A ship left harbour A and travelled for 20km on a bearing of 125°. It then changed course of 035° for a distance of 10km until it reached B. Calculate (i) the distance between A and B (ii) the bearing of B from A (iii) the bearing of A from B
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To solve this problem, we'll use basic trigonometry and geometry principles. Let's go step-by-step:
(i) Distance between A and B:
To find the total distance between A and B, we can use the Pythagorean theorem, since we have a right triangle formed by the two given distances (20km and 10km).
Using the Pythagorean theorem:
Total distance = √(20^2 + 10^2)
= √(400 + 100)
= √500
= 10√5 km (approximately)
Therefore, the distance between A and B is approximately 10√5 km.
(ii) Bearing of B from A:
To find the bearing of B from A, we need to consider the angle formed between the line connecting A and B and the line going North (0° bearing).
Since the ship travels on a bearing of 125° from A and then changes course to 035°, it means the angle between the line going North and the line connecting A and B is 125° - 35° = 90°.
Therefore, the bearing of B from A is 90°.
(iii) Bearing of A from B:
To find the bearing of A from B, we subtract the bearing of B from A from 180°.
180° - 90° = 90°.
Therefore, the bearing of A from B is 90°.
To solve this problem, we can use the concept of vectors and trigonometry. Let's break down the given information into two different vectors representing the ship's movements:
1. First vector: The ship travels 20km on a bearing of 125° from A.
2. Second vector: The ship changes course to a bearing of 035° and travels for 10km until it reaches B.
To find the answers to the given questions, we will calculate:
(i) The Distance between A and B:
To find the distance between A and B, we need to calculate the magnitude of the resultant vector formed by adding the two vectors mentioned above.
We can consider the starting point of both vectors as the origin and represent both vectors as Cartesian coordinates. Then, we can use the Pythagorean theorem to find the magnitude of the resultant vector.
Let's break down the first vector into its x and y components:
x1 = 20 * cos(125°)
y1 = 20 * sin(125°)
Next, let's break down the second vector into its x and y components:
x2 = 10 * cos(35°)
y2 = 10 * sin(35°)
Now, we can calculate the total values for x and y by summing the respective components:
x = x1 + x2
y = y1 + y2
To find the magnitude (d) of the resultant vector, we can use the Pythagorean theorem:
d = sqrt(x^2 + y^2)
(ii) The Bearing of B from A:
To find the bearing from A to B, we can use trigonometry to calculate the angle between the positive x-axis and the resultant vector.
The bearing (θ) can be found using the inverse tangent function:
θ = atan2(y, x)
(iii) The Bearing of A from B:
To find the bearing from B to A, we need to subtract 180° from the bearing of B from A obtained in (ii).
Bearing of A from B = θ + 180°
Using these steps, you can calculate the distance between A and B, the bearing of B from A, and the bearing of A from B.