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A person from the top of a hill observes a vehicle moving towards him at a uniform speed. It takes 10 minutes for the angle of depression to change from 45° to 60°. After this the time required by the vehicle to reach the bottom of the hill is
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- 12 minutes 20 seconds
- 13 minutes
- 13 minutes 40 seconds
- 14 minutes 24 seconds
Correct Option: C
AB = height of hill = h metre
Let speed of vehicle be v metre/minute.
Time taken to reach B from D = t minutes
CD = 10v metre
BD = vt metre
In ∆ ABC,
| tan 45° = | BC |
| ⇒ 1 = | metre | BC |
⇒ BC = h
= (10v + vt) metre ....(i)
In ∆ABD,
| tan 60° = | BD |
| ⇒ √3 = | vt |
⇒ h = √3 vt
⇒ 10v + vt = √3 vt
⇒ 10 = √3 t – t
⇒ 10 = t (√3 - 1)
| ⇒ t = | √3- 1 |
| = | = | (√3 - 1)(√3 + 1) | 2 |
= 5 (1.732 + 1) = 5 × 2.732
= 13.66 minutes
= 13 minutes 40 seconds
