Question

The two palm trees are of equal heights and are standing opposite each other on either side

of the river, which is 100 m wide. From a point O between them on the river the angles of

elevation of the top of the trees are 60° and 30°, respectively. Find the height of the trees

and the distances of the point O from the trees​

Answer 1

$\large\underline{\sf{Solution-}}$

Let the river be AB, so AB = 100 m.

So, one palm tree is at A and other at B.

Let AC and BD be two palm trees.

Let O be the point in between the river, such that

• OA = x m

and

• OB = 100 - x meter.

Let

• Height of the palm tree be 'h' meter.

• AC = BD = 'h' meter.

And

• Angle of elevation from point O fn the top of palm tree AC be 60° and of the top of palm tree BD be 30°

So,

$\rm :\longmapsto\:In \: \triangle \: AOC$

$\rm :\longmapsto\:tan60 \degree \: = \: \dfrac{AC}{AO}$

$\rm :\longmapsto\: \sqrt{3} = \dfrac{h}{x}$

$\bf\implies \:h = \sqrt{3} x - - - (1)$

Now,

$\rm :\longmapsto\:In \: \triangle \: BOD$

$\rm :\longmapsto\:tan30 \degree \: = \: \dfrac{BD}{BO}$

$\rm :\longmapsto\:\dfrac{1}{ \sqrt{3} } = \dfrac{h}{100 - x}$

$\rm :\longmapsto\:\dfrac{1}{ \sqrt{3} } = \dfrac{ \sqrt{3}x }{100 - x} \: \: \: \{using \: (1) \}$

$\rm :\longmapsto\:3x = 100 - x$

$\rm :\longmapsto\:3x + x= 100$

$\rm :\longmapsto\:4x= 100$

$\bf\implies \:x \: = \: 25 \: m$

On substituting the value of 'x' in equation (1), we get

$\bf\implies \:h = 25 \sqrt{3} \: m$

$\rm :\longmapsto\:Height \:of \: trees \: = \: AC = BD = 25 \sqrt{3} \: m$

$\rm :\longmapsto\:AO \: = \: 25 \: m$

$\rm :\longmapsto\:BO \: = \: AB - AO = 100 - 25 = 75 \: m$

Additional Information :-

$\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\sf Trigonometry\: Table \\ \begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\boxed{\boxed{\begin{array}{ |c |c|c|c|c|c|} \bf\angle A & \bf{0}^{ \circ} & \bf{30}^{ \circ} & \bf{45}^{ \circ} & \bf{60}^{ \circ} & \bf{90}^{ \circ} \\ \\ \rm sin A & 0 & \dfrac{1}{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{ \sqrt{3}}{2} &1 \\ \\ \rm cos \: A & 1 & \dfrac{ \sqrt{3} }{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{1}{2} &0 \\ \\ \rm tan A & 0 & \dfrac{1}{ \sqrt{3} }&1 & \sqrt{3} & \rm \infty \\ \\ \rm cosec A & \rm \infty & 2& \sqrt{2} & \dfrac{2}{ \sqrt{3} } &1 \\ \\ \rm sec A & 1 & \dfrac{2}{ \sqrt{3} }& \sqrt{2} & 2 & \rm \infty \\ \\ \rm cot A & \rm \infty & \sqrt{3} & 1 & \dfrac{1}{ \sqrt{3} } & 0\end{array}}}\end{gathered}\end{gathered}\end{gathered} \end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered}$