Question : Two posts are 2 metres apart. Both posts are on the same side of a tree. If the angles of depressions of these posts when observed from the top of the tree are 45° and 60° respectively, then the height of the tree is:
Option 1: $(3-\sqrt{3})$ metres
Option 2: $(3+\sqrt{3})$ metres
Option 3: $(-3+\sqrt{3})$ metres
Option 4: $(3-\sqrt{2})$ metres
Correct Answer: $(3+\sqrt{3})$ metres
Solution :
CD = 2 metres
BD = $x$ metre
AB = Tree = $h$ metre
From ∆ ABC,
tan 45° = $\frac{AB}{BC}$
⇒ $1 = \frac{h}{x+2}$
$\therefore h = (x + 2)$ metre .....(i)
From ∆ ABD,
$\tan 60° = \frac{AB}{BD}$
⇒ $\sqrt{3} = \frac{h}{x}$
$\therefore x = \frac{h}{\sqrt{3}}$ metre .....(ii)
Putting the value of $x$ in equation (i), we get
$h=\frac{h}{\sqrt{3}}+2$
⇒ $h(\frac{\sqrt{3}-1}{\sqrt{3}})=2$
⇒ $h= \frac{2\sqrt{3}}{\sqrt{3}-1}$
⇒ $h= \frac{2\sqrt{3}(\sqrt{3}+1)}{(\sqrt{3}-1)(\sqrt{3}+1)}$
$\therefore h=\frac{6+2\sqrt{3}}{2}=(3+\sqrt{3})$ metres
So, the height of the tree is $(3+\sqrt{3})$ metres.
Hence, the correct answer is $(3+\sqrt{3})$ metres.
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