The upper part of a tree broken by the wind makes an angle of 30° with the ground and the distance from the root to the point where the top of the tree touches the ground is 15 m. Using sine rule, find the height of the tree.
Solution
Suppose BD be the tree and the upper part of the tree is broken over by the wind at point A.
\[\text{ The total height of the tree is x + y } . \]
\[\text{ In } \bigtriangleup ABC, \]
\[\angle C = 30° \text{ and } \angle B = 90° . \]
\[ \therefore \angle A = 60° . \]
\[\text{ So, on using sine rule, we get }:\]
\[\frac{AB}{\sin30° } = \frac{BC}{\sin60° } = \frac{AC}{\sin90°}\]
\[ \Rightarrow \frac{x}{\sin30°} = \frac{15}{\sin60°} = \frac{y}{\sin90°} \]
\[So, \frac{x}{\sin 30°} = \frac{15}{\sin60°}\]
\[ \Rightarrow \frac{x}{\frac{1}{2}} = \frac{15}{\frac{\sqrt{3}}{2}}\]
\[ \Rightarrow x = \frac{15}{\sqrt{3}} = 5\sqrt{3}\]
\[\text{ Also }, \]
\[\frac{15}{\sin60° } = \frac{y}{\sin90°}\]
\[ \Rightarrow \frac{15}{\frac{\sqrt{3}}{2}} = y\]
\[ \Rightarrow y = \frac{30}{\sqrt{3}} = 10\sqrt{3}\]
\[\text{ So, the height of the tree is } x + y = 5\sqrt{3} + 10\sqrt{3} m\]
\[ = 15\sqrt{3}m\]
