Question

A tree is broken by the wind . The top struck the ground at an angle of 30° and at a distance of 10m from the root. Find the height of the tree.

Answer 1

$\bf \implies \green{{ \underline{given \div }}} \\ \\ \sf \to \: the \: top \: struck \: the \: ground \: at \: an \: angle \: = 30 \degree \\ \\ \sf \to \: distance \: 10 m \: from \: the \: root$

$\bf \to \orange{{ \underline{to \: find \div }}} \\ \\ \sf \to \: find \: the \: height \: of \: the \: tree$

$\bf \to{ \underline{solution \div }} \\ \\ \sf \to \: let \: ac \: = x \\ \\ \sf \to \: cb \: = y \\ \\ \sf \to \: ao \: = 10m \\ \\ \sf \to \: oc \: = y \\ \\ \sf \to \: in \: \triangle \: oac \: \\ \\ \sf \to \: \tan( \theta) = \frac{p}{b} \\ \\ \sf \to \: \tan( 30 \degree) = \frac{ac}{oa} \\ \\ \sf \to \: \frac{1}{ \sqrt{3} } = \frac{x}{10} \\ \\ \sf \to \: x \: = \frac{10}{ \sqrt{3} } \\ \\ \sf \to \: again \: in \triangle \: oac$

$\sf \to \: \cos( \theta) = \frac{b}{h} \\ \\ \sf \to \: \cos(30 \degree) = \frac{oa}{oc} \\ \\ \sf \to \: \frac{ \sqrt{3} }{2} = \frac{10}{y} \\ \\ \sf \to \: \sqrt{3} y = 20 \\ \\ \sf \to \: y \: = \frac{20}{ \sqrt{3} } \\ \\ \sf \to \: height \: of \: tree \: = ab = ac \: + \: cb \\ \\ \sf \to \: x \: + y \: = \frac{10}{ \sqrt{3} } + \frac{20}{ \sqrt{3} } \\ \\ \sf \to \: \frac{30}{ \sqrt{3} } \\ \\ \sf \to \: rationalise \:the \: term$

$\sf \to \: \frac{30}{ \sqrt{3} } \times \frac{ \sqrt{3} }{ \sqrt{3} } \\ \\ \sf \to \: \frac{30 \sqrt{3} }{3} \\ \\ \sf \to \: \frac{ \cancel{30} \sqrt{3} }{ \cancel{3}} = 10 \sqrt{3} \\ \\ \sf \to \: height \: of \: tree \: = 10 \sqrt{3} m$