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