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A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of 30°, which is approaching the foot of the tower with a uniform speed. Six seconds later, the angle of depression of the car is found to be 60°. What is the time taken by the car to reach the foot of the tower from this point?

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5 sec

4 sec

2 sec

3 sec

answer is D.

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Detailed Solution

Given that,
Angle of depression from top of the tower of the car at D =

300

,
Time taken by car to travel from D to C = 6 seconds.
The situation represented diagrammatically is as follows,
Question Image

Here, AE is parallel to BD and AC, AD are transversal.
We know that when two parallel lines are cut by a transversal, then the alternate angles are equal.
So,

∠ E A C = ∠ A C B and ∠ E A D = ∠ A D C

In triangle

Δ A B C

tan θ = sin θ cos θ = Perpendicular Base ⇒ tan 60 ° = Height Distance = AB BC ⇒ 3 = A B B C ⇒ B C = A B 3

In triangle

Δ A B D

tan θ = sin θ cos θ = Perpendicular Base ⇒ tan 30 ° = Height Distance = AB BD ⇒ 13 = A B B D ⇒ B D = 3 A B

Distance travelled by the car (CD) = Distance BD – Distance BC

C D = A B 3 − A B 3 ⇒ C D = A B 3 1 − 1 3 × 3 ⇒ C D = A B 3 1 − 13 ⇒ C D = A B 3 × 23 ⇒ C D = 2 A B 3 3 m

Using speed-distance relation,

S p e e d = D i s t a n c e T i m e ⇒ S p e e d = C D 6

⇒ S p e e d = 2 A B 3 3 × 6 ⇒ S p e e d = A B 3 9

Now, we have the speed of the car and the distance is

B C = A B 3

.
Calculate the time taken for the car to travel distance BC.

T i m e = A B 3 A B 3 9 ⇒ T i m e = A B 3 × 9 A B 3 ⇒ T i m e = 9 3 × 3 ⇒ T i m e = 93 = 3 s

So, the car takes 3 seconds to travel from that point to the bottom of the tower.
Therefore, the correct option is 4.

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