Question

A tree breaks due to a storm and the broken part bends down such that the top of the tree touches the ground making an angle of $30$${}^o$ with the ground at a distance of $30$ m. Then, the height of the tree is:

• A. $35$$\sqrt{3}$
• B. $60$$\sqrt{3}$
• C. $45$$\sqrt{3}$
• D. $30$$\sqrt{3}$

Answer 1

The correct option is D $30$$\sqrt{3}$

Let AB be the tree broken at point C such that

the broken part CB takes the position CO and touches the at O.

Then, OA = 30 m & $\angle$COA = 30${}^o$
In $△$OAC, tan 30${}^o$ = $\frac{AC}{OA}\Rightarrow \frac{1}{\sqrt{3}}=\frac{AC}{30}\Rightarrow AC=\frac{30}{\sqrt{3}}=10\sqrt{3}$
and,
cos 30${}^o$ = $\frac{OA}{OC}\Rightarrow \frac{\sqrt{3}}{2}=\frac{30}{OC}\Rightarrow OC=\frac{60}{\sqrt{3}}\Rightarrow OC=20\sqrt{3}$

Thus, the height of the tree $=AC+CB$
$=AC+OC$ .... ($\because$ BC = OC)
$=10$$\sqrt{3}$ + 20$\sqrt{3}$
$=30$$\sqrt{3}$ m