Two pillars of equal height and on either side of a road, which is 100 m wide. The angles of elevation of the top of the pillars are `60^@` and `30^@` at a point on the road between the pillars. Find the height of each pillar.

Options

A. 34.5 m
B. 25.4 m
C. 43.3 m
D. 75 m

Answer

C

Explanation

Let AB and CD be two pillars, each of height h metres and AC be the road that `AC = 100 m`.
Let O be the point of observation on AC.
Also, `angleAOB = 60^(@)` and `angleCOD = 30^(@)`,
`AB _|_AC` and `CD_|_AC`.
From right `DeltaOAB`, we have
`(AB)/(OA) = tan60^(@) = sqrt(3)`
`rArr h/x = sqrt(3) rArr h = sqrt(3)x".........."(i)`.
From right `DeltaOCD`, we have
`(CD)/(OC) = tan 30^(@) = 1/(sqrt(3))`.
`rARr (h)/((100 - x)) = 1/(sqrt(3)) rArr h = ((100 - x))/(sqrt(3)) "......."(ii)`
Equating the values of h from (i) and (ii) , we get
`sqrt(3)x = ((100 - x))/(sqrt(3)) rArr 3x = (100 - x) rArr 4x = 100 rArr x = 25`.
Putting `x = 25 m` in (i), we get
`h = (25xxsqrt(3)) = (25 xx 1.732) = 43.3`.
Hence, the height of each pillar is `43.3 m` and the point of observation is `25 m` away from the first pillar.