The angle of elevation of the top of an unfinished tower at a distance of 75 m from its base is 30°. How much higher must the tower be raised so that the angle of elevation of its top at the same point may be 60°?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(50\sqrt{3}\) m
The tower, the ground and the line of sight form a right triangle, so \(\tan\theta=\frac{\text{height}}{\text{distance from the base}}\). The distance stays 75 m; the height is found for each angle, and the difference is the amount by which the tower is raised.
Present height: \(\displaystyle 75\tan30^\circ=\frac{75}{\sqrt{3}}=25\sqrt{3}\) m
Height needed for \(60^\circ\): \(75\tan60^\circ=75\sqrt{3}\) m
Raise needed: \(75\sqrt{3}-25\sqrt{3}=50\sqrt{3}\) m
Shortcut: \(\tan60^\circ=3\tan30^\circ\), so from the same point the new height is three times the present height, and the raise is twice the present height: \(2\times25\sqrt{3}=50\sqrt{3}\) m.