It is found that on walking x metres towards a chimney in a horizontal line through its base, the elevation of its top changes from 30° to 60° . The height of the chimney is

Answer: Option C

Solution

Answer: Option C

Solution:

In the figure, AB is chimney and CB and DB are its shadow

$$\begin{array}{rl}& \tan 60^{\circ }=\frac{AB}{BC}=\frac{h}{BC}\\ & \Rightarrow \sqrt{3}=\frac{h}{BC}\\ & \Rightarrow BC=\frac{h}{\sqrt{3}}.......\left(\text{i}\right)\\ & \text{and}\\ & \tan 30^{\circ }=\frac{h}{DB}=\frac{h}{DB+BC}\\ & \frac{1}{\sqrt{3}}=\frac{h}{x+BC}\\ & x+BC=h\sqrt{3}\\ & \Rightarrow BC=h\sqrt{3}-x.......\left(\text{ii}\right)\\ & \text{From}\left(\text{i}\right)\text{and}\left(\text{ii}\right)\\ & \frac{h}{\sqrt{3}}=h\sqrt{3}-x\\ & \Rightarrow \frac{h}{\sqrt{3}}-h\sqrt{3}=-x\\ & x=h\sqrt{3}-\frac{h}{\sqrt{3}}\\ & x=h\left(\sqrt{3}-\frac{1}{\sqrt{3}}\right)\\ & x=h\frac{3-1}{\sqrt{3}}\\ & x=\frac{2h}{\sqrt{3}}\\ & \therefore h=\frac{\sqrt{3}}{2}x\end{array}$$