Question

A man on a cliff observes a boat, at an angle of depression 30°, which is sailing towards the shore to the point immediately beneath him. Three minutes later, the angle of depression of the boat is found to be 60°. Assuming that the boat sails at a uniform speed, determine:

1. how much more time it will take to reach the shore?
2. the speed of the boat in metre per second, if the height of the cliff is 500 m.

Answer 1

Let AB be the cliff and C and D be the two position of the boat such that ∠ADE = 30° and ∠ACB = 60° 

Let speed of the boat be X metre per minute and let the boat reach the shore after t minutes more.

In ΔABC, 

`(AB)/(BC) = tan 60^circ = sqrt(3)`

`=> h/(tx) = sqrt(3)`

In ΔADB, 

`(AB)/(DB) = tan 30^circ`

`=> (h)/(3x + tx ) = 1/sqrt(3)`

`=> (sqrt(3)t)/(3 + t) = 1/sqrt(3)`

`=>` 3t = 3 + t

∴ t = `3/2` = 1.5 minute

Also, if h = 500 m, then 

` (500)/(1.5 x)= sqrt(3)`

`=> x =  (500)/(1.5 xx 1.732)`

= 192.455 metre per minute

= 3.21 m/sec

Hence, the boat takes an extra 1.5 minutes to reach the shore.

And if the height of cliff is 500 m, the speed of the boat is 3.21 m/sec