Question

The tops of two towers of height x and y, standing on level ground, subtend angles of 30º and 60º respectively at the centre of the line joining their feet, then find x : y. [CBSE 2015]

Answer 1

Let AB and CD be the two towers and E be the mid-point of AC.

Height of the tower, AB = y

Height of the tower, CD = x

It is given that, $\angle \mathrm{AEB}=60°$ and $\angle \mathrm{CED}=30°$.

Also, AE = EC

In right ∆AEB,

$$\begin{gathered} \tan 60°=\frac{\mathrm{AB}}{\mathrm{AE}} \\ \Rightarrow \sqrt{3}=\frac{y}{\mathrm{AE}} \\ \Rightarrow \mathrm{AE}=\frac{y}{\sqrt{3}} \end{gathered}$$

In right ∆CED,

$$\begin{gathered} \tan 30°=\frac{\mathrm{CD}}{\mathrm{CE}} \\ \Rightarrow \frac{1}{\sqrt{3}}=\frac{x}{\mathrm{CE}} \\ \Rightarrow \mathrm{CE}=\sqrt{3}x \end{gathered}$$

Now, AE = CE

$$\begin{gathered} \frac{y}{\sqrt{3}}=\sqrt{3}x \\ \Rightarrow y=3x \\ \Rightarrow \frac{x}{y}=\frac{1}{3} \\ \therefore x:y=1:3 \end{gathered}$$

Hence, the ratio of x : y is 1 : 3.