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# Two pillars of equal heights stand on either side of a roadway, which is 150 m wide. At a point in the roadway between the pillars the elevations of the tops of the pillars are 60° and 30°. Find the height of the pillars and the position of the point.

**Given : **

Width of Road = 150 m

Angle of elevations of pillars = 60° and 30°

**To Find : **

Position of point from the pillars

Height of Pillars

**Solution :**

Lets take the Height tower = h

Position of point is at C.

BD = 150 m

CD = x ; BC = 150 - x

In ABC

$$\displaystyle tan\ 30° \ =\ \frac{h}{150-x} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ tan\ \theta \ =\ \frac{opposite}{hypotenusse}$$

$$\displaystyle \frac{1}{\sqrt{3}} \ =\ \frac{h}{150-x} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ tan\ 30° \ =\ \frac{1}{\sqrt{3}}$$

$150 - x \ \ = \ h √3$............................................(i)

In EDC

$$\displaystyle \begin{array}{{>{\displaystyle}l}}

tan\ 60° \ =\ \frac{h}{x} \ \ \\

\\

\sqrt{3\ } \ =\ \frac{h}{x} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ tan\ 60° \ =\ \sqrt{3} \ \

\end{array}$$

$x √3 = h$

$h = x √3............................................................. (ii)$

Substitute (ii) in (i)

$150 - x \ = \ (x√3) √3$

$150 - x \ = \ 3 x \ \ \ \ $

$150 = 3 x + x $

$150 = 4 x $

$4 x \ = \ 150 \ $

$$\displaystyle \begin{array}{{>{\displaystyle}l}}

x\ \ =\ \ \frac{150}{4} \ \\

\\

x\ \ =\ \ 37.5\

\end{array}$$

$ 150 - x \ = \ 150 - 37.5 \ = \ 112.5 $

Substitute x = 37.5 in (ii)

h = 37.5√3

so,

**Position of point from the pillars BC = 112.5 m , CD = 37.5 m.**

**Height of the Pillars h = 37.5√3 m**