Question

A man on the top of a tower observes that a car is moving directly at a uniform speed towards it. If it takes 720 seconds for the angle of depression to change from 30° to 45°, how soon will the car reach the observation tower? 

Answer 1

Let AB be the tower .

Initial position of car is C , which changes to D after 720 seconds.

In ΔADB

`"AB"/"DB" = tan45^circ`

`"AB"/"DB" = 1`

DB = AB

In ΔABC

`"AB"/"BC" = tan 30^circ`

`"AB"/"BD + DC" = 1/sqrt(3)`

`"AB"sqrt(3) = "BD + DC"`

`"AB"sqrt(3) = "AB + DC"`

`"DC" = "AB"sqrt(3) - "AB" = "AB"(sqrt(3) - 1)`

Time taken by car to travel DC distance (i.e `"AB"(sqrt(3) - 1`)) = 720 seconds

Time taken by car to travel DB distance (i.e. AB)

= `720/("AB"(sqrt(3) - 1)) xx "AB" = 720/((sqrt(3) - 1)) xx (sqrt(3) + 1)/(sqrt(3) + 1)`

= `(720(sqrt(3) + 1))/2 = 360(sqrt(3) + 1) = 360 xx 2.732 = 983.52`

Thus , the required time taken is 983.52 seconds = 984 seconds = 16 mins 24 secs.