PQ is a vertical pole with its foot Q on a level ground. A point R divides PQ such that PR:RQ=7:3. If the parts PR & RQ subtend equal angles at a point 20 m away from the foot of the pole, find the height of the pole.
Answers
Given : PQ is a vertical pole with its foot Q on a level ground.
A point R divides PQ such that PR:RQ=7:3.
PR & RQ subtend equal angles at a point 20 m away from the foot of the pole,
To find : the height of the pole.
Solution:
Tanα = QR/20
Tan(α + α) = PQ/20
PR:RQ=7:3
=> PR = 7x and RQ = 3x
=> PQ = PR + RQ = 7x + 3x = 10x
Height of pole PQ = 10x
=> Tanα = 3x/20
Tan(2α) = 10x/20
=> Tan(2α) / Tanα = 10/3
=> (2Tanα / (1 - Tan²α))/ Tanα 2/= 10/3
=> 2/ (1 - Tan²α) = 10/3
=> 6 = 10 - 10 Tan²α
=> 10 Tan²α = 4
=> Tan²α = 4 /10
=> Tan²α = 2 /5
=> Tanα = √(2 /5)
Tanα = QR/20
=> QR = 20Tanα
=> 3x = 20 √(2 /5)
=> x = 20 √(2 /5) / 3
=> 10x = 10 * 20 √(2 /5) / 3
=> height of the pole = (200/3)√(2 /5)
=> height of the pole = (40/3)√10
=> height of the pole = 42.16 m
height of the pole = 42.16 m
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