A tree breaks due to storm and the broken part bends so that the top of the tree

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A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle $30^{\circ}$ with it. The distance between the foot of the tree to the point where the top touches the ground is 12 m. Find the height of the tree.

Options:

A . 12 m

B . 4 m

C . 4

\sqrt{3}

m

D . 12

\sqrt{3}

m

Answer: Option D
:
D

The given situation can be represented by the figure below

$c o s ∠ A C B = \frac{B C}{A C} \Rightarrow c o s 30^{\circ} = \frac{B C}{A C} \Rightarrow \frac{\sqrt{3}}{2} = \frac{12}{A C} \Rightarrow A C = \frac{24}{\sqrt{3}} = 8 \times \frac{3}{\sqrt{3}} = 8 \sqrt{3} m t a n (∠ A C B) = \frac{A B}{B C} \Rightarrow t a n 30^{\circ} = \frac{A B}{12} \Rightarrow A B = \frac{12}{\sqrt{3}} = 4 \sqrt{3} m$
$∴$ The height of tree was originally AB+AC = $4 \sqrt{3} + 8 \sqrt{3} = 12 \sqrt{3} m .$

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