Question

There is a small island in 200 m wide river and a tall tree stands on the island . P and Q are points directly opposite to each other on the two banks and in the line with the tree . if the angles of elevation of the top of the tree from P and Q are 30 degree and 45 degree respectively , find the height of the tree.​

Answer 1

Answer:

P and Q are points directly opposite to each other an two banks and in the line with the tree . If the angle of elevation of the top of the tree.

Answer 2

Answer:

Let OA be the Height of the tree 'h' cm.

$\qquad\qquad\tiny\dag \: \:\underline{\sf {In \: \Delta \: POA : }}$

$:\implies \sf \tan(30) = \dfrac{Opposite \: side}{ Adjacent \: side}$

$:\implies \sf \tan(30) = \dfrac{OA}{OP}$

$:\implies \sf \dfrac{1}{ \sqrt{3} } = \dfrac{h}{OP}$

$:\implies \sf OP = \sqrt{3} \: h$

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$\qquad\qquad\tiny\dag \: \:\underline{\sf {In \: \Delta \: QOA : }}$

$:\implies \sf \tan(45) = \dfrac{Opposite \: side}{ Adjacent \: side}$

$:\implies \sf 1= \dfrac{h}{OQ}$

$:\implies \sf OQ = h$

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$\dashrightarrow\:\: \sf PQ = 200 \: m$

$\dashrightarrow\:\: \sf OP + OQ = 200 \: m$

$\dashrightarrow\:\: \sf \sqrt{3} \: h + h = 200 \: m$

$\dashrightarrow\:\: \sf h \left(\sqrt{3} + 1 \right)= 200 \: m$

$\dashrightarrow\:\: \sf h= \dfrac{200}{ \sqrt{3} + 1}$

$\qquad \: \: \: \: \: \: \tiny\dag \: \:\underline{\frak{By \: rationalising \: denominator \: we \: get : }}$

$\dashrightarrow\:\: \sf h= \dfrac{200}{ \sqrt{3} + 1} \times \dfrac{ \sqrt{3} - 1 }{ \sqrt{3} - 1 }$

$\dashrightarrow\:\: \sf h= \dfrac{200 \left( \sqrt{3} - 1 \right)}{ 3 - 1}$

$\dashrightarrow\:\: \sf h= \dfrac{200 \left( \sqrt{3} - 1 \right)}{ 2}$

$\dashrightarrow\:\: \sf h= 100 \left( \sqrt{3} - 1 \right)$

$\qquad \: \: \: \: \: \: \tiny\dag \: \:\underline{\frak{By \: taking \: \sqrt{3} \: as \: 1.732 \: we \: get : }}$

$\dashrightarrow\:\: \sf h= 100 \left(1.732 - 1 \right)$

$\dashrightarrow\:\: \sf h= 100 \left(0.732 \right)$

$\dashrightarrow\:\: \underline{ \boxed{ \sf h= 73.2 \: m}}$

$\therefore\underline{\textsf{The height of the tree is \textbf{73.2 m}}}.$

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$\begin{array}{| c | c | c | c | c | c |}\cline{1-6} \bf Angles & \bf 0^{o} & \bf 30^{o} & \bf 45^{o} & \bf 60^{o} & \bf 90^{o} \\\cline{1-6} \tt Sin \theta & 0 & \dfrac{1}{2} & \dfrac{1}{\sqrt{2}}& \dfrac{\sqrt{3}}{2}& 1\\\cline{1-6} \tt cos \theta & 1 & \dfrac{\sqrt{3}}{2} &\dfrac{1}{\sqrt{2}}&\dfrac{1}{2}&0\\\cline{1-6} \tt tan \theta & 0 & \dfrac{1}{\sqrt{3}} & 1& \sqrt{3} & \infty \\\cline{1-6} \tt cosec \theta & \infty & 2 & \sqrt{2} & \dfrac{2}{\sqrt{3}} &1\\\cline{1-6} \tt sec \theta & 1 & \dfrac{2}{\sqrt{3}} & \sqrt{2} & 2 & \infty \\\cline{1-6} \tt cot \theta & \infty & \sqrt{3} &1& \dfrac{1}{\sqrt{3}} &0\\\cline{1-6}\end{array}$