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State true or false.
From the top of a tower, the angles of depression of two objects on the same side of the tower are found to be $α$ and $β (α > β) .$ If the distance between the objects is $p$ metres, the height $h$ of the tower is given by $h = p tan α tan β tan α − tan β$ metres.

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False

True

answer is A.

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Detailed Solution

Given that the top of a tower, the angles of depression of two objects on the same side of the tower are found to be

α

and

β (α > β) .

We draw the diagram depicting the situation,
Question Image

We know,

tanθ = \frac{Perpendicular}{base}

Write equations for h in the form of tan

α

and

tan β

h = x tan α … … (1) h = (x + p) tan β … … (2)

Equate equation (1) to (2) and find x.

x tan α = (x + ρ) tan β ⇒ x (tan α − tan β) = p tan β ⇒ x = p tan β tan α − tan β

The height of the tower,

h = x tan α ⇒ h = p tan β tan α − tan β tan α ⇒ h = p tan α tan β tan α − tan β

Hence proved that the height of the tower is

p tan α tan β tan α − tan β

meters.
Therefore, the statement is true.
Hence, option 1 is correct.

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