If the angles of elevation of the top of a tower from the two points at a distance of 2m and $8 m$ from the base of the tower and in the same straight line with it are complementary, then the height of the tower is:

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3 m

6 m

5 m

4 m

answer is B.

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Detailed Solution

Given that the angles of elevation of the top of a tower from the two points at a distance of 2m

and

8 m

from the base of the tower and in the same straight line with it are complementary.
Consider the figure,
Question Image

We know,

\tan (90^{0} - θ) = cotθ

tanθ = \frac{Perpendicular}{Base}

Let A be the point 2 m away from the base of the tower and B be the point 8 m away from the base of the tower. Let the angle of elevation of the top of the tower from point A is

θ

, then the angle of elevation of the top of the tower from point B is

90 ° − θ

because the angles of elevations are given to be complementary. Let the height of the tower be. Then in triangle ABC and triangle ABD,

tan θ = H 2 … … (1)

and

⇒ tan 90 ° − θ = H 8 ⇒ cot θ = H 8 ⇒ 1 tan θ = H 8 ⇒ tan θ = 8 H … … (2)

Equation (1) and (2) imply that,

H 2 = 8 H ⇒ H 2 = 16 ⇒ H = 4

Therefore, the height of the tower is 4 m.
Hence, option 2 is correct.

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