Question : If the angle of elevation of the sun decreases from $45^\circ$ to $30^\circ$, then the length of the shadow of a pillar increases by 60 m. The height of the pillar is:
Option 1: $60(\sqrt{3}+1)$ metres
Option 2: $30(\sqrt{3}–1)$ metres
Option 3: $30(\sqrt{3}+1)$ metres
Option 4: $60(\sqrt{3}–1)$ metres
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Correct Answer: $30(\sqrt{3}+1)$ metres
Solution :
Given: If the angle of elevation of the sun decreases from $45^\circ$ to $30^\circ$, then the length of the shadow of a pillar increases by 60 m.
Let AB = height of the pole = $h$ m
CD = 60 metres.
In $\triangle ABD$,
$\tan45°=\frac{AB}{BD}$
⇒ $1=\frac{h}{BD}$
⇒ BD = $h$ m
In $\triangle ABC$,
$\tan30°=\frac{AB}{BC}$
⇒ $\frac{1}{\sqrt3}=\frac{h}{h+60}$
⇒ $h\sqrt3–h=60$
⇒ $h=\frac{60}{\sqrt3–1}=\frac{60(\sqrt3+1)}{(\sqrt3–1)(\sqrt3+1)}=30(\sqrt3+1)$ metres.
Hence, the correct answer is $30(\sqrt3+1)$ metres.
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