Question

a tree is broken by wind the top touches ground at angle of 30 degree and the distance of 12 m from root find the hight of the tree​

Answer 1

In Right Angled Triangle ABC,

• ∠B = 90°
• ∠C = 30°
• BC = 12 m

We need to find height of the tree i.e. AB + AC

$\sf{In \: \triangle \: ABC}\\\\\\\sf{tan \: 30^{\circ} = \dfrac{AB}{BC}}\\\\\\:\implies \sf{\dfrac{1}{\sqrt{3}} = \dfrac{AB}{12}}\\\\\\\\:\implies \sf{AB\sqrt{3} = 12}\\\\\\:\implies \sf{AB = \dfrac{12}{\sqrt{3} }}\\\\\\\rm{Rationalizing\:the\:denominator,}\\\\\sf{AB=\dfrac{12}{\sqrt{3}}\times \dfrac{\sqrt{3} }{\sqrt{3} }}\\\\\\\\:\implies \sf{AB = \dfrac{12\sqrt{3} }{3}}\\\\\\\\:\implies \boxed{\bf{AB = 4\sqrt{3} \:m}}$

$\tt{Now\:\:let's\:\:find\:\:AC}$

$\sf{cos\:30^{\circ} = \dfrac{BC}{AC}}\\\\\\\\:\implies \sf{\dfrac{\sqrt{3} }{2}=\dfrac{12}{AC}}\\\\\\\\\implies \sf{AC\sqrt{3} = 12 \times 2}\\\\\\\implies \sf{AC \sqrt{3} =24}\\\\\\:\implies \sf{AC = \dfrac{24}{\sqrt{3} }}\\\\\\\rm{Rationalizing \:the\:denominator,}\\\\\\:\implies \sf{AC=\dfrac{24}{\sqrt{3} } \times \dfrac{\sqrt{3} }{\sqrt{3} }}\\\\\\\\:\implies \sf{AC=\dfrac{24\sqrt{3} }{3}}\\\\\\:\implies \boxed{\bf{AC=8\sqrt{3}\: m}}$

$\sf{Length\:of\:tree=AB+AC}\\\\\\\dashrightarrow \: \sf{Length\:of\:tree=4\sqrt{3} + 8\sqrt{3} }\\\\\\\longrightarrow \: \tt{\bold{\underline{\large{Length\:of\:tree= 8\sqrt{3} \:m}}}}$