Question

Two persons on the same side of a tall building notice the angle of elevation of the
top of the building to be 30 and 60 respectively. If the height of the building is
72 m. find the distance between the two persons -(/3= 1.73)​

Answer 1

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

• Let the height of a tall building be AD, = 72 meter.

and

• Let two persons on the same side of hill be at point B and C respectively from the bottom of the building.

• Let BC = 'x' meter and AB = 'y' meter.

According to statement,

• ∠ABD = 60°

and

• ∠ ACD = 30°.

Now,

We have

• AD = 72 m

• AB = 'y' m

• BC = 'x' m

• ∠ ABD = 60°

• ∠ ACD = 30°

Consider,

$\rm :\longmapsto\:In \: \triangle \: ABD$

$\rm :\longmapsto\:tan60 \degree \: = \dfrac{AD}{AB}$

$\rm :\longmapsto\: \sqrt{3} = \dfrac{72}{y}$

$\rm :\longmapsto\:y = \dfrac{72}{ \sqrt{3} }$

$\rm :\longmapsto\:y = \dfrac{72}{ \sqrt{3} } \times \dfrac{ \sqrt{3} }{ \sqrt{3} }$

$\rm :\longmapsto\:y = \dfrac{72 \sqrt{3} }{3}$

$\bf\implies \:y \: = \: 24 \: \sqrt{3} \: m \sf \: \: - - - (1)$

Now,

$\rm :\longmapsto\:In \: \triangle \: ACD$

$\rm :\longmapsto\:tan30 \degree \: = \dfrac{AD}{AC}$

$\rm :\longmapsto\:\dfrac{1}{ \sqrt{3} } = \dfrac{72}{x + y}$

$\rm :\longmapsto\:x + y = 72 \sqrt{3}$

$\rm :\longmapsto\:x + 24 \sqrt{3} = 72 \sqrt{3} \: \: \: \: \: \{ using \: (1) \: \}$

$\rm :\longmapsto\:x = 48 \sqrt{3}$

$\rm :\longmapsto\:x \: = \: 48 \times 1.73$

$\bf\implies \:x = 83.04 \: meter$

$\overbrace{\underline{\boxed{\bf\:Hence, \: distance \: between \: persons \: is \: 83.04 \: m}}}$

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