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Q.

A pole $5 m$ high is fixed on the top of a tower. The angle of elevation of the top of the pole as observed from a point A on the ground is $60 °$ and the angle of depression of the point $A$ from the top of the tower is $45 °$ , then find the height of the tower.

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a

5.83 m

b

7.83 m

c

6.83 m

d

8.83 m

answer is B.

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Detailed Solution

Given,
Height of the pole, h1 = 5m.
Angle of elevation of the top of the pole,

θ_{1} = 60 °

.
And the angle of depression of the top of the tower,

θ_{2} = 45 °

.
Consider the figure,
Question Image

Question Image

Let h2 be the height of the tower and d be the distance of the point on ground from the base of the tower.
Since the angle of elevation is equal to the angle of depression, the relation between height of tower and distance between point on ground and tower is given by,

\tan 45^{0} = \frac{h_{2}}{d}

\Rightarrow 1 = \frac{h_{2}}{d}

\Rightarrow h_{2} = d

The relation between the height of tower plus the height of pole on top and the distance of point on ground and tower is given by,

tan 60 ° = h 2 + 5 d ⇒ tan 60 ° = h 2 + 5 h 2 ⇒ 3 = 1 + 5 h 2 ⇒ h 2 = 5 3 − 1 ⇒ h 2 = 5 0.732 ⇒ h 2 = 6.83

Therefore, the height of the tower is 6.83 m.
Therefore, option 2 is correct.

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