Question

If the angle of elevation of the sun changes from ${30}^{\circ }to{45}^{\circ }$, the length of the shadow of a pillar decreases by 20 meters. The height of the pillar is:

• A. $20(\sqrt{3}-1)m$
• B. $20(\sqrt{3}+1)m$
• C. $10(\sqrt{3}-1)m$
• D. $10(\sqrt{3}+1)m$

Answer 1

The correct option is D $10(\sqrt{3}+1)m$

In $△ABC,\tan {30}^o=\frac{h}{20+x}\frac{1}{\sqrt{3}}=\frac{h}{20+x}...\left(2\right)$

In $△BDC,\tan {45}^o=\frac{h}{x}h=x...\left(1\right)$

Put (1) in (2)

$\frac{1}{\sqrt{3}}=\frac{h}{20+h}$

$\therefore 20+h=h\sqrt{3}\therefore h\sqrt{3}-h=20$

$\Rightarrow h(\sqrt{3}-1)=20$

$\Rightarrow h=\frac{20}{\sqrt{3}-1}\times \frac{\sqrt{3}+1}{\sqrt{3}+1}=\frac{20(\sqrt{3}+1)}{2}=10(\sqrt{3}+1)m$

$\Rightarrow h=10(\sqrt{3}+1)m$