Question

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 ${60}^0$ and ${30}^0$; find the height of the pillars and the position of the point.

Answer 1

Let AB and CD be the two towers of height $h,m$ each.
And, let P be a point in the roadway BD such that BD = 150 m. < APS =${60}^0$ and < CPD = ${30}^0$
In $△ABP$,
$AB/BP=tan{60}^0$
$BP=h/tan{60}^0$
$BP=h/\sqrt{3}$
In $△CDP,$
$CD/DP=tan{30}^0$
$PD=\sqrt{3}h$
Now,$150=BP+PD$
$150=\sqrt{3}h+h/\sqrt{3}$
$h=150/(\sqrt{3}+1/\sqrt{3})=150/2.309$
$h=64.95m$
Thus, the height of the pillars are 64.95 m each.
Now,
The point is $BP=h/\sqrt{3}$ from the first pillar.
Which is a distance of $64.95/\sqrt{3}$ from the first pillar.
Thus, the position of the point is $37.5m$ from the first pillar.