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Is the following true or false?

A flagstaff stands on the top of a tower. At a point distant d from the base of the tower, the angles of elevation of the top and lower of the flagstaff are $β$ and $α .$ Then the height of the flagstaff is $d tan β − t a n α .$

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True

False

answer is A.

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Detailed Solution

Given that,
The distance between point and base of the tower = d.
The angle of elevation of the top of the flagstaff and lower of flagstaff =

β

and

α .

We can calculate the angle of elevation by the formula

tan θ = P e r p e n d i c u l a r B a s e

.
We draw the schematic diagram,
Question Image

The distance BC from the triangle

Δ B C D

tan α = D C B C ⇒ tan α = D C d ⇒ D C = d tan α ...... (1)

The distance DC from the triangle

Δ A B C

tan β = A D + D C B C ⇒ tan β = A D + D C d ....... (2)

Substituting the value of DC from the equation (1) in equation (2),

⇒ tan β = A D + d tan α d ⇒ d tan β = A D + d tan α ⇒ A D = d tan β − tan α

Thus, the height of the flagstaff will be

d tan β − tan α

which is true.
Therefore the correct option is 1.

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