A bird is flying at a height of \(20\sqrt{3}\) m. At a particular moment, the angles of depression of the top and the bottom of a tree from the bird are \(30^\circ\) and \(60^\circ\) respectively. What is the height of the tree?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{40}{\sqrt{3}}\) m
An angle of depression is measured down from the horizontal line through the eye. The 60° line to the foot of the tree gives the horizontal distance; the 30° line to the top then gives how far the bird is above the top of the tree.
Horizontal distance: \(\tan 60^\circ=\displaystyle\frac{20\sqrt{3}}{d}\), so \(d=\displaystyle\frac{20\sqrt{3}}{\sqrt{3}}=20\) m
Height of the bird above the top of the tree: \(20\tan 30^\circ=\displaystyle\frac{20}{\sqrt{3}}\) m
Height of the tree \(\displaystyle=20\sqrt{3}-\frac{20}{\sqrt{3}}=\frac{60-20}{\sqrt{3}}=\frac{40}{\sqrt{3}}\) m
