From the top of a hill 200 m high, the angles of depression of the top and bottom of a pillar are 30° and 60° respectively. Find the height of the pillar and its distance from the hill.

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Solution

Let $A B$ be the hill and $D E$ be the pillar. Draw $C D$ ⊥ $A B$ .
Thus, we have:
$A B = 200$ m, ∠ $B E A = 60^{o}$ and ∠ $B D C = 30^{o}$
Now, let $A E = x$ m such that $C D = x$ m and let $D E = h$ m such that $A C = h$ m.

In the right ∆ $A E B$ , we have:
$\frac{A B}{A E} = \tan 60^{o} = \sqrt{3}$

⇒ $\frac{200}{x} = \sqrt{3}$
⇒ $x = \frac{200}{\sqrt{3}} = 115.47$ m
Now, in the right ∆ $B D C$ , we have:
$\frac{B C}{C D} = \tan 30^{o} = \frac{1}{\sqrt{3}}$

⇒ $\frac{(200 - h)}{x} = \frac{1}{\sqrt{3}}$

By putting $x = \frac{200}{\sqrt{3}}$ in the above equation, we get:
$\frac{(200 - h) \sqrt{3}}{200} = \frac{1}{\sqrt{3}}$
⇒ $600 - 3 h = 200$
⇒ $3 h = 400$
⇒ $h = \frac{400}{3} = 133.33$ m

We now have:
Height of the pillar = 133.33 m
Distance of the pillar from the hill = 115.47 m

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