Question

A boy standing on a horizontal plane finds a bird flying at a distance of 100 m from him at an elevation of 30°. A girl standing on the roof of 20 metre high building, finds the angle of elevation of the same bird to be 45°. Both the boy and the girl are on opposite sides of the bird. Find the distance of bird from the girl.

Answer 1

Let O be the position of the bird, B be the position of the boy and EG be the builiding at which G is the position of the girl.
Let `OL, BF` and `GM , OL`. Then
`BO = 100m, angleOBL = 30^(@)`,
`FG = 20 m` and `angleOGM = 45^(@)`
From right `DeltaOLB`, we have
`(OL)/(BO) = sin 30^(@) rArr (OL)/(100 m) = 1/2`
`rArr OL = 100 m xx 1/2 = 50 m` .
`OM = OL - ML = OL - FG = 50m - 20 m = 30m`.
From right `DeltaOMG`, we have
`(OM)/(OG) = sin 45^(@) = 1/(sqrt(2)) rArr OG = sqrt(2) xx OM = sqrt(2) xx 30 m`
`rArr OG = 30 xx 1.41 m = 42.3 m`.
Distance of the bird from the girl `= 42.3 m`.