Question

From the top of a lighthouse, the angle of depression of two ships on the opposite sides of it is observed to be 30° and 60°. If the height of the lighthouse is h meters and the line joining the ships passes through the foot of the lighthouse, show that the distance between the ships is `(4"h")/sqrt(3)` m

Answer 1

A and C be the position of two ships.

Let AB be x and BC be y.

Distance between the two ships is x + y

In the right ∆ABD, tan 60° = `"BD"/"AB"`

`sqrt(3) = "h"/x`

x = `"h"/sqrt(3)`  ...(1)

In the right ∆BCD,

tan 30° = `"BD"/"BC"`

`1/sqrt(3) = "h"/y`

y = `sqrt(3)`h

Distance between the two ships (x + y) = `"h"/sqrt(3) + sqrt(3)"h"`

= `("h" + 3"h")/sqrt(3)`

= `(4"h")/sqrt(3)`

Hence it is verified.