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Q.

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. 3 minutes later the angle of depression of the boat is found to be 60°. Assuming that the boat sails at a uniform speed, the time taken by the boat to reach the shore is


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a

1.5min

b

4.5min

c

6min

d

6.5min  

answer is B.

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Detailed Solution

It is given a man on a cliff observes a boat at an angle of depression is 30° .The boat is sailing towards the shore to the point beneath him. After 3 minutes the angle of depression of the boat is 60°.
According to the given data, the figure has been drawn below.
Question ImageLet AB be the cliff, speed be s, total time taken be T, total distance BD be X, height of cliff is h and ∠BAC=30° and ∠BAD=60°.
We know speed=distancetime.
⇒s=XT
⇒X=sT
Let the distance covered in 3 min is  x=3s.
In the figure CD represents the distance x.
From ΔABC, we get
tanθ= Perpendicular base    tan30°=BCAB  ⇒13=BCAB     ∵tan 30 ° = 1 3    ⇒AB=BC3......(1) From the figure,
BD =BC+CD ⇒BC=BD-CD              =X-x….(2)
Now equation 1 becomes,
⇒AB=3(X-x)......(3)
From ΔABD, we get
tanθ= Perpendicular base  
tan60°=BDAB  ⇒3=XAB    ∵tan 60 ° = 3  
 ⇒AB=X3.............(4) From (3),(4)  ⇒3(X-x)=X3  ⇒X=3(X-x)  ⇒X=3X-3(3s) ……{x=3s}
⇒3X-X=9s
⇒2X=9  ⇒2(sT)=9s ….{X=sT}
⇒ 2T=9
 ⇒T=92 
          =4.5 min
So, the time required to reach the shore is 4.5 min.
Hence, the correct option is 2.
 
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