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Two pillars of equal height stand on either side of the roadway which is 30 m wide. At a point in the roadway between the pillars, the elevations of the tops of the pillars are $60 °$ and $30 °$ . What will be the heights of the pillars and the position of the point?

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1532 m

, 7.5m

1534 m

, 7.0m

1732 m

, 7.2m

1734 m

, 6.9m

answer is A.

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Detailed Solution

Given that,
The width of the road =

30

m
The angles made by towers =

60 °

, and

30 °

We draw the diagram,
Question Image

The height of the equal pillars AB and CD = y
The width of the road

B D = 30

B E = x

then,

D E = 30 − x

Using the identity

tan θ = Perpendicular Base

Δ A B E

tan θ = AB BE

Substituting BE=x, AB= y, and

θ

60 °

in the equation

tan θ = AB BE

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

Using the identity

tan θ = Perpendicular Base

Δ C D E

tan θ = CD DE

Substituting DE=

30 − x

, CD=y and

θ

30 °

in the equation

tan θ = CD DE

⇒ tan 30 ° = y 30 − x ⇒ 13 = y 30 − x [∵ tan 30 ° = 13] ⇒ 30 − x = 3 y

The value of y,
Substituting

x

y 3

in the equation

30 − x = 3 y

⇒ 30 − y 3 = 3 y ⇒ 30 = 3 y + y 3 ⇒ y = 3034 ⇒ y = 1532 m

Thus, the height of the equal pillars will be

1532 m .

Substituting

y = 1532 m

in equation

x = y 3

⇒ x = 15323 ⇒ x = 7.5 m

Thus the heights of both pillars will be

1532 m

, and the position of the point will be

7.5

m.
Therefore the correct option is 1.

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Two pillars of equal height stand on either side of the roadway which is 30 m wide. At a point in the roadway between the pillars, the elevations of the tops of the pillars are 60 ° and 30 ° . What will be the heights of the pillars and the position of the point?

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Two pillars of equal height stand on either side of the roadway which is 30 m wide. At a point in the roadway between the pillars, the elevations of the tops of the pillars are $60 °$ and $30 °$ . What will be the heights of the pillars and the position of the point?