Question

Two boats approaching a light house in mid sea from opposite directions observe the angle of elevation of the top of the light house as 30° and 45° respectively. If the distance between the two boats is 150m, find the height of the light house. 

Answer 1

Let the position of the two boats be at points A and C. Let BD be the lighthouse of height h.
Let AD = x. Then, CD = 150 - x 
In ΔBAD, 

`tan45^circ = "BD"/"AD"`

⇒ `1 = "h"/"X"`

⇒ h = X  ...(1)

In ΔBDC,

`tan30^circ = "BD"/"DC"`

⇒ `1/sqrt(3) = "h"/(150 - X)` 

⇒ `150 - "X" = sqrt(3)"h"`  ....(2)

From (1) and (2), 

`150 - "h" = sqrt(3)"h"`

`150 = (sqrt(3) + 1)"h"`

`"h" = 150/(sqrt(3) + 1) xx (sqrt(3) - 1)/(sqrt(3) - 1)`

= `(150(sqrt(3) - 1))/(3 - 1)`

= `75(sqrt(3) - 1)`

= `75 xx 0.732 = 54.9`

Thus, the height of the light house is 54. 9 m.