question_answer

From the top of a pillar of height 20 m the angles of elevation and depression of the top and bottom of another pillar are \[30{}^\circ \] and \[45{}^\circ \]respectively. The height of the second pillar (in metres) is

A)  \[\frac{20}{\sqrt{3}}(\sqrt{3}-1)\]          

B)  \[\frac{20}{\sqrt{3}}(\sqrt{3}+1)\]

C)  \[20\sqrt{3}\]

D)  \[\frac{20}{\sqrt{3}}\]

Correct Answer: B

Solution :

Let SP and RQ be two pillars in

which SP = 20 m

Here,

Let \[ST\bot RQ\]

Then,    \[\angle RST=30{}^\circ \] and

Let        \[RT=x,\]\[SP=TQ=20\,\,\text{m}\]

Also,     SP = PQ

As         \[\theta =45{}^\circ \]\[\Rightarrow \]\[PQ=ST=20\,\,\text{m}\]

In \[\Delta SRT,\]\[\tan 30{}^\circ =\frac{x}{ST},\]\[\frac{x}{20}=\frac{1}{\sqrt{3}}\]\[\Rightarrow \]\[x=\frac{20}{\sqrt{3}}\]

\[\therefore \]Height of pillar \[RQ=RT+TQ\]

\[=20+\frac{20}{\sqrt{3}}=\frac{20\,\,(\sqrt{3}+1)}{\sqrt{3}}\]