Two poles of equal heights are standing opposite each other on either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 60° and 30°, respectively. Find the height of the poles and the distances of the point from the poles.
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Answer:
If two poles of equal heights are standing opposite each other on either side of the road, which is 80 m wide, and from a point between them on the road, the angles of elevation of the top of the poles are 60° and 30° respectively, then the height of the poles are 20√3 m and the distances of the point from the poles is ...
Let us consider the two poles of equal heights as AB and DC and the distance between the poles as BC.
From a point O, between the poles on the road, the angle of elevation of the top of the poles AB and CD are 60° and 30° respectively.
Trigonometric ratio involving angles, distance between poles and heights of poles is tan θ.
Let the height of the poles be x
Therefore AB = DC = x
In ΔAOB,
tan 60° = AB/BO
√3 = x / BO
BO = x / √3 ....(i)
In ΔOCD,
tan 30° = DC / OC
1/√3 = x / (BC - OB)
1/√3 = x / (80 - x/√3) [from (i)]
80 - x/√3 = √3x
x/√3 + √3x = 80
x (1/√3 + √3) = 80
x (1 + 3) / √3 = 80
x (4/√3) = 80
x = 80√3 / 4
x = 20√3
Height of the poles x = 20√3 m.
Distance of the point O from the pole AB
BO = x/√3
= 20√3/√3
= 20
Distance of the point O from the pole CD
OC = BC - BO
= 80 - 20
= 60
The height of the poles is 20√3 m and the distance of the point from the poles is 20 m and 60 m
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Verified answer
Let us consider the two poles of equal heights as AB and DC and the distance between the poles as BC.
From a point O, between the poles on the road, the angle of elevation of the top of the poles AB and CD are 60° and 30° respectively.
Trigonometric ratio involving angles, distance between poles and heights of poles is tan θ.
Let the height of the poles be x
Therefore AB = DC = x
In ΔAOB,
tan 60° = AB/BO
√3 = x / BO
BO = x / √3 ....(i)
In ΔOCD,
tan 30° = DC / OC
1/√3 = x / (BC - OB)
1/√3 = x / (80 - x/√3) [from (i)]
80 - x/√3 = √3x
x/√3 + √3x = 80
x (1/√3 + √3) = 80
x (1 + 3) / √3 = 80
x (4/√3) = 80
x = 80√3 / 4
x = 20√3
Height of the poles x = 20√3 m.
Distance of the point O from the pole AB
BO = x/√3
= 20√3/√3
= 20
Distance of the point O from the pole CD
OC = BC - BO
= 80 - 20
= 60
The height of the poles is 20√3 m and the distance of the point from the poles is 20 m and 60 m.