Question

The height of the tower is 100 m. When the angle of elevation of the sun changes from ${30}^{\circ }to{45}^{\circ }$, the shadow of the tower becomes x metres less. The value of x is :

• A. $100m$
• B. $100\sqrt{3}m$
• C. $100(\sqrt{3}-1)m$
• D. $\frac{100}{\sqrt{3}}m$

Answer 1

The correct option is C $100(\sqrt{3}-1)m$

Let AB be the height of the tower.
Given:
The angle of elevation changed from ${30}^o$ to ${45}^o$.

When angle of elevation changed from ${30}^o$ to ${45}^o$, the length of the shadow also reduced by $xm$.

Then, let us consider the remaining shadow as $ym$.

The trigonometric ratio that connects the height of the tower and shadow of the tower is $tan\theta$.

In $\Delta ABC$ , $tan{45}^{\circ }=\frac{AB}{BC}=\frac{100}{y}$
$1=\frac{100}{y}(\because tan{45}^{\circ }=1)$
$y=100m$ -----(i)

In $\Delta ABD,tan{30}^{\circ }=\frac{AB}{BD}=\frac{100}{(y+x)}$
We know that, $tan{30}^o=\frac{1}{\sqrt{3}}$ and $y=100m$.
Then, $\frac{1}{\sqrt{3}}=\frac{100}{(100+x)}\Rightarrow 100+x=100\sqrt{3}\Rightarrow x=100(\sqrt{3}-1)m$

Thus, the reduced shadow length is $100(\sqrt{3}-1)m$.