Let AB is a building of height 60m and CD is a light house of height h m. The angle of elevation and angle of depression of the top and bottom of a light house from top of building are 30° and 60° respectively.
∠CAE = 30° and ∠EAD = 60°
∠ADB = ∠EAD = 60° (Alternate angle)
Draw BD ∣∣ AE
∴ ∠AEC = 90° (Corresponding angle)
∠ABD + ∠BDE = 90° + 90° = 180°
∴ AB ∣∣ DE
So, ABDE is a rectangle.
DE = AB = 60m
and CE = (h – 60)m
From right angled ΔAEC,
\(\tan 30° = \frac{CE}{AE}\)
\(\frac 1{\sqrt 3} = \frac {h - 60}{BD}\) [∵ AE = BD]
BD = 3(h – 60)m …..(i)
From right angled ΔABD,
\(\tan 60° = \frac{AB}{BD}\)
\(\sqrt 3 = \frac{60}{BD}\)
\(BD = \frac{60}{\sqrt 3} = \frac{60\times \sqrt 3}{\sqrt 3 \times \sqrt 3}\)
= 20√3 .........(ii)
Put the value in equation (ii) from equation. (i),
√3(h – 60) = 20√3
h – 60 = \(\frac{20√3}{√3}\) = 20
h = 20 + 60 = 80 m
Hence, height of light house = 80 m
(i) Difference in height between light house and building = 80 – 60 = 20 m
(ii) Distance between light house and building BD = 20√3 m
