Two maritime vessels navigate on opposing flanks of a navigational beacon. Elevation angles toward the beacon's peak measured from these vessels register as 30° and 45° respectively. Given a beacon height of 100 m, determine the separation distance between both vessels :_________?
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Answer: C. 273 m
Explanation
Trigonometric tangent functions relate the beacon height to the horizontal distances from both vessels. The distance for the thirty-degree angle equals the height multiplied by the square root of three. Summing the two individual calculated distances from the beacon yields the total separation between vessels. Right-angled triangle geometry provides the foundation for solving elevation and distance problems. The tangent of forty-five degrees equals one, making the corresponding distance equal to the height. Square root approximations, such as 1.732 for the square root of three, facilitate decimal calculations.
Maritime navigation relies on trigonometric positioning to determine distances from coastal lighthouses. Surveyors use elevation angles and known heights to measure inaccessible terrain separations accurately. Aviation and radar systems apply identical triangulation principles for tracking object locations. Option A represents only the distance corresponding to the thirty-degree angle triangle. Option B incorrectly assumes equal distances on both sides of the navigational beacon. Option D overestimates the total sum by using an incorrect trigonometric distance factor.
- Subject
- Mathematics
- Category
- Mathematics
- Topic
- —
- Difficulty
- medium
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