A person standing on the bank of ariver observes that the angle of ele

1.	A person standing on the bank of ariver observes that the angle of elevationof the top of a tree standing on theopposite bank is 60° .When he moves 40maway from the bank he finds the angleof elevation to be 30°. Find theheight of the tree and the width ofthe river.
Answer» $\blue{\bold{\underline{\underline{Answer:}}}}$ $\green{\tt{\therefore{Height\:of\:tree=20\sqrt{3}\:m}}}$ $\green{\tt{\therefore{Width\:of\:river=20\:m}}}$ $\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$ $\green{\underline \bold{<klux>GIVEN</klux>: }} \\ \tt: \implies length \: DC = 40 \: m \\ \\ \tt: \implies First \: angle \: of \: elevation = 60 \degree \\ \\ \tt: \implies Second\: angle \: of \: elevation = 30 \degree \\ \\ \red{\underline \bold{To \: Find: }} \\ \tt: \implies Height \: of \: tree = ? \\ \\ \tt: \implies Width \: of \: river =?$ • ACCORDING to given QUESTION : $\bold{In \:right \: angled \: \triangle \: ABD} \\ \tt: \implies tan \: \theta = \frac{p}{b} \\ \\ \tt: \implies tan \:30 \degree = \frac{AB}{BD} \\ \\ \tt: \implies \frac{1}{ \sqrt{3} } = \frac{AB}{40 + x} \\ \\ \tt: \implies 40 + x = AB \sqrt{3} \\ \\ \tt: \implies x = AB \sqrt{3} - 40 - - - - - (1) \\ \\ \bold{In \: right \: angled \: \triangle \: ABC} \\ \tt: \implies tan \: \theta = \frac{p}{b} \\ \\ \tt: \implies tan \: 60 \degree = \frac{AB}{BC} \\ \\ \tt: \implies \sqrt{3} = \frac{AB}{x} \\ \\ \tt: \implies x = \frac{AB}{ \sqrt{3} } - - - - - (2)$ $\text{From \: (1) \: and \: (2)} \\ \tt: \implies \frac{AB}{ \sqrt{3} } = AB \sqrt{3} - 40 \\ \\ \tt: \implies AB= 3AB - 40 \sqrt{3} \\ \\ \tt: \implies 40 \sqrt{3} = 2AB \\ \\ \tt: \implies AB = \frac{40 \sqrt{3} }{2} \\ \\ \green{\tt: \implies AB = 20 \sqrt{3} \: m } \\ \\ \text{Putting \: value \: of \: ab \: in \: (2)} \\ \tt: \implies x = \frac{20 \sqrt{3} }{ \sqrt{3} } \\ \\ \green{\tt: \implies x = 20 \: m} \\ \\ \green{\tt\therefore Height \: of \: tree \: is \: 20 \sqrt{3 } \: m} \\ \\ \green{\tt\therefore Width \: of \: river \: is \: 20 \: m}$

A person standing on the bank of ariver observes that the angle of elevationof the top of a tree standing on theopposite bank is 60° .When he moves 40maway from the bank he finds the angleof elevation to be 30°. Find theheight of the tree and the width ofthe river.

Answer»

$\blue{\bold{\underline{\underline{Answer:}}}}$

$\green{\tt{\therefore{Height\:of\:tree=20\sqrt{3}\:m}}}$

$\green{\tt{\therefore{Width\:of\:river=20\:m}}}$

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\green{\underline \bold{<klux>GIVEN</klux>: }} \\ \tt: \implies length \: DC = 40 \: m \\ \\ \tt: \implies First \: angle \: of \: elevation = 60 \degree \\ \\ \tt: \implies Second\: angle \: of \: elevation = 30 \degree \\ \\ \red{\underline \bold{To \: Find: }} \\ \tt: \implies Height \: of \: tree = ? \\ \\ \tt: \implies Width \: of \: river =?$

• ACCORDING to given QUESTION :

$\bold{In \:right \: angled \: \triangle \: ABD} \\ \tt: \implies tan \: \theta = \frac{p}{b} \\ \\ \tt: \implies tan \:30 \degree = \frac{AB}{BD} \\ \\ \tt: \implies \frac{1}{ \sqrt{3} } = \frac{AB}{40 + x} \\ \\ \tt: \implies 40 + x = AB \sqrt{3} \\ \\ \tt: \implies x = AB \sqrt{3} - 40 - - - - - (1) \\ \\ \bold{In \: right \: angled \: \triangle \: ABC} \\ \tt: \implies tan \: \theta = \frac{p}{b} \\ \\ \tt: \implies tan \: 60 \degree = \frac{AB}{BC} \\ \\ \tt: \implies \sqrt{3} = \frac{AB}{x} \\ \\ \tt: \implies x = \frac{AB}{ \sqrt{3} } - - - - - (2)$

$\text{From \: (1) \: and \: (2)} \\ \tt: \implies \frac{AB}{ \sqrt{3} } = AB \sqrt{3} - 40 \\ \\ \tt: \implies AB= 3AB - 40 \sqrt{3} \\ \\ \tt: \implies 40 \sqrt{3} = 2AB \\ \\ \tt: \implies AB = \frac{40 \sqrt{3} }{2} \\ \\ \green{\tt: \implies AB = 20 \sqrt{3} \: m } \\ \\ \text{Putting \: value \: of \: ab \: in \: (2)} \\ \tt: \implies x = \frac{20 \sqrt{3} }{ \sqrt{3} } \\ \\ \green{\tt: \implies x = 20 \: m} \\ \\ \green{\tt\therefore Height \: of \: tree \: is \: 20 \sqrt{3 } \: m} \\ \\ \green{\tt\therefore Width \: of \: river \: is \: 20 \: m}$