Answer: Step 1: Break down the problem
We have a boat traveling from island A to island B with a speed of 30 km/h. The bearing from island A to island B is 045°, and the distance between the two islands is 50 km. There is a steady current of 5 km/h at a bearing of 315°.
Step 2: Convert bearings to angles
We need to convert the bearings to angles measured from the north (0°). The bearing from island A to island B is 045°, which is equivalent to an angle of 45°.
The bearing of the current is 315°, which is equivalent to an angle of 225° (since 315° - 360° = -45°, and then adding 180° to get the angle measured from the north).
Step 3: Resolve the current into components
We need to resolve the current into its north-south and east-west components:
Current (north-south component) = 5 km/h × sin(225°) = -2.5 km/h (since sin(225°) = -0.5) Current (east-west component) = 5 km/h × cos(225°) = -4.33 km/h (since cos(225°) = -0.866)
Step 4: Calculate the resultant velocity
We need to find the resultant velocity of the boat, which is the vector sum of the boat's velocity and the current's velocity.
Let's assume the boat's velocity is 30 km/h at an angle of 45°. We can resolve this velocity into its north-south and east-west components:
Boat's velocity (north-south component) = 30 km/h × sin(45°) = 21.21 km/h (since sin(45°) = 0.707) Boat's velocity (east-west component) = 30 km/h × cos(45°) = 21.21 km/h (since cos(45°) = 0.707)
Now, we can add the components of the boat's velocity and the current's velocity:
Resultant velocity (north-south component) = 21.21 km/h - 2.5 km/h = 18.71 km/h Resultant velocity (east-west component) = 21.21 km/h - 4.33 km/h = 16.88 km/h
Step 5: Find the course
We can use the resultant velocity components to find the course the boat must travel to offset the flow of current:
tan(θ) = (east-west component) / (north-south component) = 16.88 km/h / 18.71 km/h ≈ 0.902
θ ≈ arctan(0.902) ≈ 42.3°
So, the boat must travel on a course of approximately 042.3° to offset the flow of current.
Step-by-step explanation: