Question

33. A flagstaff stands on a tower. At a point distance 60 m from the base of the tower
the top of the flagstaff makes an angle of 60°from the base and the tower makes an angle of 30°
at that very point. Find the height of the flagstaff.​

Answer 1

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

Let assume that

• Height of tower, AC be 'x' meter

and

• Height of flagstaff, CD be 'y' meter.

It is given that,

• A point B is 60 meter away from the base of the tower, A.

Now, we have

• AB = 60 m

• AC = x m

• CD = y m

• ∠CBA = 30°

• ∠DBA = 60°

Consider,

$\rm :\longmapsto\:In \: \triangle \: ABC$

$\rm :\longmapsto\:tan \: 30 \degree \: = \dfrac{AC}{AB}$

$\rm :\longmapsto\:\dfrac{1}{ \sqrt{3} } = \dfrac{x}{60}$

$\rm :\implies\:x = \dfrac{60}{ \sqrt{3} }$

$\rm :\longmapsto\:x = \dfrac{60}{ \sqrt{3} } \times \dfrac{ \sqrt{3} }{ \sqrt{3} }$

$\rm :\longmapsto\:x = \dfrac{60 \sqrt{3} }{3}$

$\bf\implies \:x = 20 \sqrt{3} \: m$

Now,

$\rm :\longmapsto\:In \: \triangle \: ADB$

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

$\rm :\longmapsto\: \sqrt{ 3} = \dfrac{x + y}{60}$

$\rm :\longmapsto\:60 \sqrt{3} = 20 \sqrt{3} + y$

$\rm :\longmapsto\:y = 60 \sqrt{3} - 20 \sqrt{3}$

$\bf\implies \:y = 40 \sqrt{3} \: m$

$\bf\implies \:Height \: of \: flagstaff = 40 \sqrt{3} \: meter.$

Additional Information :-

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