Answer:
Explanation:
To calculate the distance between boats A and B and the bearing of A from B, we can use trigonometry and vector addition.
a. Distance between A and B:
We can treat the paths of boats A and B as vectors. The position vector of boat A, denoted as vector A, is (15 km, 020°), and the position vector of boat B, denoted as vector B, is (14 km, 290°).
We need to convert the angles from degrees to radians to use trigonometric functions. The formula for converting degrees to radians is:
Radians = Degrees * (π / 180)
So, we convert the angles:
Angle A = 020° * (π / 180) ≈ 0.3491 radians
Angle B = 290° * (π / 180) ≈ 5.0615 radians
Now, we can calculate the components of vector A and vector B:
Vector A = (15 km * cos(0.3491), 15 km * sin(0.3491)) ≈ (14.5089 km, 5.2684 km)
Vector B = (14 km * cos(5.0615), 14 km * sin(5.0615)) ≈ (-10.6063 km, 8.0072 km)
To find the distance between A and B, we can use the distance formula:
Distance = √((Δx)^2 + (Δy)^2)
Distance = √((14.5089 km - (-10.6063 km))^2 + (5.2684 km - 8.0072 km)^2) ≈ 29.17 km
b. Bearing of A from B:
To find the bearing of A from B, we can use the inverse tangent (arctan) function. The bearing is the angle measured clockwise from the north direction.
Bearing of A from B = arctan((Δx / Δy)) in degrees
Bearing of A from B = arctan((-10.6063 km / 8.0072 km)) in degrees ≈ -51.92°
Since bearings are typically measured from the north direction clockwise, we need to adjust the bearing to be positive:
Bearing of A from B = 360° - 51.92° ≈ 308.08°
So, to two decimal places:
a. The distance between A and B is approximately 29.17 km.
b. The bearing of A from B is approximately 308.08°.