QuestionA ship is sailing east. At one point, the bearing of a submerged rock is . After the ship has sailed 13.7 mi , the bearing of the rock has become . Find the distance of the ship from the rock at the latter point.
The distance is approximately mi.
(Do not round until the final answer. Then round to the nearest tenth as needed.)
Studdy Solution
STEP 1
1. The ship is initially sailing east.
2. The initial bearing of the rock from the ship is .
3. After sailing 13.7 miles, the bearing of the rock changes to .
4. We need to find the distance from the ship to the rock at the latter point.
STEP 2
1. Convert bearings to angles in a standard coordinate system.
2. Use the Law of Cosines to find the distance from the ship to the rock.
STEP 3
Convert the bearings to angles in a standard coordinate system:
- Initial bearing of corresponds to an angle of from the north.
- Final bearing of corresponds to an angle of from the north.
To use these in a standard coordinate system (where east is ), we convert:
- Initial angle:
- Final angle:
STEP 4
Use the Law of Cosines to find the distance from the ship to the rock:
The Law of Cosines states:
Where:
- miles (distance sailed),
- is the distance from the ship to the rock at the latter point,
- is the angle between the two bearings.
Calculate the angle :
Substitute into the Law of Cosines:
Since the second term becomes zero due to the distance being zero initially, we simplify:
The distance of the ship from the rock at the latter point is:
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