A sailboat leaves port and sails for 7 miles NE and then 9 miles NW. If the speed remains constant, what bearing would take the sailboat home?

To determine the bearing that would take the sailboat home, we need to understand the concept of bearings and how they relate to the given information.

Bearings are the direction measured in degrees clockwise from the north. In this case, we can imagine the starting point as the origin (port), and the sailboat is moving in a northeast (NE) and then northwest (NW) direction.

To find the bearing that would take the sailboat home, we can follow these steps:

1. Visualize the movement of the sailboat on a coordinate plane: Take the starting point as the origin (0,0) and plot the movements of the sailboat.
- The sailboat sails 7 miles in a northeast direction. This can be represented as a vector (7,7) since NE is 45 degrees from both the north and east directions.
- The sailboat then sails 9 miles in a northwest direction. This can be represented as a vector (-9,9) since NW is 45 degrees from both the north and west directions.
- The final position of the sailboat can be calculated by adding these two vectors together: (7,7) + (-9,9) = (-2,16).

2. Find the bearing of the final position: To find the bearing from the starting point to the final position, we need to calculate the angle it forms with the north direction. We can use the arctan function to determine this angle.

- The arctan function can be applied to the final position vector (-2, 16). The formula is arctan(y/x), where y represents the change in the north direction (16) and x represents the change in the east direction (-2).

- Using this formula, the bearing can be calculated as arctan(16/-2) = -82.87 degrees.

3. Adjust the bearing to a standard compass bearing: The bearing obtained in the previous step (-82.87 degrees) is measured clockwise from the north. Compass bearings, however, are measured clockwise from the reference direction, which is usually the east or west.

- To convert the bearing to a compass bearing, we need to adjust it by adding or subtracting 90 degrees, depending on the reference direction.

- In this case, the reference direction is the east, so we add 90 degrees to the bearing: -82.87° + 90° = 7.13°.

Therefore, the sailboat would need to follow a bearing of approximately 7.13 degrees (in a clockwise direction) to return home.