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+2 votes
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The angles of elevation and depression of the top and bottom of a lighthouse from the top of a building, 60 m high, are 30º and 60º respectively. Find
(i) the difference between the heights of the lighthouse and the building.
(ii) distance between the lighthouse and the building.

2 Answers

+1 vote
by (17.0k points)
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Best answer

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

+4 votes
by (266k points)

Solution is.............

by (10 points)
+2
Very nice

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