A boat moves from a point A on the bearing 150¡ã to another point B 10km away.it then changes direction on being south 45¡ã east for a distance of 12km to another point C.

1.. How far is the boat from the starting.
2.. What is the bearing of C from A.

I see a skinny triangle with sides 10 and 12 with an angle of 165° between them

If d is the distance between A and B
d^2 = 10^2+12^2 - 2(10)(12)cos165
...
d = ...

Once you have d, use the sine law to find the angle of the triangle at A
sinA/12 = sin165/d
...

To solve this problem, we can use the concepts of trigonometry and vector addition.

1. To find how far the boat is from the starting point, we need to find the horizontal and vertical components of the boat's motion and then use the Pythagorean theorem.

- The first leg of the boat's journey is on a bearing of 150° for a distance of 10km. To find the horizontal component (x), we can use the cosine function: cos(150°) = x / 10km. Rearranging the equation, we find that x = 10km * cos(150°).
- To find the vertical component (y), we can use the sine function: sin(150°) = y / 10km. Rearranging the equation, we find that y = 10km * sin(150°).
- The second leg of the boat's journey is on a bearing of south 45° east (135°) for a distance of 12km. This leg only contributes to the vertical component (y), so we do not need to consider the horizontal component (x).
- To find the total vertical distance, we add the vertical components from both legs: y_total = y (from leg 1) + y (from leg 2).
- To find the total distance from the starting point, we use the Pythagorean theorem: distance = √(x^2 + y_total^2).

2. To find the bearing of point C from point A, we need to find the angle that the line AC makes with the north direction.

- We already know the horizontal and vertical distances from point A to point C. Let's call them X and Y, respectively.
- The bearing is the angle that the line AC makes with the north direction, so we can use the arctan function to find this angle: bearing = arctan(Y / X).
- Note that the arctan function returns an angle between -90° and 90°, so we need to adjust the bearing based on the location of point C relative to point A. If point C is to the east of point A, the bearing will be positive. If point C is to the west of point A, the bearing will be negative.

By plugging in the values and performing the calculations, we can find the answers to both questions.