Form a point on the same plane along the horizontal line passes through the foot of a chimney, the angle of elevation of the top of the chimney is `30^(@)` and the angle of elevation of the top of the chimney is `60^(@)` at a point on the same straight line proceeding 50 metres nearer to the chimney. Calculate the height of the chimney.

Explanation

Let AB be the chimney, from the point D the angle of elevation of the top A of the chimney is `30^(@)` and from the point C, 60 metres nearer to the foor B of the chimney, the angle of elevation of the top A of the chimney is `60^(@)`.
As per qestion, DC = 50 m, `angleADB = 30^(@) and angleACB = 60^(@)`.
Now, from the right-angled triangle ABC, we get,
`tan 60^(@) = (AB)/(BC)` [by definition ]
or, `sqrt(3) =(AB)/(BC) " "or, " "BC = (AB)/(sqrt(3))`.........(1)
Again, from the right-angled triangle ABD, we get,
`tan 30^(@) =(AB)/(BD)` [ by definition]
or, `(1)/(sqrt(3) ) =(AB)/(AD) " " or, " " sqrt(3)AB = BD`
or, `sqrt(3)AB = BC + CD " " or, " " sqrt(3)AB = (AB)/(sqrt(3))+50` [from (1) ]
or, `sqrt(3)AB - (AB)/(sqrt(3)) = 50`
or, AB`(sqrt(3) - (1)/(sqrt(3))) = 50 " "or, " "AB xx (2)/(sqrt(3)) = 50`
or, AB=`(50sqrt(3))/(2) " "or, " " AB=25sqrt(3)`
Hence the height of the chimney is `25sqrt(3)`m.