An observer standing at a point P on the top of a hill near the sea-shore notices that the angle of depression of a ship moving towards the hill in a straight line at a constant speed is 300 . After 45 minutes , this angle becomes 450. If T (in minutes) is the total time taken by the ship to move to a point in the sea where the angle of depression from P of the ship is 600, then T is equal to :

  • Option 1)

    45\left ( 1+\frac{1}{\sqrt{3}} \right )

  • Option 2)

    45\left ( 1+\sqrt{3}\right )

  • Option 3)

    45\left ( 1+\frac{2}{\sqrt{3}} \right )

  • Option 4)

    45\left ( 2+\frac{1}{\sqrt{3}} \right )

 

Answers (1)

As we learnt in 

Height and Distances -

The height or length of an object or the distance between two distant objects can be determined with the help of trigonometric ratios.

-

Time taken to travel between any points is proportional to distance covered.

Hence, \frac{T_{AC}}{T_{AB}}= \frac{AC}{AB} \: \: \left [ \because uniform \: speed \right ]

In \Delta PQA, \frac{PQ}{AQ}= tan 30^{\circ}\Rightarrow AQ= \sqrt{3}h \: \: \: \cdot (1)

Similarly, from \: \Delta PBQ, \: tan45^{\circ}= \frac{PQ}{BQ}\Rightarrow BQ=h \: \: \: \cdot (2)

Similarly, from \: \Delta PQC, \: tan60^{\circ}= \frac{PQ}{CQ}\Rightarrow CQ=\frac{h}{\sqrt{3}} \: \: \: \cdot (3)

Now, \frac{AC}{AB}=\frac{AQ-CQ}{AQ-BQ}=\frac{\sqrt{3}h-\frac{h}{\sqrt{3}}}{\sqrt{3}h-h}=\frac{{\sqrt{3}-\frac{1}{\sqrt{3}}}}{\sqrt{3}-1}

                                       =\frac{({\sqrt{3}-\frac{1}{\sqrt{3}})}^{\sqrt{3}+1}}{(\sqrt{3}-1)(\sqrt{3}+1)} =\frac{3-1+\sqrt{3}-\frac{1}{\sqrt{3}}}{2}

                                       =\frac{2+\frac{2}{\sqrt{3}}}{2}=1+\frac{1}{\sqrt{3}}

Thus, T_{AC}=T_{AB}\times \left ( 1+\frac{1}{\sqrt{3}} \right )

T=45\left ( 1+\frac{1}{\sqrt{3}} \right )


Option 1)

45\left ( 1+\frac{1}{\sqrt{3}} \right )

This option is correct

Option 2)

45\left ( 1+\sqrt{3}\right )

This option is incorrect

Option 3)

45\left ( 1+\frac{2}{\sqrt{3}} \right )

This option is incorrect

Option 4)

45\left ( 2+\frac{1}{\sqrt{3}} \right )

This option is incorrect

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