question_answer

The angle of elevation of the top of a tower from a point on the ground is 45°. Moving 21m directly towards the base of the tower, the angle of elevation changes to 60°. What is the height of the tower, (to the nearest meter)?

A) 48 m                

B) 49 m

C) 50 m                

D) 51 m

Correct Answer: C

Solution :

In \[\Delta PBC,\] \[\tan {{60}^{o}}=\frac{h}{x}\Rightarrow \frac{h}{x}=\sqrt{3}\]\[\Rightarrow x=\frac{h}{\sqrt{3}}\] In \[\Delta PAC,\] \[\tan \,{{45}^{o}}=\frac{h}{21+x}=1\Rightarrow h=21+x\] \[\Rightarrow h=21+\frac{h}{\sqrt{3}}\] [from equation (i)] \[\Rightarrow h\left( 1-\frac{1}{\sqrt{3}} \right)=3\] \[\Rightarrow h=\frac{21\sqrt{3}}{(\sqrt{3}-1)}\times \frac{(\sqrt{3}+1)}{(\sqrt{3}+1)}\] \[\Rightarrow h=\frac{21\sqrt{3}(\sqrt{3}+1)}{2}=\]