question_answer

The angle of elevation of the top of a tower from the top of a house is \[{{60}^{o}}\] and the angle of depression of its base is\[{{30}^{o}}\]. If the horizontal distance between the house and the tower be \[12\,\,m\], then the height of the tower is

A) \[48\sqrt{3}m\]               

B) \[16\sqrt{3}m\]

C) \[24\sqrt{3}m\]               

D) \[\frac{16}{\sqrt{3}}m\]

Correct Answer: B

Solution :

Let \[AE\] be the tower and \[BD\] be the house. Then,                 \[BD=h\,\,m\]                 \[AE=H+h\]                 \[\angle ABC={{60}^{o}}\]                 \[\angle BED={{30}^{o}}\] and        \[DE=BC=12\,\,m\] In\[\Delta BDE\],                 \[\tan {{30}^{o}}=\frac{BD}{DE}=\frac{h}{12}\] \[\Rightarrow \]               \[h=12\tan {{30}^{o}}\] \[\Rightarrow \]               \[h=\frac{12}{\sqrt{3}}\] In\[\Delta ACB\],                 \[\tan {{60}^{o}}=\frac{AC}{BC}=\frac{H}{12}\] \[\Rightarrow \]               \[H=12\sqrt{3}\]. \[\therefore \]Height of tower\[=H+h\]                 \[=12\sqrt{3}+\frac{12}{\sqrt{3}}\]                 \[=16\sqrt{3}m\]