Question:
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 degree and 30 degree , respectively. Find the height of the poles and the distances of the point from the poles.
Answer:
Let AB and DE are two poles and given AB = DE
Let BD = 80 is the road.
Let there is a point C on the road.
Let CD = x then BC = 80 - x
Now from triangle EDC,
tan 30 = ED/CD
=> 1/√3 = ED/x
=> x = ED√3 ........1
Again from triangle ABC
tan60 = AB/BC
=> √3 = AB/(80 - x)
=> AB = √3(80 - x)
=> AB = 80√3 - √3x
=> AB = 80√3 - √3*ED*√3 ( x = ED√3)
=> AB = 80√3 - 3ED
=> AB = 80√3 - 3AB (ED = AB)
=> AB + 3AB = 80√3
=> 4AB = 80√3
=> AB = 80√3/4
=> AB = 20√3
So AB = ED = 20√3
So height of the pole is 20√3 m.
Now from equation 1
x = ED√3
=> x = 20√3*√3
=> x = 20*3
=> x = 60
So CD = 60
and BC = 80 - x = 80 - 60
=> BC = 20
So distances of the point from the poles are 60 m and 20 m.
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