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From a point on the level ground, the angle of elevation of the top of a tower is $30 °$ . On proceeding 30 m towards the tower the angle of elevation becomes $60 °$ . What will be the height of the tower?

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53

152

153

103

answer is C.

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Detailed Solution

Given,
Angle of elevation

(θ) = 30 °

Distance moved towards the tower

y = 30

m
The new angle of elevation from the ground

(ϕ) = 60 °

We draw a schematic diagram and get a relation connecting the height of the tower and the angle of elevation

θ

Δ P Q S

,
PQ be the tower and QS be the ground.

x

be the distance from the bottom of the tower to the position at which the angle of elevation of the top of the tower is

60 °

.
In

Δ P Q S

we use

tan θ = h e i g h t b a s e

tan θ = P Q Q S ⇒ tan θ = P Q Q R + R S ⇒ tan θ = h x + 30

Substituting

θ

30 °

tan θ = h x + 30

⇒ tan 30 ° = h x + 30 ⇒ 13 = h x + 30 [∵ tan 30 ° = 13] ⇒ h = x + 30 3

Considering

Δ P Q R

and get another relation connecting the angle of elevation

ϕ

and the height of the tower,

tan ϕ = P Q Q R ⇒ tan ϕ = h x

Substituting

ϕ

60 °

tan ϕ = h x

⇒ tan 60 ° = h x ⇒ 3 = h x [∵ tan 60 ° = 3] ⇒ h = 3 x

Equating the relations

h = x + 20 3

and

h = 3 x

⇒ x + 30 3 = 3 x ⇒ x + 30 = 3 × 3 x ⇒ x + 30 = 3 x ⇒ 30 = 3 x − x

⇒ 30 = 2 x ⇒ 302 = x ⇒ x = 15

Substituting

x = 15 m

h = 3 x

⇒ h = 3 × 15 ⇒ h = 153 m

Thus the height of the tower will be

153

m.
Therefore the correct option is 3.

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From a point on the level ground, the angle of elevation of the top of a tower is 30 ° . On proceeding 30 m towards the tower the angle of elevation becomes 60 ° . What will be the height of the tower?

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From a point on the level ground, the angle of elevation of the top of a tower is $30 °$ . On proceeding 30 m towards the tower the angle of elevation becomes $60 °$ . What will be the height of the tower?