From a point on the ground, the angle of elevation of the top of a tower is 30°. After moving 20 m closer to the tower in a straight line, the angle of elevation becomes 45°. Find the height of the tower.

Let the initial horizontal distance from the point to the base of the tower be x metres, and height of tower be h metres. From first position: tan30° = h / x ⇒ 1/√3 = h / x ⇒ h = x/√3 …(1) From second position: distance = x − 20 tan45° = h / (x − 20) = 1 ⇒ h = x − 20 …(2) Equate (1) and (2): x − 20 = x/√3 Bring x terms together: x − x/√3 = 20 x(1 − 1/√3) = 20 x = 20 / (1 − 1/√3) You can rationalise, but we actually only need h. From (2): h = x − 20. Compute: x = 30 + 10√3 (after simplifying) So h = x − 20 = (30 + 10√3) − 20 = 10 + 10√3