The angle of elevation of the top of a chimney from the top of a tower is 60° and the angle of depression of foot of the chimney from the top of the tower is 30°. If the height of the tower is 40 m, then find the height of the chimney.

Correct option is D

Solution:

Let the height of the building be h m.

In Triangle APC:

From the triangle formed with the top of the tower to the top of the chimney:
​$$\tan(60°) = \frac{PC}{AP}\\\ \\\sqrt{3} = \frac{h - 40}{x} .........\tag{1}$$​​

In Triangle ABD:

From the triangle formed with the top of the tower to the foot of the chimney:
​$$\tan(30°) = \frac{AB}{BD}\\\ \\\frac{1}{\sqrt{3}} = \frac{40}{x} .........\tag{2}$$​​

Solving Equations (1) and (2):

From equation (2):
​$$x = 40\sqrt{3}$$​​
Substitute x into equation (1):
​$$\sqrt{3} = \frac{h - 40}{40\sqrt{3}}\\\ \\h - 40 = 40\sqrt{3} \cdot \sqrt{3}\\\ \\h - 40 = 40 \cdot 3\\\ \\h = 120 + 40 \\ \\h = 160$$​​

The height of the chimney is 160m, ensuring compliance with pollution control norms.