question_answer

If a flagstaff of 6 m high placed on the top of a tower throws a shadow of \[2\sqrt{3}\]m along the ground, then the angle that the sun makes with the ground is

A)  \[45{}^\circ \] 

B)  \[60{}^\circ \]

C)  \[30{}^\circ \] 

D)  \[90{}^\circ \]

Correct Answer: B

Solution :

Let OA and AB be the shadows of tower OR and flagstaff RC respectively on the grounds.

Let the sun makes an angle \[\phi \]with the ground.

Let                    \[OA=x\]

In right angled \[\Delta OAR,\]

\[\tan \phi =\frac{h}{x}\]                        ?(i)

and in \[\Delta OBQ,\]\[\tan \phi =\frac{h+6}{x+2\sqrt{3}}\]                    ?(ii)

From Eqs. (i) and (ii),

\[\therefore \]                  \[\frac{h}{x}=\frac{h+6}{x+2\sqrt{3}}\]

\[h\,\,(x+2\sqrt{3})=x\,\,(h+6)\]

\[2\sqrt{3}h=6x\]

\[x=\frac{h}{\sqrt{3}}\]\[\Rightarrow \]\[\frac{h}{x}=\sqrt{3}\]

\[\Rightarrow \]   \[\tan \phi =\sqrt{3}\]     [from Eq. (i)]

\[\Rightarrow \]   \[\phi =60{}^\circ \]