Solve any two sub-question:

A person standing on the bank of a river observes that the angle of elevation of the top of a tree standing on the opposite bank is , When he moves 40 m away form the bank, he finds the angle of elevation to be . Find the height of the tree and the width of the river.

Consider the above diagram where AB is the height of tree with point A as the top of the tree

Point B base of the tree

Initially the person is at point C therefore BC is the width of the river and the person observes the angle of elevation to be 60° i.e. ∠ACB = 60°

The person moves 40 m away from the bank of the rives thus the new position of person is point D and CD = 40m

From D the person observes the angle of elevation to be 30° i.e. ∠ADB = 30°

Consider ∆ABC

tan60° = AB/BC

∴AB = BC√3 …(i)

Consider ∆ABD

tan30° = AB/BD

Using CD = 40 m

(BC + 40) = AB√3

∴ AB = (BC + 40)/√3 …(ii)

From (i) and (ii)

BC√3 = (BC + 40)/√3

∴ BC√3×√3 = BC + 40

∴ 3BC = BC + 40

∴ 2BC = 40

∴ BC = 20 m

Therefore width of the river = BC = 20 m

Substituting BC in equation (i)

AB = 20√3

∴ AB = 20×1.73

∴ AB = 34.6 m

Therefore height of tree = AB = 34.6 m

Hence height of tree is 34.6 meters and width of river is 20 meters