question_answer

A vertical tower stands on a horizontal land and is surmounted by a vertical flag staff of height 12 metres. At a point on the plane, the angle of elevation of the bottom and the top of the flag staff are respectively \[{{45}^{o}}\] and \[{{60}^{o}}\]. Find the height of tower.

A)  \[6\left( \sqrt{3}+4 \right)\,m\]           

B)  \[6\left( \sqrt{3}+1 \right)\,m\]

C)         \[7\left( \sqrt{3}+1 \right)\,m\]   

D)         None of these

Correct Answer: B

Solution :

 Let AB be the tower of height h metre and BC be the height of flag staff surmounted on the tower, Let the point of the plane be D at a distance x metre from the foot of the tower. In  \[\Delta ABD,\]             \[\tan {{45}^{o}}=\frac{AB}{BD}\Rightarrow \frac{h}{x}=1\Rightarrow x=h\]     ?.(i) In  \[\Delta ADC,\] \[\tan {{60}^{O}}=\frac{AC}{AD}\Rightarrow \sqrt{3}=\frac{12+h}{x}\Rightarrow \sqrt{3}x=12+h\]\[x=\frac{12+h}{\sqrt{3}}\times \frac{\sqrt{3}}{\sqrt{3}}=\frac{\sqrt{3}}{3}\,(12+h)\]     ?..(ii) From(1) and (2),  \[h=\frac{\sqrt{3}}{3}(12+h)\Rightarrow h=6\left( \sqrt{3}+1 \right)\] So, the height of tower \[=6\left( \sqrt{3}+1 \right)\,m\]