question_answer

The angle of elevation of the top of a hill at the a of a tower is \[60{}^\circ \]and the angle of elevation of the top of the tower at the foot of the hill is \[30{}^\circ .\]If the tower is 50 m tall, the height of the hill is

A)  \[100\sqrt{3}\,\,\text{m}\]

B)  \[50\sqrt{3}\,\,\text{m}\]

C)  \[150\,\,\text{m}\]

D)  \[\frac{150}{\sqrt{3}}\,\,\text{m}\]

Correct Answer: C

Solution :

Let height of the hill be h metre.

In \[\Delta CDB,\]

\[\tan 30{}^\circ =\frac{CD}{BD}\]

\[\frac{1}{\sqrt{3}}=\frac{50}{BD}\]

\[\Rightarrow \]               \[BD=50\sqrt{3}\,\,\text{m}\]

Again in \[\Delta ABD,\]

\[\frac{AB}{BD}=\tan 60{}^\circ \]

\[\Rightarrow \]\[\frac{AB}{50\sqrt{3}}=\sqrt{3}\]

\[AB=50\sqrt{3}\times \sqrt{3}\]\[\Rightarrow \]\[h=150\,\,\text{m}\]