Question

A vertical tower stands on a horizontal plane and is surmounted by a vertical flag-staff. At a point on the plane 70 metres away from the tower, an observer notices that the angles of elevation of the top and the bottom of the flag-staff are respectively ${60}^{\circ }and{45}^{\circ }$. Find the height of the flag-staff and that of the tower.

Answer 1

Let BC be the tower of height x m and AB be the flag staff of height y, 70 m away from the tower, makes an angle of elevation are 60° and 45° respectively from top and bottom of the flag staff.

Let AB = y m, BC = x m and CD = 70 m.

$\angle BDC=45$ and $\angle ADC=60$

So we use trigonometric ratios.

In a triangleBCD,

$tanD=\frac{BC}{CD}$

$tan45=\frac{x}{70}$

$1=\frac{x}{70}$

$x=70$

Again in a triangle ADC,

$tanD=\frac{AD+BC}{CD}$

$tan60=\frac{y+x}{70}$

$\sqrt{3}=\frac{y+x}{70}$

$70\sqrt{3}=70+y$

$y=70(\sqrt{3}-1)$

$y=51.24$

Hence the height of flag staff is 51.24m and height of tower is 70m.