Question

A vertical tree is broken by the wind. The top of the tree touches the ground and
makes an angle ${30}^{\circ }$ with it. If the top of the tree touches the ground $30$m away from its foot, then find the actual height of the tree.

Answer 1

Let $C$ be the point at which the tree is broken and let the top of the tree touch the ground at $A$.

Let $B$ denote the foot of the tree.

Given $AB=30$m and $\angle CAB={30}^{\circ }$

In the right angled $△CAB$,

$\tan {30}^{\circ }=\frac{BC}{AB}$

$\Rightarrow BC=AB\tan {30}^{\circ }$

$\therefore BC=\frac{30}{\sqrt{3}}$

$=10\sqrt{3}m$ (1)

Now, $\cos {30}^{\circ }=\frac{AB}{AC}$

$\Rightarrow AC=\frac{AB}{\cos {30}^{\circ }}$

So,

$AC=\frac{30\times 2}{\sqrt{3}}=10\sqrt{3}\times 2=20\sqrt{3}$m. (2)

Thus, the height of the tree $=BC+AC=10\sqrt{3}+20\sqrt{3}$

$=30\sqrt{3}$m.