Question

A tower of height is located exactly opposite to a tower of height on a straight road. From the top of , if the angle of depression of the foot of is twice the angle of elevation of the top of , then the width (in m) of the road between the feet of the towers and is

Answer 1

Let height of first tower is h
and height of second tower is H.
elevation angle of tower , $\theta$
so, depression angle of tower , $2\theta$

situation is shown in figure,

see triangle ABC,
$tan2\theta=\frac{H}{d}$
$\frac{2tan\theta}{1-tan^2\theta}=\frac{H}{d}$.....(1)

from triangle CDE ,
$tan\theta=\frac{H-h}{d}$......(2)

put equation (2) in equation (1),
2{(H-h)/d}/{1 - (H-d)²/d²} = H/d

or, 2(H - h)d/{d² - (H-h)²} = H/d

or, 2(H - h)d² = Hd² - H(H - h)²

or, d²{H - 2(H - h)} = H(H - h)²

or, d²{H - 2H + 2h} = H(H - h)²

or, d²{2h - H} = H(H - h)²

or, d² = H(H - h)²/(2h - H)

hence, d = (H - h) × √{H/(2h - H)}