95
A man standing on the deck of a ship, which is $10$ m above water level, observes the angle of elevation of the top of a hill as $60^\circ$ and the angle of depression of the base of the hill as $30^\circ$. Calculate the distance of the hill from the ship and the height of the hill.
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Let $(h + 10)$ be the height of the hill and $x$ be the distance between base of the hill and the ship
$\frac{h}{x} = \tan 60^\circ \implies h = \sqrt{3}x \dots (i)$
$\frac{10}{x} = \tan 30^\circ = \frac{1}{\sqrt{3}} \implies x = 10\sqrt{3} \dots (ii)$
$\therefore h = 30$ [ from $(i)$ and $(ii)$]
Height of the hill $= 30 + 10 = 40$ m
Distance between the hill and the ship $= 10\sqrt{3}$ m
$\frac{h}{x} = \tan 60^\circ \implies h = \sqrt{3}x \dots (i)$
$\frac{10}{x} = \tan 30^\circ = \frac{1}{\sqrt{3}} \implies x = 10\sqrt{3} \dots (ii)$
$\therefore h = 30$ [ from $(i)$ and $(ii)$]
Height of the hill $= 30 + 10 = 40$ m
Distance between the hill and the ship $= 10\sqrt{3}$ m