Multiple choice

On the same side of a tower two objects are located. Observed from the top of the tower their angles of depression are $\displaystyle 45^{0}$ and $\displaystyle 60^{0}$. If the height of the tower is $150\ m$ the distance between the objects is

  1. $63.5$ $m$
  2. $76.9$ $m$
  3. $86.7$ $m$
  4. $90$ $m$
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A Correct answer
Explanation

Let h = 150. Distance 1 = h/tan(45) = 150. Distance 2 = h/tan(60) = 150/sqrt(3) = 150/1.732 = 86.6. The distance between objects is 150 - 86.6 = 63.4 m.

AI explanation

Let h be the height of the tower and d1 and d2 be the horizontal distances to the two objects from the base. Given the 45 degree angle of depression, the triangle is isosceles, making d1 equal to h, which is 150 m. For the second object with a 60 degree angle of depression, tan 60 degrees equals h over d2, so d2 equals 150 divided by 1.732, which is approximately 86.6 m. The distance between the objects is d1 minus d2, which equals 150 minus 86.6, giving a result of 63.5 m.

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