A Coast Guard cutter detects an unidentified ship at a distance of 16.7 km in the direction 13.8° east of north. The ship is traveling at 24.9 km/h on a course at 39.2° east of north. The Coast Guard wishes to send a speedboat to intercept the vessel and investigate it. If the speedboat travels 54.2 km/h, in what direction should it head? Express the direction as a compass bearing with respect to due north.
To find the direction in which the speedboat should head, we need to determine the angle between the course of the ship and the intercept course of the speedboat. Here's how you can do it step by step:
1. Start by drawing a diagram to visualize the problem. Draw a coordinate system with the north direction labeled as "N" and the east direction labeled as "E".
2. From the Coast Guard cutter, draw a vector 16.7 km long at an angle of 13.8 degrees east of north. Label this vector as "Ship".
3. From the tip of the Ship vector, draw another vector representing the ship's velocity. This vector should be 24.9 km/h long and at an angle of 39.2 degrees east of north. Label this vector as "Ship Velocity".
4. Now, draw a vector from the Coast Guard cutter towards the ship's location after a certain time, representing the potential position of the ship in the future. The length of this vector is determined by the ship's velocity and the time it takes for the speedboat to intercept it. Since the time is not given, we can assume it takes the speedboat t hours to reach the ship.
5. Draw a vector representing the speedboat's velocity, which is 54.2 km/h. Label this vector as "Speedboat Velocity".
6. Now, draw a vector from the Coast Guard cutter towards the potential ship position, equal in length to the speedboat velocity multiplied by the time t.
7. The angle between the vectors "Ship Velocity" and "Speedboat Velocity" represents the angle that the speedboat needs to head in order to intercept the ship. Use vector addition to subtract the vector "Ship Velocity" from the vector "Speedboat Velocity" to find their resultant vector.
8. Measure the angle of the resultant vector relative to the north direction using a protractor. This angle is the direction in which the speedboat should head.
9. Convert the angle to a compass bearing by adding the angle to 360 degrees if it is negative or subtracting it from 360 degrees if it is positive. For example, if the angle is -20 degrees, the compass bearing would be 360 - 20 = 340 degrees.
By following these steps, you should be able to determine the direction in which the speedboat should head to intercept the ship.