Question

A vertical pole more than $100ft$ high consists of two portions, the lower being $1/3$ of the whole. lf the upper portion subtends an angle ${\tan }^{-1}\left(1/2\right)$ at a point distant 40 ft. from the foot of the pole, the height of the pole is

• A. $105$
• B. $120$
• C. $135$
• D. $150$

Answer 1

Answer: (B)

$120$

$tan=1/2$
$tan\alpha =\frac{h}{3+40}$
$tan(\alpha +\theta )=\frac{h}{40}$
$\frac{tan\alpha }{3}=\frac{tan\alpha +tan\theta }{1-tan\alpha tan\theta }$
$\frac{1+2tan\alpha }{2-tan\alpha }=\frac{tan\alpha }{3}$
$3+6tan\alpha =2tan\alpha -tan\alpha$
$ta{n}^2\alpha +4tan\alpha +3=0$
$tan\alpha =-1,tan\alpha =-3$
$1=\frac{h}{120}$
$h=120$