Question

An observer point for ships moving in the sea 500m above the sea level. The person manning this point observes the angle of depression of twc boats as 45° and 30°. Find the distance between the boats when they are on the same side of the observation point and when they are on opposite sides of the observation point. 

Answer 1

Case 1: When the boats are on same side of the observation point. 

Let the position of the two ships be C and D. Let A be the point of observation. 
AB = 500 m 
In ΔBAC, 

`tan45^circ = "AB"/"BC"`

⇒ `1 = 500/"BC"`

⇒ BC = 500  ....(1)

In ΔABD, 

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

⇒ `1/sqrt(3) = 500/"BD"`

⇒ `"BD" = 500sqrt(3)`   ...(2)

From (1) and (2), 
`"CD" = "BD" - "BC" = 500(sqrt(3) - 1) = 500 xx 0.732 = 366`

Thus, in this case, the distance between the boats is 366 m. 

Case 2: When the boats are on different side of the observation point. 

Let the position of the two ships be A and C. Let B be the point of observation. 
In ΔBAD, 

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

⇒ `1 = 500/"AD"`

⇒ AD = 500 ....(1)

In ΔBDC,

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

⇒ `1/sqrt(3)  = 500/"DC"`

⇒ `"DC" = 500sqrt(3)` ....(2)

From (1) and (2), 

`"AC" = "AD" + "DC" = 500 (1 + sqrt(3)) = 500 xx 2.732 = 1366`

Thus, in this case, the distance between the boats is 1366 m.