Question

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 20m from it, the upper part of the pole subtends an angle whose tangent is 1/2. The possible heights of the pole are -

• A. $20m$ and $20\sqrt{3}$m
• B. $20m$ and $60m$
• C. $16m$ and $48m$
• D. None of these

Answer 1

The correct option is B $20m$ and $60m$

Now $\tan \beta =\frac{h}{60}$, $tan(\alpha +\beta )=\frac{h}{20}$

$tan(\alpha +\beta )=\frac{tan\alpha -tan\beta }{1-tan\alpha tan\beta }$ = $\frac{h}{20}$

=> $\frac{\frac{1}{2}+\frac{h}{60}}{1-\frac{h}{120}}$ = $\frac{h}{20}$

$\frac{60+2h}{120-h}=\frac{h}{20}$

$h^2-80h+1200=0$

By solving this we get h = $20m$ or $60m.$