For a bird flying at a height of 20\(\sqrt {3}\) m, the angle of depression to the top of a tree and the bottom of the tree at a particular moment is 30° and 60° respectively.  What is the height of the tree?

\(\frac{40}{\sqrt {3}}\)

Let E is the bird and AB is the tree.

In ΔEBC  →  Tan 60° = \(\frac{CE}{BC}\)

\(\sqrt {3}\) = \(\frac{20}{\sqrt {3}}\)

BC = 20

In ΔAED  →  Tan 30° = \(\frac{ED}{AD}\)

\(\frac{1}{\sqrt {3}}\) = \(\frac{ED}{20}\)   [AD = BC]

ED = \(\frac{20}{\sqrt {3}}\)

Height of the tree = EC - ED = 20\(\sqrt {3}\) - \(\frac{20}{\sqrt {3}}\) = \(\frac{40}{\sqrt {3}}\)