question_answer

From the top of a cliff 50m high, the angles of depression of the top and bottom of a tower are observed to be \[30{}^\circ \] and\[45{}^\circ \]. The height of tower is

A) 50 m

B) \[50\sqrt{3}m\]

C) \[50(\sqrt{3}-1)m\]

D) \[50\left( 1-\frac{\sqrt{3}}{3} \right)m\]

Correct Answer: D

Solution :

[d] Let height of the tower be h m and distance between tower and cliff be x m.

\[\therefore \,\,\,CD=h,\,\,\,BD=x\]

In \[\Delta ABD,\,\,\,\tan 45{}^\circ =\frac{AB}{BD}\]

or \[1=\frac{50}{x}\]

\[x=50\]                                                ? (i)

In \[\Delta AEC\]

\[tan\,\,30{}^\circ =\frac{AE}{EC}=\frac{AB-EB}{EC}=\frac{AB-DC}{BD}\]

\[(\because \,\,\,\,EB=DC,\,\,EC=BD)\]

\[\frac{1}{\sqrt{3}}=\frac{50-h}{x}\] or \[x=50\sqrt{3}-h\sqrt{3}\]

or \[50=50\sqrt{3}-h\sqrt{3}\] or \[h\sqrt{3}=50\sqrt{3}-50\]

or \[h=\frac{50(\sqrt{3}-1)}{\sqrt{3}}=50\left( 1-\frac{1}{\sqrt{3}} \right)\]

\[\therefore \,\,\,\,\,h=50\left( 1-\frac{\sqrt{3}}{3} \right)\]