THREE TOWNS A,,B,C ARE SITUATED SO THAT |AB|=100KM . THE BEARING OF B FROM A IS 060 AND THE BEARING OF C FROM A IS 290 DEGREES . CALCULATE THE DISTANCE OF C FROM A .

Question

THREE TOWNS A,,B,C ARE SITUATED SO THAT |AB|=100KM . THE BEARING OF B FROM A IS 060 AND THE BEARING OF C FROM A IS 290 DEGREES . CALCULATE THE DISTANCE OF C FROM A .

Answer 1

As always, draw a diagram.

use the law of sines

AC/sin50° = 100/sin70°

Answer 2

To calculate the distance of town C from town A, we can use the Law of Cosines.

Step 1: Draw a diagram. A is the starting point, B is 100 km away from A at a bearing of 060 degrees, and C is also some distance away from A at a bearing of 290 degrees.

Step 2: Identify the triangle formed by A, B, and C.

Step 3: In triangle ABC, use the Law of Cosines:

c^2 = a^2 + b^2 - 2ab * cos(C)

Here, c represents the distance between A and C, a represents the distance between A and B (which is 100 km), b represents the distance between B and C, and C represents the angle between sides a and b.

Step 4: Substitute the known values into the equation:

c^2 = 100^2 + b^2 - 2(100)(b) * cos(290 degrees)

Step 5: Simplify the equation:

c^2 = 10000 + b^2 - 200b * cos(290 degrees)

Step 6: Calculate the cosine value of 290 degrees:

cos(290 degrees) = cos(360 degrees - 70 degrees) = cos(70 degrees)

Step 7: Substitute the cosine value into the equation:

c^2 = 10000 + b^2 - 200b * cos(70 degrees)

Now, we need to know the value of b to continue with the calculation.

Answer 3

To calculate the distance of town C from town A, we can use the Law of Cosines. This law relates the lengths of the sides of a triangle to the cosine of one of its angles.

First, let's draw a diagram to visualize the situation described in the question:

```
B
/|\
/ | \
/ | \
/60°|290°\
A---100km---C
```

Here, point A represents town A, B represents town B, and C represents town C. AB has a length of 100 km. The bearing of town B from A is 060 degrees, and the bearing of town C from A is 290 degrees.

In this diagram, we have a triangle ABC, and we want to find the length of side AC.

Now, let's apply the Law of Cosines:

c^2 = a^2 + b^2 - 2ab * cos(C)

In this equation:
- c represents the side we want to find (AC).
- a and b represent the known sides (AB = 100 km and BC).
- C represents the angle opposite to side c (angle ABC).

Let's calculate each part of the equation:

a = AB = 100 km

To calculate side BC, we need to find the remaining angle of triangle ABC (angle BAC) since we have the bearing of town B from A (060 degrees) and the bearing of town C from A (290 degrees).

The bearing of town B from A is given as 060 degrees. This indicates that angle BAC is 180 degrees minus the bearing, which gives us 180 - 60 = 120 degrees.

Now, we can find angle ABC by subtracting 120 degrees from 290 degrees:

ABC = 290 - 120 = 170 degrees

Now, we calculate side BC using the Law of Cosines:

b = BC
C = ABC = 170 degrees
a = AB = 100 km

Plugging these values into the Law of Cosines equation:

AC^2 = 100^2 + BC^2 - 2 * 100 * BC * cos(170°)

Now, we need to solve this equation to find AC.