Question

the angle of elevation of an object from a point 200 meters above a lake is 30 degrees and the angle of depression of it's image in the lake is 45 degrees. Find the height of the object above the lake.

Answer 1

Let h and d represent the height of the object above the lake and its horizontal distance from the observer, respectively.

Looking at the reflection of the object in the lake's surface is equivalent to observing the object at distance h below the lake's surface, or observing it from 200 m below the lake's surface. Considering the latter case, we have
  (h+200)/d = tan(45°)
  (h -200)/d = tan(30°)
Solving these for d and equating the results gives
  (h+200)/tan(45°) = (h -200)/tan(30°)
Solving for h, we get
  h(1/tan(30°) -1/tan(45°)) = 200(1/tan(45°) +1/tan(30°))
  h = 200(tan(45°) +tan(30°))/(tan(45°) -tan(30°))
  h ≈ 746.41

The object is about 746.4 meters above the lake.