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Calculating Heights and Distances

The angle of ...

Question

The angle of elevation of the top of a tower from the bottom and top of a building of height $h m t$ are $α$ and $β$ respectively. The height of the tower is

A

\frac{h sin α cos β}{sin (α - β)}

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B

∣ ∣ ∣ \frac{d sin α cos β}{cos (α + β)} ∣ ∣ ∣

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C

∣ ∣ ∣ \frac{d cos α + sin β}{cos (α - β)} ∣ ∣ ∣

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D

∣ ∣ ∣ \frac{d sin α - sin β}{sin (α - β)} ∣ ∣ ∣

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Solution

The correct option is A $\frac{h sin α cos β}{sin (α - β)}$
Given,
$∠ A B D = β, ∠ E C D = α, B C = h, A B = E C$

$⟹ A B = \frac{A D}{t a n β}$

$⟹ E C = \frac{E D}{t a n α} = \frac{A D - h}{t a n α}$

We know that $A B = E C$

$\frac{A D}{t a n β} = \frac{A D - h}{t a n α}$

Height of tower $A D = \frac{h t a n β}{t a n β - t a n α}$

$⟹ A D = \frac{h s i n α c o s β}{s i n (α - β)}$

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Calculating Heights and Distances

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