Question

A tree standing on a horizontal plane is leaning towards east. At two points situated at distances a and b exactly due west on it, the angles of elevation of the top are respectively α and β. Prove that the height of the top from the ground is (b-a)tanαtanβ/tanα-tanβ

Answer 1

Solution:

Let AB be the tree, P and Q be the points where it is observed.

Draw AC perpendicular to the ground.

Here, BP =a, BQ =b

Let AC = h and BC = x

=> In ΔACP, tan α = AC/PC

=> tan α = h/ (x+a)

Therefore,

x + a = h/ tan α

∴ x = h/tan α - a ........(1)

Similarly in ΔACQ, tan β = AC/QC

=> tan β =  h / ( x + b)

Therefore,

x+b = h / tan β

∴x = h/tanβ - b ............(2)

=> h/tan α - a = h/tan β - b ( comparing (1) and (2) )

∴h/tan α - h/tan β = -b + a

h(tan β - tan α / tan α × tan β) = -(b-a)

∴h = -(b-a) * tan α * tan β / ( tan β - tan α)

∴h = (b-a) tan α * tan β / ( tan α - tan β )