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 60°. When he moves 40 m away from the bank, he finds the angle of elevation to be 30°. Find:

1. the height of the tree, correct to 2 decimal places,
2. the width of the river.

Answer 1

(i)

In ΔBCD,

tan 60° = `"Perpendicular"/"Base"`

In ΔACD,

tan 30° = `"Perpendicular"/"Base"`

⇒ `1/sqrt3 = (CD)/(AC)`

⇒ AC = `CD sqrt3`

From figure,

AC = AC − BC

⇒ 40 = `CD sqrt3 - (CD)/sqrt3`

⇒ `(3CD - CD)/sqrt3 = 40`

⇒ `(2CD)/sqrt3 = 40`

⇒ `2CD = 40sqrt3`

⇒ `CD = (40sqrt3)/2`

⇒ CD = 20 × 1.732

= 34.64 m

Hence, height of tree = 34.64 metres.

(ii)

From part (i), we get:

BC = `(CD)/sqrt3`

BC = `(20sqrt3)/sqrt3`

BC = 20 m

Hence, width of river = 20 metres.