Question

An aeroplane at an altitude of 1800 m finds that two boats are sailing towards it in the same direction. The angles of depression of the boats as observed from the aeroplane are 60° and 30° respectively. Find the distance between the two boats. `(sqrt(3) = 1.732)`

Answer 1

C and D are the positions of the two boats.

Let the distance between the two boats be “x”

Let BC = y

∴ BD = (x + y)

In the right ∆ABC, tan 30° = `"AB"/"BD"`

`1/sqrt(3) = 1800/(x + y)`

x + y = `1800sqrt(3)`

y = `1800sqrt(3) - x`  ...(1)

In the right ∆ABC, tan 60° = `"AB"/"BC"`

`sqrt(3) = 1800/y`

y = `1800/sqrt(3)`  ...(2)

From (1) and (2) we get

`1800/sqrt(3) = 1800sqrt(3) - x`

1800 = `1800 xx 3 - sqrt(3)x`

`sqrt(3)x` = 5400 – 1800

x = `3600/sqrt(3)`

= `(3600 xx sqrt3)/(sqrt3 xx sqrt(3)`

= `(3600 xx sqrt(3))/3`

= 1200 × 1.732

= 2078.4 m

Distance between the two boats = 2078.4 m