# The height of a tower is 100 m. When the angle of elevation of the sun changes from 30° to 45°, the shadow of the tower becomes x metres less. The value of x is.

A. $100 \mathrm{~m}$ B. $100 \sqrt{3} m$ C. $100(\sqrt{3}-1) m$ D. $\frac{100}{\sqrt{3}} m$ Answer: Option C
Let AB be tower and AB = 100 m and angles of elevation of A at C and D are 30° and 45° respectively and CD = x Let BD = y $\begin{array}{l}\text{Now in right }\ \triangle\text{ADB},\\ \tan\theta=\frac{\text{ Perpendicular }}{\text{ Base }}=\frac{AB}{DB}\\ \tan45^{\circ}=\frac{100}{y}\\ \Rightarrow1=\frac{100}{y}\\ \Rightarrow y=100\\ \text{Similarily in right }\triangle\text{ACB},\\ \tan 30^{\circ}=\frac{A B}{C B} \\ \Rightarrow \frac{1}{\sqrt{3}}=\frac{100}{y+x} \\ \Rightarrow \frac{1}{\sqrt{3}}=\frac{100}{100+x} \\ \Rightarrow 100+x=100 \sqrt{3} \\ \Rightarrow x=100 \sqrt{3}-100 \\ \Rightarrow x=100(\sqrt{3}-1) m \end{array}$