question_answer

A vertical pole consists of two parts, the lower part being one third of the whole. At a point in the horizontal plane through the base of the pole and distance 20 meters form it, the upper part of the pole subtends an angle whose tangent is \[\frac{1}{2}.\] the possible heights of the pole are

A) 20 m and \[20\sqrt{3}\]

B) 20 m and 60 m

C) 16 m and 48 m

D) None of these  

Correct Answer: B

Solution :

[b] From the given figure we have \[\frac{H}{3d}=\tan \alpha \] and \[\frac{H}{d}=\tan \beta \] \[\tan (\beta -\alpha )=\frac{1}{2}=\frac{\frac{H}{d}-\frac{H}{3d}}{1+\frac{{{H}^{2}}}{3{{d}^{2}}}}\] \[\Rightarrow {{H}^{2}}-4dH+3{{d}^{2}}=0\] \[\Rightarrow {{H}^{2}}-80H+3(400)=0\] \[\Rightarrow H=20\] and 60 m