Multiple choice

$A$ and $B$ are two stations due north and south of a tower of height $25m$. The angles of depression of the stations from the top of the towe $r$ observed to be $30^{0}$ and $45^{0}$ respectively. The distance between the two stations is

  1. $25(\sqrt{3}+1)$
  2. $25(\sqrt{3}-1)m$
  3. $25\sqrt{3}m$
  4. $25(2+\sqrt{3})m$
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A Correct answer
Explanation

The horizontal distance to the station at 30 degrees is 25tan(60 degrees) = 25sqrt(3) m. The distance to the station at 45 degrees is 25tan(45 degrees) = 25 m. Since the stations are on opposite sides of the tower, their separation is 25(sqrt(3) + 1) m.

AI explanation

Because the stations are north and south of the tower, they are in opposite directions. The distance to station A is 25 / tan(30), which equals 25 * sqrt(3) meters, and the distance to station B is 25 / tan(45), which equals 25 meters. The total distance between the stations is the sum of these distances. Adding them yields 25 * sqrt(3) plus 25, which factors to 25(sqrt(3) + 1). The result is 25(sqrt(3) + 1).

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