The angle of elevation of the top of a tower from a point on the ground is 30°. On moving 20 m closer to the tower, the angle of elevation becomes 45°. Find the height of the tower.

ATQ, Let height of tower = h, initial horizontal distance = x. From first position: tan 30° = h/x ⇒ 1/√3 = h/x ⇒ h = x/√3 From second position: distance = x − 20, angle = 45°: tan 45° = h/(x − 20) = 1 ⇒ h = x − 20 Equate: x − 20 = x/√3 x(1 − 1/√3) = 20 x[(√3 − 1)/√3] = 20 x = 20√3 / (√3 − 1) Rationalize: x = 20√3(√3 + 1) / [(√3 − 1)(√3 + 1)] = 20√3(√3 + 1) / (3 − 1) = 10√3(√3 + 1) = 10(3 + √3) Height h = x − 20 = 10(3 + √3) − 20 But from h = x/√3: h = [10(3 + √3)] / √3 = 10(3 + √3)/√3 × (√3/√3) = 10(3√3 + 3)/3 = 10(√3 + 1) So height of tower = 10(√3 + 1) m.