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Aman observes a car from the top of a tower, which is moving towards the tower with a uniform speed. If the angle of depression of the car changes from to in 12 minutes, find the time taken by the car now to reach the tower.

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16 minutes 20 seconds

15 minutes 55 seconds

16 minutes 23 seconds

15 minutes 20 seconds

answer is A.

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Detailed Solution

Given that, a man observes a car from the top of a tower, which is moving towards the tower with a uniform speed, the angle of depression of the car changes from

in 12 minutes.
We know that,

\tan θ = \frac{Opposite}{Adjacent}

.
Angle of depression and angle of elevation are equal.
Draw the diagram depicting the situation.
Question Image

\tan 30 ° = \frac{h}{12 v + vt}

\Rightarrow 12 v + vt = \frac{h}{\tan 30 °}

And,

\tan 45 ° = \frac{h}{vt}

\Rightarrow vt = \frac{h}{\tan 45 °}

The distance between C and D,

12 v = (12 v + vt) - vt

⇒ 12 v = h tan 30 ° − h tan 45 ° ⇒ 12 v = h 3 − h ⇒ 12 v = h (3 − 1) ⇒ v = h (3 − 1) 12

The distance between A and D,

v t = h tan 45 ° ⇒ h (3 − 1) 12 t = h ⇒ t = D 3 − 1 ⇒ t = 16.39 minutes ⇒ t = 16 minutes + (0 .39 × 60) seconds ⇒ t = 16 minutes 23 seconds

Thus, the time taken by car to reach the tower is 16 minutes 23 seconds.
Hence the correct option is 1.

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