Question : On the ground, there is a vertical tower with a flagpole on its top. At a point 9 metres away from the foot of the tower, the angles of elevation of the top and bottom of the flagpole are 60° and 30°, respectively. The height of the flagpole is:
Option 1: $5\sqrt{3}$ metres
Option 2: $6\sqrt{3}$ metres
Option 3: $6\sqrt{2}$ metres
Option 4: $6\sqrt{5}$ metres
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Correct Answer: $6\sqrt{3}$ metres
Solution : In the figure, the flagpole is AB, BC is the tower and CD is the base. In $\triangle$ACD, $⇒\tan 60°=\frac{\text{AC}}{\text{DC}}$ $⇒\sqrt{3}=\frac{\text{AC}}{9}$ $⇒\text{AC}=9\sqrt{3}$ metres In $\triangle$BCD, $⇒\tan 30°=\frac{\text{BC}}{\text{DC}}$ $⇒\frac{1}{\sqrt{3}}=\frac{\text{BC}}{9}$ $⇒\text{BC}=\frac{9}{\sqrt{3}} = 3\sqrt{3}$ metres $⇒\text{AC = AB + BC}$ $⇒\text{AB = AC – BC}$ $⇒\text{AB} = 9\sqrt{3} - 3\sqrt{3}=6\sqrt{3}$ metres Hence, the correct answer is $6\sqrt{3}$ metres.
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