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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