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At a point on the level ground, the angle of elevation of the top of a tower is $45 °$ . On moving 20 m towards the tower, the angle of elevation becomes $60 °$ .What is the height of the tower?

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47.32m

48.32m

47m

48m

answer is A.

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Detailed Solution

Given that,
The angle of elevation from the point on the ground

θ = 45 °

The angle of elevation after moving

20

m towards the tower

ϕ = 60 °

We draw the schematic diagram,
Question Image

We use the trigonometric identity,

tan θ = p e r p e n d i c u l a r b a s e

The distance DC form the

Δ A C D

⇒ tan 60 ° = A C D C ⇒ 3 = A C D C tan 60 ° = 3 ⇒ A C = 3 D C ........ (i)

The distance DC form the

Δ A B C

⇒ tan 45 ° = A C B D + D C ⇒ 1 = A C 20 + D C ⇒ 20 + D C = A C ........ (i i)

Substituting the value of AC from the equation (1) in (2)

⇒ D C + 20 = 3 D C ⇒ 3 D C − D C = 20 ⇒ D C 3 − 1 = 20 ⇒ D C = 20 3 − 1

On rationalizing we get,
Question Image

From (ii),

⇒ A C = 27.32 + 20 ⇒ A C = 47.32

Thus, the height of the tower will be

47.32

m.
Therefore the correct option is 1.

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