, mark a point 4km to the east of A and label it as B.
- Next, we know that Village C is 3km due south of B. So, mark a point 3km directly below B and label it as C.
Step 2: Calculate the position of Village D
- We are told that Village D is 4km S 50 W from C. This means that Village D is located 4km in the direction of 50 degrees west from C.
- To find the position of D, start at C and move 4km in the direction of 50 degrees west. Mark this point as D.
Step 3: Calculate the distance between A and D
- The distance between two points can be calculated using the Pythagorean theorem. In this case, we can find the distance between A and D by calculating the hypotenuse of the right-angled triangle formed by A, C, and D.
- Using the distances AC and CD, we can apply the Pythagorean theorem to find AD.
- AC = 3km (from C to B) + 4km (from B to A) = 7km
- CD = 4km (given)
- AD = √(AC^2 + CD^2) = √(7^2 + 4^2) = √49 + 16 = √65 ≈ 8.06km
Step 4: Calculate the bearing of A from D
- To find the bearing of A from D, we need to determine the angle formed between north and the line joining A and D.
- We can use trigonometry to calculate this angle.
- Using the distances AC and CD, we can find the angle C in the triangle ACD.
- Since we know the lengths of the sides AC (7km) and CD (4km), we can use the tangent function.
- tan(C) = AC/CD = 7/4
- C = tan^(-1)(7/4) ≈ 60.9 degrees
- The bearing of A from D is equal to the bearing of north from D minus the angle C.
- The bearing of north from D is 180 degrees (opposite direction) so the bearing of A from D is 180 - 60.9 = 119.1 degrees.
Therefore, the distance of Village A from Village D is approximately 8.06km, and the bearing of A from D is approximately 119.1 degrees.