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From the top of a $120 m$ high tower, a man observes two cars on the opposite sides of the tower and in straight line with the base of tower with angles of depression as $60 °$ and $45 °$ . Find the distance between the two cars.

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139.28 m

169.28 m

189.28 m

179.28 m

answer is D.

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Detailed Solution

Height of tower, h=120m
Angle of depression from top to one of the car,

θ_{1} = 60 °

And angle of depression from the top to another car,

θ_{2} = 45 °

Consider the figure,
Question Image

Let d be the distance between the two cars and x be the distance between the car and the tower.
Consider,

∆ ADB

tan θ = p e r p e n d i c u l a r b a s e ⇒ tan 60 ° = 120 x ⇒ x = 120 tan 60 ° ⇒ x = 1203 ∵ tan 60 ° = 3

Consider,

∆ ABC

tan θ = p e r p e n d i c u l a r b a s e ⇒ tan 45 ° = 120 d − x ⇒ d − x = 120 tan 45 ° ⇒ d − x = 120 ∵ tan 45 ° = 1 ⇒ d − 1203 = 120 ⇒ d = 120 + 1203 ⇒ d = 120 + 69.28 ⇒ d = 189.28

Thus, the distance between the two cars is 189.28 m.
Therefore, option 4 is correct.

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