To answer the question, let's start by drawing a diagram to better understand the given information.
(a) Drawing the diagram:
To create the diagram, follow these steps:
1. Draw three points representing towns A, B, and C.
2. Label the distance between A and B as 15km.
3. Label the distance between A and C as 90km.
4. Use bearings to add the angles to the diagram. The bearing of B from A is 075, which means we need to draw a line from A to B with an angle of 75 degrees clockwise from the north. Similarly, the bearing of C from A is 310, which means we need to draw a line from A to C with an angle of 310 degrees clockwise from the north.
The diagram should look like this:
```
A ----------------------- B
\
\
\
\
\
\
\
\
\
C
```
(b) Finding the distance between B and C:
To find the distance between B and C, we can use the cosine rule. The cosine rule states that in a triangle ABC, where sides a, b, and c are opposite angles A, B, and C, respectively, the following equation holds:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, we need to find the distance between B and C, which is the side opposite angle A in triangle ABC. Let's label this distance as BC.
Using the cosine rule, we have:
BC^2 = AB^2 + AC^2 - 2 * AB * AC * cos(A)
Substituting the known values:
BC^2 = 15^2 + 90^2 - 2 * 15 * 90 * cos(75)
Calculating this equation will give you the square of the distance between B and C. To find the actual distance, take the square root of the result.
(c) Finding the bearing of C from B:
To find the bearing of C from B, we need to determine the angle that the line from B to C makes with the north direction.
Using basic trigonometry, we can find this angle by calculating the inverse tangent of the ratio of the horizontal and vertical distances between B and C. Let's label this angle as angle CBD.
tan(BDC) = (BC / AC)
Finally, to find the bearing of C from B, subtract angle CBD from the bearing of B from A (075 in this case) and adjust for any negative angles by adding 360 if needed.