Question : If the angle of elevation of the Sun changes from 30° to 45°, the length of the shadow of a pillar decreases by 20 metres. The height of the pillar is:
Option 1: $20(\sqrt{3}-1)$ m
Option 2: $20(\sqrt{3}+1)$ m
Option 3: $10(\sqrt{3}-1)$ m
Option 4: $10(\sqrt{3}+1)$ m
Correct Answer: $10(\sqrt{3}+1)$ m
Solution : In $\Delta$ ABC $\tan 45°=\frac{AB}{BC}$ ⇒ $1=\frac{h}{BC}$ ⇒ $BC=h=AB$ In $\Delta$ ABD $\tan 30°=\frac{AB}{BD}$ ⇒ $\frac{1}{\sqrt3}=\frac{h}{h+20}$ ⇒ $h+20=h\sqrt{3}$ ⇒ $h=\frac{20}{\sqrt{3}–1}$ ⇒ $h=\frac{20}{\sqrt{3}–1}×\frac{\sqrt{3+1}}{\sqrt{3+1}}$ ⇒ $h=10(\sqrt{3}+1)$ m Hence, the correct answer is $10(\sqrt{3}+1)$ m.
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