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Question
Two pillars of equal heights stand on either side of a roadway, which is 150 m wide. At a point in the roadway between the pillars the elevations of the tops of the pillars are 60° and 30°; find the height of the pillars and the position of the point.
Solution
Let AB and CD be the two towers of height h m.
Let P be a point in the roadway BD such that BD = 150 m, ∠APB = 60° and ∠CPD = 30°
In ΔABP,
`(AB)/(BP) = tan 60^circ`
`=> BP = h/(tan 60^circ) = h/sqrt(3)`
In ΔCDP,
`(CD)/(DP) = tan 30^circ`
`=> PD = 1/sqrt(3)`
Now, 150 = BP + PD
`=> 150 = h/sqrt(3) + 1/sqrt(3)`
∴ `h = 150/(sqrt(3) + 1/sqrt 3)`
= `150/2.309`
= 64.95 m
Hence, height of the pillars is 64.95 m.
The point is `(BP)/sqrt(3)` from the first pillar.
That is the position of the point is `64.95/sqrt(3) m` from the first pillar.
The position of the point is 37.5 m from the first pillar.
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