the height of the cliff

Let the height of the cliff be h m and the horizontal distance from the cliff to the tower be x m.

Height of tower = 20 m.

Angle of depression to the bottom of the tower = 60°.

Therefore:

tan60° = h/x

√3 = h/x

x = h/√3 ...(1)

Angle of depression to the top of the tower = 45°.

Vertical difference = h − 20

Therefore:

tan45° = (h − 20)/x

1 = (h − 20)/x

x = h − 20 ...(2)

Equating (1) and (2):

h/√3 = h − 20

h = √3h − 20√3

h(√3 − 1) = 20√3

h = 20√3/(√3 − 1)

= 30 + 10√3

≈ 47.32 m

Answer:

Height of the cliff ≈ 47.32 m.

Angles of depression equal the corresponding angles of elevation. Use tanθ = vertical height difference / horizontal distance for the lines of sight to the bottom and top of the tower, then eliminate the common horizontal distance.