Two ships A and B leave port Y at the same time. A sailed on a bearing of 26 degrees for 30km while B sailed due west of Y for 10km, calculate
(a) The distance of line AB
(b) The bearing of A from B
(c) The bearing of B from A
Assuming that by "bearing" you are using the notation that North = 0° and clockwise is positive, then you have a triangle
ABO, where O is the origin and angle BOA = 116°
use the cosine law:
AB^2 = 10^2 + 30^2 - 2(10)(30)cos116
find AB
then use the sine-law to find the other angles.
To solve this problem, we can use basic trigonometry and vector analysis. Let's go step by step to find the solutions:
(a) The distance of line AB:
We can find the distance of line AB by using the Pythagorean theorem. The base of the triangle (side AB) is the horizontal distance sailed by ship B, which is 10 km. The other two sides are formed by the vertical and horizontal components of ship A's journey.
The vertical component can be calculated using the trigonometric function sine. We have the angle of 26 degrees and the hypotenuse of 30 km. So, the vertical component is given by:
Vertical component = 30 km * sin(26 degrees)
The horizontal component can be calculated using the trigonometric function cosine. Similarly, we have the angle of 26 degrees, and the hypotenuse of 30 km. So, the horizontal component is given by:
Horizontal component = 30 km * cos(26 degrees)
Now, we can find the total distance AB using the Pythagorean theorem:
Distance AB = sqrt((Horizontal component)^2 + (Vertical component)^2)
Plug in the values for the horizontal and vertical components to get the final distance AB.
(b) The bearing of A from B:
To find the bearing of A from B, we need to determine the angle between the line AB and the west direction. We can use the inverse tangent (arctan) function to find this angle using the vertical and horizontal components.
First, calculate the sine of the angle:
sin(angle) = Vertical component / Distance AB
Then, calculate the angle itself by taking the inverse tangent:
angle = arctan(Vertical component / Horizontal component)
The bearing of A from B will be the angle in degrees measured from the west direction. Subtract this angle from 180 degrees to get the bearing.
(c) The bearing of B from A:
To find the bearing of B from A, we can use the same logic as above. However, this time we will use the horizontal and vertical components of B relative to A.
First, calculate the sine of the angle:
sin(angle) = Vertical component of B / Distance AB
Then, calculate the angle itself by taking the inverse tangent:
angle = arctan(Vertical component of B / Horizontal component of B)
The bearing of B from A will be the angle in degrees measured from the west direction. Add this angle to 180 degrees to get the bearing.
By following these steps, you should be able to calculate the answers to all three parts of the question.